1307.1980.tex








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%%%%%%%%%%%%%%%%% KPZ (Kardar-Parisi-Zhang) en dimension $\ge$ 3 
%                  Partie I - Bornes mono-�chelles
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\begin{document}

\vspace*{1.5cm}
\begin{center}
{\Large \bf Diffusive limit for 3-dimensional KPZ equation. (1) PDE estimates.}


\end{center}

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\vspace{2mm}
\begin{center}
{\bf  Jacques Magnen$^a$ and J\'er\'emie Unterberger$^b$}
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\vskip 0.5 cm
\centerline {$^a$Centre de Physique Th\'eorique,\footnote{Laboratoire associ\'e au CNRS UMR 7644}
Ecole Polytechnique,} 
\centerline{91128 Palaiseau Cedex, France}
\centerline{jacques.magnen@cpht.polytechnique.fr}
\vskip 0.5 cm
\centerline {$^b$Institut Elie Cartan,\footnote{Laboratoire 
associ\'e au CNRS UMR 7502} Universit\'e de Lorraine,} 
\centerline{ B.P. 239, 
F -- 54506 Vand{\oe}uvre-l\`es-Nancy Cedex, France}
\centerline{jeremie.unterberger@univ-lorraine.fr}

\vspace{2mm}
\begin{quote}

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\footnotesize
{We study in the present series of articles the Kardar-Parisi-Zhang  (KPZ) equation
$$ \partial_t h(t,x)=\nu\Del h(t,x)+\lambda  |\nabla h(t,x)|^2 +\sqrt{D}\eta(t,x), \qquad x\in\R^d $$
in $d\ge 3$ dimensions in the perturbative regime, i.e. for $\lambda>0$ small enough and an initial condition 
$h_0=h(t=0,\cdot)$ such that $\lambda|\nabla h_0|=o(1)$. The forcing term $\eta$ in the right-hand side is
 a regularized white noise.
We  prove a large-scale diffusive limit for the solution, in particular a heat-kernel behaviour for the covariance in a parabolic scaling. The proof is generally based on perturbative
estimates obtained by a multi-scale cluster expansion, and a rigorous implementation of K. Wilson's renormalization group
scheme; it  extends to  equations in the KPZ universality class,
$$ \partial_t h(t,x)=\nu\Del h(t,x)+\lambda  V(\nabla h(t,x)) +\sqrt{D}\eta(t,x), \qquad x\in\R^d $$
for a large class of  isotropic, convex deposition rates $V\ge 0$. Our expansion is meant to be a rigorous substitute for the response field formalism whenever the latter makes predictions.  An important part of the proof relies however
 more specifically  on
a priori bounds for the solutions of the KPZ equation, coupled with  large deviation estimates.

\medskip
The present article is dedicated to PDE estimates. Our results extend those previously obtained for the noiseless
equation. We prove in particular a comparison principle for sub- and supersolutions of the KPZ equation in an
adapted functional space containing unbounded functions, which may be of independent interest for the  study
of  viscous Hamilton-Jacobi equations in general. The comparison to the linear heat equation through a
Cole-Hopf transform is an essential ingredient in the proofs, and our results are accordingly valid only for
a function $V$ with at most quadratic growth at infinity. 
}
\end{quote}

\vspace{4mm}
\noindent

 
 \medskip
 \noindent {\bf Keywords:}
  KPZ equation, viscous Hamilton-Jacobi equations, maximum principle, response formalism, constructive field theory, renormalization, cluster expansions.

\smallskip
\noindent
{\bf Mathematics Subject Classification (2010):}  35B50, 35B51, 35D40, 35K55, 35R60,  35Q82, 60H15,  81T08, 81T16, 81T18, 
82C41.


\newpage
\tableofcontents

%\medskip




\section{General introduction}



The KPZ equation \cite{KPZ} is a stochastic partial differential equation describing the growth by normal deposition of an interface  in $(d+1)$ space dimensions, see e.g. \cite{BarSta,Car}. 
By definition the time evolution of the height $h(t,x)$, $x\in\R^d$, is given by
\BEQ \partial_t h(t,x)=\nu\Del h(t,x)+\lambda  \left( \sqrt{1+|\nabla h(t,x)|^2}-1\right) +\sqrt{D}\eta(t,x), \qquad x\in\R^d \label{eq:KPZ0} \EEQ
 where $\eta(t,x)$ is some (possibly regularized) white noise.
 The gradient $|\nabla h|$ (the slope of the interface) is assumed to remain throughout small so that the evolution makes physically sense, precluding
e.g. any overhang, so that the non-linear term $\sqrt{1+|\nabla h(t,x)|^2}-1\simeq \half |\nabla h(t,x)|^2$ is essentially quadratic; using this approximation gives the most common
form of this equation in the literature. Following these preliminary remarks, we shall call  KPZ equation any equation of the type
\BEQ \partial_t h(t,x)=\nu\Del h(t,x)+\lambda   V(\nabla h(t,x))) +\sqrt{D}\eta(t,x), \qquad x\in\R^d \label{eq:KPZ} \EEQ
where the {\em deposition rate} $V$ is  isotropic and convex (hence $V(\nabla h(t,x))=a+b|\nabla h(t,x)|^2+\ldots$ around $0$, with $b\ge 0$). The interest is generally in the large-scale limit of this equation, for $t$ large. 
A well-known naive rescaling argument gives some ideas about the dependence on the dimension of this limit. 
 Namely, the linearized equation, a stochastic heat equation which is a particular instance of {\em Ornstein-Uhlenbeck
 process}, 
\BEQ \partial_t \phi(t,x)=\nu \Del \phi(t,x)+ \sqrt{D} \eta(t,x), \quad (t,x)\in\R_+\times\R^d \label{eq:O.-U.} \EEQ
 is invariant under the  rescaling $\phi(t,x)\mapsto \phi^{\eps}(t,x):= \eps^{-d_{\phi}}
\phi(\eps^{-1} t,\eps^{-\half} x)$, where
\BEQ d_{\phi}:= \half(\frac{d}{2}-1) \EEQ
is the {\em scaling dimension} of the field $\phi$ (or rather {\em half} the scaling dimension, in the
physicists' convention); we used here the equality in distribution,
$\eta(\eps^{-1} t,\eps^{-\half}x)\stackrel{(d)}{=} \eps^{\half(1+\frac{d}{2})}\eta(t,x)$. 
Assuming that $\phi$ is a solution of the KPZ equation instead (say, with quadratic
deposition rate $|\nabla \phi|^2$) yields after rescaling 
\BEQ \partial_t \phi^{\eps}(t,x)=\nu \Del \phi^{\eps}(t,x)+ \eps^{d_{\phi}} \frac{\lambda}{2}|\nabla \phi^{\eps}(t,x)|^2 + \sqrt{D} \eta^{\eps}(t,x), \label{eq:power-counting-vertex-2} \EEQ
where $\eta^{\eps}\stackrel{(d)}{=}\eta$.
 For $d>2$,  the scaling exponent $d_{\phi}$ is $>0$, and the non-linear term scaling coefficient, $\eps^{d_{\phi}}$, vanishes in the limit $\eps\to 0$; in other terms, 
the KPZ equation is sub-critical at large scales in $\ge 3$ dimensions and believed to behave like the corresponding linearized equation up to a redefinition
(called {\em renormalization}) of the {\em diffusion constant} $\nu$ and of the {\em noise strength} $D$. Let $M>1$
(hereafter called {\em scale parameter}) be some fixed arbitrary parameter (e.g. $M=2$). More
precisely, according
to the general scheme due to K. Wilson \cite{Wilb,Wilc}, the fluctuations of the solution field at time scale
of order $\eps^{-1}\approx M^j$ and space scale of order $\eps^{-\half}\approx M^{j/2}$ should be approximately
governed for $j$ large by a linearized equation with scale-dependent coefficients $\nu^j,D^j$ $(j\ge 0)$, themselves  solutions of 
a certain complicated but explicit discrete dynamical system.  We prove this diffusive limit in the present series of articles
following Wilson's scheme, thus establishing on firm mathematical ground old predictions of
physicists. The 
space dimension $d$ does not really matter as long as $d\ge 3$, and we shall refer to this equation as $\KPZ_3$.

\medskip

Let us emphasize the differences  with the one-dimensional $\KPZ_1$ equation -- traditionally considered with a quadratic term $|\nabla h(t,x)|^2$ in the right-hand side 
making it partially integrable \cite{DEHP,PraSpo,Joh} --, which is perhaps more familiar to the mathematical community due to the striking progress made by several groups 
in the study of that equation
in the last years, see e.g. \cite{ACQ,Cor,Sasspo,TraWid}. Recent work \cite{Hai}, using  arguments coming from PDE's and rough paths, shows that $\KPZ_1$ is a.s. well-defined for {\em short
time}; showing existence beyond some short random time is possible only through  the Cole-Hopf transformation 
$h\mapsto \exp(h)$ which turns it into
a linear equation. Note that $h$ represents  the free energy of  directed polymers in random environment \cite{Cor}. On the other hand (reversing the previous scaling argument) it seems impossible to solve $\KPZ_3$ for short time (short scale)
unless $\eta$ is regularized; the most natural regularization here (due to the origin of this problem in real-world statistical physics) would be  to use a simple lattice cut-off. However (the choice of regularization scheme
being essentially irrelevant) we choose here for convenience a regularization by space convolution with a smooth 'bump' function. 

\medskip


What we consider here is the opposite regime, namely the  {\em 
large-scale}
limit of KPZ. The large-scale limit of $\KPZ_1$ is an extremely interesting, partially understood object;
 it is a strongly-coupled theory, solved by means of sophisticated methods coming from mathematical physics and integrable systems (Bethe Ansatz, Tracy-Widom distribution of the largest eigenvalue of random matrices, free fermions and
 determinantal processes,
 matrix formulations... see e.g. \cite{Cor} for a detailed review). Here we consider instead $\KPZ_3$,
 and prove  a rather trivial limit, furthermore in a non-physical dimension since a three-dimensional interface requires a four-dimensional space (note that $\KPZ_2$ is believed by
perturbative QFT arguments to be strongly coupled at large scales \cite{Bar,Car} and its large-scale limit is not at all understood). As in dimension 1, the exponential of the solution represents  the free energy of  directed polymers in random environment when $V$ is quadratic; several authors
discussed the associated polymer measure, see e.g.  \cite{ImbSpe,CarHu, ComYos}; the paper
\cite{ComYos} proves that the  regime  we consider $(d\ge 3, \lambda\ll 1)$ is a weak disorder
regime for the polymer measure. Their
results are not unrelated to ours --  we also use a random path representation --, but the analytic and probabilistic methods (martingale theory viz. multi-scale
analysis), the object (study of the polymer measure viz. law of the KPZ field) and the range of applicability (limited to the quadratic case in the above cited papers) differ
widely.
   
\medskip   
   
We believe that the interest of our result lies in the wide scope of applicability of our perturbation methods. Many dynamical problems in statistical physics, modelized as  interacting particle systems,  or
as  parabolic SPDEs heuristically derived by some mesoscopic limit, have been  turned into a functional integral form using the so-called {\em response} (or {\em Martin-Siggia-Rose}) 
{\em formalism} and
studied by using standard perturbative expansions originated from quantum field theory (QFT); for reviews see e.g. \cite{Car} or \cite{AltBen}. Despite the lack of mathematical rigour,
this formalism yields a correct description of  the qualitative behaviour of such dynamical problems in the large scale limit. Here we provide an efficient way of making this approach
rigorous. 

Let us here comment on the specific difficulties of dynamical problems from a QFT viewpoint, taking $\KPZ_3$ with quadratic non-linearity as an example but meaning to be general. The MSR formalism -- which may be viewed
as a careless Girsanov transform in
infinite dimension --  consists in rewriting formally the solution
$h$ of the KPZ equation as the first marginal of the measure $\frac{1}{Z} {\cal D}h{\cal D}\tilde{h} e^{- \int dt dx
\left( {\cal L}_0(h,\tilde{h})(t,x)+{\cal L}_{int}(h,\tilde{h})(t,x)\right)}$,
 where ${\cal L}_0(h,\tilde{h})(t,x)=\II \tilde{h}(t,x)(\partial_t-\nu\Del) h(t,x)+\frac{D}{2} \tilde{h}^2(t,x)$ is the free (quadratic) part, and
${\cal L}_{int}(h,\tilde{h})(t,x)=\II \frac{\lambda}{2} \tilde{h}(t,x) |\nabla h(t,x)|^2$ is the interaction term. The "infinite-dimensional Lebesgue measures" 
$"{\cal D}h"\, =\, "\prod_{(t,x)}dh(t,x)"$ and $"{\cal D}\tilde{h}"\, =\, "\prod_{(t,x)}d\tilde{h}(t,x)"$ do not exist, but the product ${\cal D}h{\cal D}\tilde{h}e^{-\int{\cal L}_0}$ can be given
a sense (using some regularization procedure) as a Gaussian measure. The problem is, it is a {\em complex-valued} Gaussian measure. The Feynman diagram perturbative approach consists
in expanding $\exp -\int {\cal L}_{int}$ into a series and making a clever resummation of some truncation of it; it is non rigorous since it yields $n$-point functions in terms
of an asymptotic expansion in the coupling parameter $\lambda$ which is divergent in all interesting cases. Constructive approaches developed by mathematical physicists from the 70es
have developed sophisticated, systematic truncation methods making it possible to control the error terms. The partial resummations are interpreted in the manner of K. Wilson  as a
scale-by-scale renormalization of the parameters $\nu,\Del,\lambda$ of the Lagrangian ${\cal L}_0+{\cal L}_{int}$. However, these approaches are  for the moment 
essentially limited to equilibrium statistical
physics models for which  the Lagrangian is {\em real} and  bounded below by some {\em non-degenerate quadratic form}  in the fields.  Here the absolute value of the measure involves a
degenerate weight $e^{-\frac{D}{2} \int dt dx \tilde{h}(t,x)^2}$ from which $h$ is absent. Less systematic approaches have been carried out successfully 
in diverse cases where only the propagator (here, $e^{t\nu\Del}$) needs to be renormalized and the state space is finite-dimensional (and thus, despite the similarities of the issues, not connected to the response
formalism): diffusive limit of random walks in random environments  \cite{BricKup},
proof of the KAM theorem \cite{BricGawKup}. One should also cite the article by Bricmont, Kupiainen and Lin
\cite{BKL} which gives long-time asymptotics of sub-critical non-linear {\em deterministic} PDE's by using successive
rescalings; the methods of the latter article  have inspired   analysts working on the blow-up of non-linear PDE's, and
also the  authors of the present article, although the successive re-elaborations necessitated by the incorporation
of the noise term have made the influences difficult to recognize.
\medskip


The reader will find a description of the results and of the main steps of the proof in the three upcoming articles,
(2) {\em Generalized PDE estimates through Hamilton-Jacobi-Bellman formalism}; (3) {\em The multi-scale expansion}. We shall mainly be concerned in this first part with PDE estimates for the {\em noiseless} or
{\em homogeneous KPZ equation}, and some preliminary applications to estimates for the noisy equation.

\medskip

Here is a brief outline of the article. Section 2 is concerned with {\em bounded} solutions of the homogeneous
KPZ equation, relying in particular on the comparison principle for non-linear parabolic PDE's. Some results are
derived with little effort from those already existing in the literature; on the other hand, the bounds on the
gradient and on the higher derivatives of the solution, see Theorems \ref{th:decay-nabla-L1} and \ref{th:bound-higher-derivatives}, may not be
found elsewhere. The really original material starts in section 3 with the search for solutions in  new spaces
of unbounded functions with good averaging properties, called hereafter ${\cal H}_{\alpha}^{\lambda}$ or
${\cal W}^{1,\infty;\lambda}_{\alpha}$ (see sections 3.2 and 3.3). 
Since we aim at proving that the solutions
of the KPZ  equation are essentially obtained from those of the linearized equation by applying perturbation theory, 
such spaces should contain the stationary solution $\phi$ of the  Ornstein-Uhlenbeck process (\ref{eq:O.-U.}),
and actually they do, as follows from a non-standard large-deviation theory for random variables
with heavy-tailed distributions (see Appendix B). Note that $\phi$
is space-translation invariant in law, hence unbounded. We prove in particular a comparison principle
(see Theorem \ref{th:comparison})  for sub- and supersolutions of the homogeneous KPZ equation in these spaces,
implying existence and unicity for viscosity solutions, which are proved to be classical.  The study of
the full inhomogeneous equation (\ref{eq:KPZ}) is much more difficult and will  be carried out only starting
from the third article in the series.  However, it can be done without using perturbation theory provided one uses a scale
cut-off, i.e. one selects the fluctuations of the solution on time ranges of order $M^j$ and space ranges
of order $M^{j/2}$ for some integer $j\ge 0$.  Such cut-offs must be implemented in some coherent way both
on the propagator $G=(\partial_t-\nu\Del)^{-1}$ and on the noise $\eta$. We show in
section 4 how to solve uniquely and bound along essentially the same lines as in section 3 some 
{\em $j$-scale infra-red cut-off}  inhomogeneous equation with a  force
term $g$ of scale $j$, satisfying Assumption \ref{assumption:etaj}.  Finally sections 5 and 6 are appendices containing
multi-scale decompositions of $G$ and $\eta$ and large-deviation estimates, implying the applicability
of the general arguments developed in  section 4 to the case of the noisy KPZ equation (\ref{eq:KPZ}).





%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\section{Model and notations}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

We consider throughout the present article either the {\em homogeneous} (or {\em noiseless})  equation
\BEQ \partial_t h=\nu\Del h+\lambda V(\nabla h) \label{eq:1.1}\EEQ
where $\lambda>0$ is a fixed, arbitrary constant,
or the {\em infra-red cut-off, inhomogeneous} equation,
\BEQ \partial_t h=\nu\Del h-\eps h+V(\nabla h)+g, \label{eq:1.2}\EEQ
where the constant $\eps=M^{-j}$ ($M>1$, $j\ge 0$) is an infra-red cut-off of scale $j$.  The assumptions on $V$ are the following:


\begin{Assumption} \label{assumptions}
The deposition rate $V$ satisfies the following assumptions,
\begin{itemize}
\item[(1)] $V$ is $C^2$;
\item[(2)] $V$ is isotropic, i.e. $V(\nabla h)$ is a function of $y=|\nabla h|$; by abuse of notation we shall
consider $V$ either as a function of $\nabla h$ or of $y$;
\item[(3)] $V$ is convex; 
\item[(4)] $V(0)=0$ and  $V(y)\ge 0$ for all $y\ge 0$;
\item[(5)] (quadratic growth at infinity) $V(y)\le y^2$ for all $y\ge 0$.
\end{itemize}
\end{Assumption}

It follows immediately from Assumptions (1), (2) and (4) that $V(y)=O(y^2)$ near $y=0$.  Assumption (5) is  thus equivalent (up to a redefinition of the constant $\lambda$) to requiring that $V$ has quadratic growth at infinity.

As for the force term $g$, it is assumed to be regular enough and have good averaging properties, depending on
the cut-off scale $j\approx -\log\eps$; the regularized white noise  $\eta$  (as shown in  Appendix A) satisfies
these properties (see sections 4 and 5).

\medskip


Assumption (3)  is a key assumption to get a time decay of the
gradient of the solution, and is also used in the proof of the comparison theorem for unbounded solutions;  Proposition \ref{prop:decay-gradient} (ii), (iii) hold under a stronger assumption.
Assumption (5) allows a comparison of the solutions to those of  the usual KPZ equation corresponding to $V(y)=y^2$, which is linearizable.  For the proof of the diffusive limit, we shall also
need very weak estimates on the derivatives of any order of $V$, satisfied for all 'reasonable' functions. 


\medskip
{\bf Notations.}  The scale parameter is an arbitrary constant $M>1$ fixed once and for all. The notation:
$f(u)\lesssim g(u)$, resp. $f(u)\gtrsim g(u)$ means: $|f(u)|\le C|g(u)|$, resp. $|f(u)|\ge C|g(u)|$, where $C>0$ is an unessential constant (depending only on $d,\nu,D$ and $M$). 
Similarly, $f(u)\approx g(u)$ means: $f(u)\lesssim g(u)$ and $g(u)\lesssim f(u)$.
We denote by $L^p$, $p\in[1,\infty]$  the usual Lebesgue spaces with associated norm $||\ ||_{p}$,  by ${\cal W}^{1,\infty}$ the Sobolev space of bounded functions with bounded generalized derivative, and by $C^{1,2}$ the space of functions which are $C^1$ in time and $C^2$ in space. The positive, resp. negative part of a function $f$ is denoted
by $f_+$, resp. $f_-$; by definition, $f^+,f^-\ge 0$, $f=f^+-f^-$ and $f^+ f^-=0$. The {\em oscillation} $\osc_{\Omega} f$
of a continuous function $f$ on a domain $\Omega$ is defined as  $\sup_{\Omega}f-\inf_{\Omega}f$; the {\em average} 
$\frac{1}{|\Omega|} \int_{\Omega} f$ of $f$
on a bounded domain $\Omega$ is denoted by $\fint_{\Omega}f$. The space of
lower, resp. upper semicontinuous functions on a domain $\Omega$ is denoted by $LSC(\Omega)$, resp. $USC(\Omega)$.




%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%


\section{Bounds for the homogeneous equation: the case of a bounded initial condition}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


We consider in this section the homogeneous equation,
\BEQ \partial_t h=\nu\Del h+\lambda V(|\Del h|)  \label{eq:KPZdet} \EEQ
with initial condition $h_0(x)=h(0,x)$ in ${\cal W}^{1,\infty}$.  One finds in the literature a detailed
study of the particular case $V(y)=y^q$, $q>1$. Most basic results (including existence), based on the
principle of maximum or on a short-time series expansion of the mild solution, depend very little
on the precise form of $V$, provided it is regular enough and, say, polynomially bounded. We quickly review them
now and leave it to the reader to check that they extend to a rate $V$ satisfying Assumptions \ref{assumptions} (1),
(2), (4). 

By \cite{AmoBen} and \cite{BenLau}, 
the Cauchy problem has a unique, global solution $u$ which is classical for positive times, that is,
 $u\in C([0,+\infty)\times\R^d)\cap C^{1,2}((0,\infty)\times\R^d)$.
The comparison principle, in the form proved by Kaplan \cite{Kap} for classical, bounded solutions of
non-linear parabolic equations on unbounded spatial domains, implies that $h_t\ge 0$ for all
$t\ge 0$ (resp. $h_t\le 0$ for all $t\ge 0$) if
$h_0\ge 0$  (resp. $h_0\le 0$) and  yields the  a priori estimates
\BEQ ||h_t||_{\infty}\le ||h_0||_{\infty},\qquad ||\nabla h_t||_{\infty}\le ||\nabla h_0||_{\infty} \qquad 
(t\ge 0). \label{eq:a-priori-estimates}\EEQ

\medskip


We now prove time-decay estimates of the solution for various norms, emphasizing those which are not
straightforward extension of previously known results for $V(y)=y^q$. Such estimates
 come roughly from three different sources. Generally speaking,  constants appearing in the inequalities deteriorate
 when $\nu\to 0$ whenever parabolic estimates are involved  (see below); Proposition \ref{prop:decay-gradient}
 (ii), (iii) is an outstanding exception.


\medskip

We recall here briefly for non-specialists the {\em maximum principle } and the {\em comparison principle} 
for parabolic PDE's, in a weak
form which is sufficient for section 2. Standard references on the subject are e.g. \cite{Eva}, \cite{CraIshLio},
\cite{Bar}.


\begin{Proposition}[maximum and comparison principle]
Let $u(t,x)$, $(t,x)\in[0,T]\times\R^d$ be a classical solution of the parabolic PDE $\partial_t u(t,x)=\nu\Del u(t,x)+
W(t,x,\nabla u(t,x))$, where $W$ is a smooth function, bounded in any subset of the form $\R\times\R^d\times K$,
$K\subset\R^d$ compact. Assume that $\sup_{[0,T]\times\R^d} |u|<\infty$ and $\sup_{[0,T]\times\R^d}
|\nabla u|<\infty$. Then:
\begin{itemize}
\item[(i)] (weak maximum principle) $\forall t\in[0,T],\ ||u_t||_{\infty}\le ||u_0||_{\infty}$.
\item[(ii)] (weak comparison principle) let $\bar{U}$, resp. $\underline{U}$ be a super-, resp. sub-solution of the above PDE, namely, $\bar{U},\underline{U}\in C^{1,2}([0,T]\times\R^d)$ and $\partial_t \bar{U}(t,x)\ge \nu\Del \bar{U}(t,x)+
W(t,x,\nabla \bar{U}(t,x)),\partial_t \underline{U}(t,x)\le\nu\Del \underline{U}(t,x)+
W(t,x,\nabla \underline{U}(t,x)).$ Assume $\bar{U}_0\ge\underline{U}_0$. Then $\bar{U}_t\ge\underline{U}_t$ for all $t\ge 0$.
\end{itemize}
\end{Proposition}

Note that the above proposition extends under appropriate monotonicity hypothesis to parabolic PDE's of the form
$\partial_t u(t,x)=\nu\Del u(t,x)+
W(t,x,u(t,x),\nabla u(t,x))$. However, it is precisely the absence of dependence of $W$ on $u(t,x)$ that makes
two-sided a priori estimates like (\ref{eq:a-priori-estimates}) so easy.


%%%%%%%%%%

\subsection{Comparison to the linear heat equation}

%%%%%%%%%%%%%%



Assumptions \ref{assumptions} (4)-(5), $0\le V(y)\le y^2$, allows (as we shall presently see) a direct 
{\em comparison with the linear heat equation} if either  $h_0\ge 0$
or  $h_0\le 0$. Bounds for signed initial conditions  follow then from the comparison principle: namely, letting
$\bar{h}$, resp. $\underline{h}$ be the solution of (\ref{eq:KPZdet}) with initial condition $h^+_0$, resp. $-h^-_0$,
one has
\BEQ \underline{h}\le 0, \bar{h}\ge 0; \qquad \qquad \underline{h}\le h\le \bar{h}.\EEQ 
Also, $t\mapsto ||\bar{h}_t||_1$ is increasing, while $t\mapsto ||\underline{h}_t||_1$ is decreasing.

Considering first $\underline{h}$,  the comparison
principle allows one to bound the solution of (\ref{eq:KPZdet}) by the solution of the linear heat equation
with same initial condition, namely, 
\BEQ |\underline{h}(t)|\le e^{t\nu\Del}h_0^-. \label{eq:h-}\EEQ
We now turn to $\bar{h}$ and bound similarly the solution of (\ref{eq:KPZdet}) with positive initial condition by
the solution $u$ of the standard KPZ equation, $\partial_t u=\nu\Del u+\lambda |\nabla u|^2$ with the same initial
condition. The exponential 
transformation $w:=e^{\frac{\lambda}{\nu}u}-1$ turns it into the linear equation $\partial_t w=\nu\Del w$, with
positive initial condition $w_0=e^{\frac{\lambda}{\nu}h^+_0}-1$. The inequality $x\le \frac{\nu}{\lambda}(e^{\frac{\lambda}{\nu}x}-1),\ x\ge 0$ yields
\BEQ ||\bar{h}_t||_{\infty}\le||u_t||_{\infty}\le \frac{\nu}{\lambda} ||w_t||_{\infty}.\EEQ

\medskip
To go further, we assume $w_0\in L^1$ and use the following standard parabolic estimates \cite{Sch} for $q=1$.

\begin{Proposition}[parabolic estimates] There exist constants $C_k$, $k=0,1,\ldots$ depending only on $d$ such that,
for every regular enough function $f_0:\R^d\to\R$ and $p\ge q\ge 1$,
\BEQ ||\nabla^k e^{t\nu\Del}f_0||_{p}\le C_k (\nu t)^{-\frac{d}{2}(\frac{1}{q}-\frac{1}{p})-\frac{k}{2}}||f_0||_q,\qquad k\ge 0. 
\label{eq:parabolic-estimates} \EEQ
\end{Proposition}


Let $\mu$ be the Lebesgue measure on $\R^d$. The well-known identity $\int f(u(x))dx=\int_0^{+\infty} \mu(u>a)f'(a)da$ valid for  $u:\R^d\to\R_+$ measurable and $f:\R_+\to\R$ smooth
 yields for $f(u)=\frac{\nu}{\lambda}(e^{\frac{\lambda}{\nu}h^+_0}-1)$
\BEA  ||\bar{h}_t||_{1}&\le & \frac{\nu}{\lambda} ||w_t||_1\le \frac{\nu}{\lambda} ||w_0||_1 \nonumber\\
 &\lesssim& \int h^+_0(x) {\bf 1}_{h^+_0(x)\le \nu/\lambda} dx + \int_{\nu/\lambda}^{+\infty}
 \mu(h^+_0>a) e^{\frac{\lambda}{\nu} a} da \nonumber\\
  &\lesssim & ||h^+_0||_1 \left(1+\int_{\nu/\lambda}^{+\infty} \frac{e^{\frac{\lambda}{\nu}a}}{a}
 da \right)\nonumber\\
  &\lesssim& ||h^+_0||_1 e^{\frac{\lambda}{\nu}||h^+_0||_{\infty}}\EEA
 so 
 \BEQ \bar{I}_{\infty}:=||h^+_0||_1 e^{\frac{\lambda}{\nu}||h^+_0||_{\infty}}  \label{eq:barIinfty}\EEQ
 is an upper bound for $\sup_{t\ge 0} ||\bar{h}_t||_1$ (see \cite{LauSou}, Proposition 2 (iii)); 
 at the same time, one gets
\BEQ ||\bar{h}_t||_{\infty} \le \frac{\nu}{\lambda} ||w_t||_{\infty}\lesssim \frac{\nu}{\lambda}
||w_0||_1 t^{-d/2} \lesssim \bar{I}_{\infty} t^{-d/2}.  \EEQ


On the other hand, (\ref{eq:h-}) gives immediately if $h_0^-\in L^1$
\BEQ ||\underline{h}_t||_{\infty}\lesssim ||h_0^-||_1 t^{-d/2}.\EEQ

\medskip

Thus one has shown a global bound for the $L^1$-norm, and a time-decay in $O(t^{-d/2})$ for the sup-norm of solutions of (\ref{eq:KPZdet}) with arbitrary integrable
initial condition $h_0\in {\cal W}^{1,\infty}\cap L^1$.




%%%%%%%%%%%%%%%%%%%%

\subsection{Time-decay of solutions of viscous Hamilton-Jacobi equations}

%%%%%%%%%%%%%%%%%%%%%%%



A second series of results is a particular case of the more general  {\em time-decay of the  gradients of solutions of viscous Hamilton-Jacobi
equations}, which can itself be seen as (1) an extension to non-linear equations of the standard parabolic
estimates; (2) or a multi-dimensional extension of the decay of  solutions of scalar 
conservation laws, see \cite{GilGueKer}, \cite{BenLau}, \cite{BenBenLau}, or \cite{BenKarLau}, section 3 for further
results concerning in particular a single-sided bound on the Hessian.  Generally speaking, such results rely on
 convexity assumptions on $V$. Here we shall only state the following estimate, 
which is an extension of  \cite{BenLau}, Theorem 1. In Proposition \ref{prop:decay-gradient}, by exception, constants, explicit and
implicit (i.e. hidden by the sign $\lesssim$) are $\nu$-independent.


\begin{Proposition}[time-decay of the gradient] \label{prop:decay-gradient}
\begin{itemize}
\item[(i)] If $yV'(y)-V(y)\ge 0$ , then \BEQ ||\nabla h_t||_{\infty}\lesssim ||h_0||_{\infty} (\nu t)^{-\half}. \label{eq:decay-gradient1} \EEQ    
\item[(ii)] Under the stronger assumption 
\BEQ yV'(y)-V(y)\ge C\min(y^2,y^q),\qquad y\ge 0 \label{eq:stronger-assumption}\EEQ
for some constant $C>0$ and some exponent $q\in(1,2]$, one has 
\BEQ ||\nabla h_t||_{\infty}\lesssim  \left(\frac{||h_0||_{\infty}/\lambda}{t}\right)^{1/q}, \qquad
t\le \frac{||h_0||_{\infty}}{\lambda}  \label{eq:1.14}\EEQ
\BEQ  ||\nabla h_t||_{\infty}\lesssim  \left(\frac{||h_0||_{\infty}/\lambda}{t}\right)^{1/2}, \qquad
t\ge \frac{||h_0||_{\infty}}{\lambda}  \label{eq:1.15}\EEQ
\item[(iii)] Under the even stronger assumption 
\BEQ yV'(y)-V'y)\ge Cy^2 \label{eq:even-stronger-assumption}, \EEQ
 one has for all $t\ge 0$
 \BEQ |\nabla h_t(x)|\lesssim \left(\frac{|h_t(x)|/\lambda}{t}\right)^{1/2}, \qquad x\in\R^d.
 \label{eq:pointwise-decay-gradient} \EEQ
 Hence in particular
\BEQ  ||\nabla h_t||_{\infty}\lesssim  \left(\frac{||h_0||_{\infty}/\lambda}{t}\right)^{1/2}. \label{eq:decay-gradient2}\EEQ 
\end{itemize}
\end{Proposition}

Note that condition (i), $yV'(y)-V(y)\ge 0$ is a consequence of the convexity of $V$ (see  Assumption \ref{assumptions} (3)). On the other hand,  the hypothesis (\ref{eq:stronger-assumption}) in (ii) holds true for functions $V(y)$ that behave like $y^2$ for $y$ small, and like
$y^q$, $1<q\le 2$ for $y$ large; the stronger hypothesis (\ref{eq:even-stronger-assumption}) in (iii) for
functions that behave like $y^2$ both for $y$ small and $y$ large. Note that the decay in (\ref{eq:decay-gradient1}) is produced by the diffusion term $\nu\Del$ in the equation, so it might be called a generalized parabolic estimate; while (\ref{eq:1.14},\ref{eq:1.15})
or (\ref{eq:decay-gradient2}) are diffusion-independent effects of the non-linear term in the equation, and would
also hold true for viscosity solutions of the first-order Hamilton-Jacobi equation obtained by letting $\nu\to 0^+$.
\medskip


{\bf Proof.} We first rescale $h$ and $x$ by letting $x\to x'=\nu^{-\half} x$, $h\to u=\frac{\lambda}{\nu} h$ 
so that $\nabla_x h=\left(\frac{\lambda^2}{\nu}\right)^{-\half} \nabla_{x'}u$, and
$W(y)=\frac{\lambda^2}{\nu} V(\left(\frac{\lambda^2}{\nu}\right)^{-\half}y)$, so that the equation for $u$
\BEQ \partial_t u=\Del u +W(|\nabla u|) \EEQ
is independent of the parameters $\nu,\lambda$. 

\begin{itemize}
\item[(i)] 
Referring to the proof of Lemma 3 in \cite{GilGueKer}, from which \cite{GilGueKer},
Theorem 2 follows immediately, and letting directly $\eps=0$,  we see that $\frac{|\nabla u|^2}{\theta^2(u)}$ is a 
super-solution for the parabolic
operator
\BEQ \widetilde{\cal N}(w):=\Del w+b\cdot \nabla w+cw^2+ew-\partial_t w,\EEQ
where  $c:=2\theta(u)\theta''(u)$, $e=-2(W(\nabla u)-\nabla u\cdot W'(\nabla u))$
(note that for $V$ homogeneous, $V(|\nabla u|)=|\nabla u|^q$,  $\widetilde{\cal N}(\frac{|\nabla u|^2}{\theta^2(u)})={\cal N}(\frac{|\nabla u|^2}{\theta^2(u)})$ where $\cal N$ \cite{GilGueKer} has
instead of $ew$  a sum of two terms, $2(q-1)\theta^{q-1}(u)\theta'(u)w^{1+q/2}-2\frac{\theta'(u)}{\theta(u)}
H(\nabla u)w$, with $H(\xi)=W(\xi)-\xi\cdot\nabla W(\xi)+(q-1)|\xi|^q$, the first term $2(q-1)\theta^{q-1}(u)\theta'(u)
w^{1+q/2}$ compensating the last term $(q-1)|\xi|^q$ in $H$). Choose $\theta=\theta_1$ as in \cite{GilGueKer},
eq. (24), so that $c=-1$. Now, for $V$ isotropic, $e=-2\frac{\theta'(u)}{\theta(u)}(W(y)-yW'(y))$, $y=|\nabla u|$;
this is $\le 0$ under the assumptions (\ref{assumptions}). Hence $t^{-1}$ is a sub-solution of $\widetilde{\cal N}$, and the
comparison principle yields $|\nabla u_t(x)|^2\le \frac{\theta_1^2(u_0(x))}{t}$. Now $||\theta_1||_{\infty}\le 
2||u_0||_{\infty}$. Scaling back to the original variables $h,x$ yields  the first bound, $||\nabla h_t||_{\infty}\lesssim ||h_0||_{\infty} (\nu t)^{-\half}.$  



\item[(ii)] The second bound is an 
extension of \cite{BenLau}. Exactly as in (i), one may assume that $u_0\le 0$. Up to an overall change of sign, $u\mapsto -u$, we are in the
conditions of \cite{BenLau}, Theorem 1, with $p=2$, except that $V$ is not necessarily a power function. Letting again $\eps=0$, the function $\Theta$ in eq. (20) p. 2005 is here $\Theta(r)=\Theta(y^2)=2y^2\frac{d}{dy^2}W(y)-W(y)=yW'(y)-W(y)$;
by assumption, 
\BEQ \Theta(r)\gtrsim \min(r,\left(\frac{\lambda^2}{\nu}\right)^{1-\frac{q}{2}} r^{q/2})=
r {\bf 1}_{r\le \frac{\lambda^2}{\nu}}+\left( \frac{\lambda^2}{\nu}\right)^{1-\frac{q}{2}} r^{\frac{q}{2}}
{\bf 1}_{r>\frac{\lambda^2}{\nu}}.\EEQ
 Eq. (14) p. 2003 implies
\BEQ {\cal L}w+Cv^{-2}\Theta(v^2 w)w\le 0\EEQ
with ${\cal L}=\partial_t-\Del$ up to some gradient term vanishing on functions $h$ which are independent of $x$ (see eq. (10) p. 2002 for a precise definition), $v:=\sqrt{u}$, $w=|\nabla v|^2$. Now  $v\le ||u_0||_{\infty}^{\half}$, hence
\BEA &&  v^{-2}\Theta(v^2 w)w\ge w^2 \qquad (w\lesssim \left( \frac{\nu||u_0||_{\infty}}{\lambda^2}\right)^{-1}),
\nonumber\\ && \qquad \qquad 
v^{-2}\Theta(v^2 w)w\ge  \left( \frac{\nu||u_0||_{\infty}}{\lambda^2}\right)^{\frac{q}{2}-1} w^{1+\frac{q}{2}} \qquad (w\gtrsim \left(\frac{\nu ||u_0||_{\infty}}{\lambda^2}\right)^{-1}) \nonumber\EEA
so ${\cal N}w\le 0$, where ${\cal N}$ is the parabolic differential operator
\BEQ {\cal N}:h\mapsto {\cal L}h+C N(h),\qquad N(h)=h^2 {\bf 1}_{h\le \left(\frac{\nu||u_0||_{\infty}}{\lambda^2}\right)^{-1}}+
\left(\frac{\nu||u_0||_{\infty}}{\lambda^2}\right)^{\frac{q}{2}-1} h^{1+\frac{q}{2}} {\bf 1}_{h> \left(
\frac{\nu||u_0||_{\infty}}{\lambda^2}\right)^{-1}}.\EEQ
Note that $N$ is an increasing function. The comparison principle thus implies that $w\le h$ if ${\cal N}h\ge 0$.
Such a function $h=h(t)$ is easily constructed by solving the ordinary differential equations $\partial_t h=-
\left(\frac{\nu||u_0||_{\infty}}{\lambda^2}\right)^{\frac{q}{2}-1} h^{1+\frac{q}{2}}$ for $h\le 
\left(\frac{\nu||u_0||_{\infty}}{\lambda^2}\right)^{-1}$, $\partial_t h=-h^2$ for $h\ge 
\left(\frac{\nu||u_0||_{\infty}}{\lambda^2}\right)^{-1}$, yielding up to unimportant constants $h(t)=
\left(\frac{\nu||u_0||_{\infty}}{\lambda^2}\right)^{\frac{2}{q}-1} t^{-\frac{2}{q}}$ $(t\le
\frac{\nu||u_0||_{\infty}}{\lambda^2})$,
$h(t)=\frac{1}{t}$ $(t\ge \frac{\nu||u_0||_{\infty}}{\lambda^2})$. This gives bounds for $||\nabla v_t||_{\infty}$ by taking the square-root,
and then bounds for $||\nabla u_t||_{\infty}$ by noting that $\nabla u=2u^{1/2}\nabla v$ and $u^{\half}_t(x)\le ||u_0||_{\infty}^{\half}$ (see \cite{BenKarLau}, proof of Proposition 3.1), namely,
\BEQ ||\nabla u_t||_{\infty}\lesssim \left(\frac{\nu}{\lambda^2}\right)^{-\half} \left( \frac{\nu||u_0||_{\infty}}{\lambda^2 t}\right)^{1/q}, \qquad t\le \frac{\nu ||u_0||_{\infty}}{\lambda^2};\EEQ
\BEQ ||\nabla u_t||_{\infty}\lesssim \left(\frac{||u_0||_{\infty}}{t}\right)^{\half},\qquad
 t\ge \frac{\nu ||u_0||_{\infty}}{\lambda^2}.\EEQ
 Hence (\ref{eq:1.14},\ref{eq:1.15}) by rescaling. 
 
 \item[(iii)] Under the stronger assumption (\ref{eq:even-stronger-assumption}), the previous computations
 yields $h(t)=\frac{1}{t}$ for all $t>0$. Hence $|\nabla u_t(x)|\lesssim \left( \frac{|u_t(x)|}{t}\right)^{\half}$. Eq. (\ref{eq:pointwise-decay-gradient}), and then (\ref{eq:decay-gradient2}),  follow by rescaling and using
 the a priori bound $||h_t||_{\infty}\le || h_0||_{\infty}$.


\end{itemize}
\hfill\eop


%%%%%%%%%%%%%%%%%%%%%%

\subsection{Bounds through integral representation of mild solutions}

%%%%%%%%%%%%%%%%%%%%%%%%%




The third source of results is the integral form of the equation,
\BEQ h_t=e^{t\nu\Del}h_0+\lambda\int_0^t e^{(t-s)\nu\Del} V(\nabla h_s)ds, \label{eq:integral-form} \EEQ
the solutions of which, traditionally called {\em mild solutions}, are not necessary twice differentiable in space.
(\ref{eq:integral-form}) is used to prove local-in-time well-posedness of the equation, while the  a priori estimates
(\ref{eq:a-priori-estimates}) imply  global existence \cite{AmoBen}. We shall not come back to this; instead,  we give an application to the
proof of various bounds for the gradient and for higher derivatives of the solution. Generally speaking,
$\nu$-dependent constants
(throughout denoted by $C$ and possibly varying from line to line) come out of the computations everywhere. From \cite{AmoBen}, the solution
obtained by iterating (\ref{eq:integral-form}),
\BEQ h^{(0)}_t=h_0,\qquad h^{(k+1)}_t=e^{t\nu\Del}h^{(k)}_0+\lambda\int_0^t e^{(t-s)\nu\Del} V(\nabla h^{(k)}_s)ds
\qquad (k\ge 0) \label{eq:iterative} \EEQ
 in search for a fixed point is obtained as a converging series for
$t<T_1^*$, \BEQ T_1^*=C(\lambda||\nabla h_0||_{\infty})^{-2} \EEQ
 for some constant $C$, and shown to be uniformly smooth: namely, for every $k\ge 0$,
$||\nabla^k h_t||_{\infty}\le C_k||\nabla^k h_0||_{\infty},\ 0\le t\le T_1^*$ provided the initial solution has
bounded derivatives of  order $\le k$. By an appropriate choice of $C$ one may assume that $C_2=C_3=2$. This, in turn, shows, using the a priori bound, $||\nabla h_t||_{\infty}\le ||\nabla h_0||_{\infty}$,  that the solution at any later time also has bounded
derivatives of arbitrary order. We are interested here in quantitative bounds that can be shown to be close to
optimal in some case where explicit computations are possible (see next paragraph).

\bigskip


We shall give two different results. Recall $\bar{I}_{\infty}=||h^+_0||_1 e^{\frac{\lambda}{\nu} ||h^+_0||_{\infty}}$
(see (\ref{eq:barIinfty}). Our first result uses hypothesis (\ref{eq:even-stronger-assumption}), $yV'(y)-V(y)\ge Cy^2$
(see Proposition \ref{prop:decay-gradient} (iii)).


\begin{Theorem}[decay in $L^1$-norm of the gradient] \label{th:decay-nabla-L1}
Assume $h_0\in W^{1,\infty}\cap L^1$ and let $h_t$ be the solution of the KPZ equation (\ref{eq:KPZdet}), with
$V$ satisfying the hypothesis (\ref{eq:even-stronger-assumption}). Then 
\BEQ ||\nabla h_t||_1\lesssim \max(J_{\infty},J_{\infty}^{1+\frac{1}{d}}) (1+t)^{-\half},\qquad J_{\infty}=
\sup\left(1,\frac{||\nabla h_0||_1}{||h_0||_1},\lambda ||\nabla h_0||_{\infty} (1+O(\lambda ||h_0||_{\infty}))
\right) ||h_0||_1 e^{C\lambda ||h_0||_{\infty}}. \label{eq:decay-nabla-L1}\EEQ
\end{Theorem}

The time-decay in $O(t^{-\half})$ of $||\nabla h_t||_1$ is shown in \cite{BenKarLau} for $V(y)=y^2$, but with a constant $J_{\infty}$
which is roughly $e^{\bar{I}_{\infty}}$ and thus far from optimal (see p. 1290 and 1291). The emphasis there was on the
asymptotic convergence for $t\to \infty$ of the solution to a multiple of the heat-kernel (see Theorem 2.3 (a)),
 an interesting result in itself to which we do not come back here. 

\medskip


{\bf Proof.}

Our proof, based on intuition derived from the explicit computations of the next paragraph, shows that there are
different time regimes for $||\nabla h_t||_1$. Initially (i) the $L^1$-norm of the gradient may increase (as is the
case for the $L^1$-norm of the solution when the initial condition is positive); for later times (iii) it decreases like
the square-root of time. There
also appears a regime (ii) for intermediate times, during which the $L^1$-norm of the gradient is shown to be essentially
constant.

These three regimes come from the three essentially different bounds one has on $||\nabla h_t||_{\infty}$; namely,
(i) $||\nabla h_t||_{\infty}\le ||\nabla h_0||_{\infty}$ by the comparison principle; (ii) $||\nabla h_t||_{\infty}\lesssim \sqrt{\frac{||h_0||_{\infty}}{\lambda}} t^{-\half}$ by Proposition \ref{prop:decay-gradient} (iii); (iii) 
\BEQ ||\nabla h_t||_{\infty}\lesssim ||h_{t/2}||_{\infty} t^{-\half}\lesssim I_{\infty} t^{-(d+1)/2},
\label{eq:d+1/2} \EEQ
 as follows from a combination of Proposition \ref{prop:decay-gradient} (i) and of the parabolic estimates
developed in the lines following eq. (\ref{eq:barIinfty}), where 
\BEQ I_{\infty}=
||h_0||_1 e^{C\lambda||h_0||_{\infty}} \label{eq:Iinfty} \EEQ
is an upper bound for $\sup_{t\ge 0} ||h_t||_1$ (in order to get not too complicated formulas, we avoid the unpleasant task of optimizing the constants, and choose $C$ large enough).

\begin{itemize}
\item[(i)] For $t$ small one uses the trivial bound (i), $||\nabla h_t||_{\infty}\le ||\nabla h_0||_{\infty}$,
and applies the iterative scheme (\ref{eq:iterative}) in uniform time slices $[T_0^*,T_1^*]=[0,T_1^*],[T_1^*,T_2^*]=
[T_1^*,2T_1^*],\ldots,[T_{n_0-1}^*,T_{n_0}^*]=[(n_0-1)T_1^*,n_0 T_1^*]$ where $n_0\approx \lambda ||h_0||_{\infty}$,
so that $||\nabla h_0||_{\infty}\approx \sqrt{\frac{||u_0||_{\infty}}{\lambda}} (T_{n_0}^*)^{-\half}$. At some time
comparable with $T_{n_0}^*$,  the bound (ii) on $||\nabla h_t||_{\infty}$ becomes better.  We let
$M_n^{(k)}:=\sup_{[T_n^*,T_{n+1}^*]} ||\nabla u_t^{(k)}||_1$ and $M_n:=\sup_{[T_n^*,T_{n+1}^*]} ||\nabla u_t||_1=
\lim_{k\to\infty} M_n^{(k)}$. By (\ref{eq:iterative}) and the parabolic estimates recalled in (\ref{eq:parabolic-estimates}),
\BEQ M_n^{(k+1)}\lesssim ||\nabla h_{T_n^*}||_1+\lambda \sup_{t\in[T_n^*,T_{n+1}^*]} \int_{T_n^*}^t
(t-s)^{-\half} ||(\nabla h^{(k-1)}(s))^2||_1 ds, \label{eq:M1}\EEQ
together with the interpolation inequality, 
\BEQ ||(\nabla h^{(k-1)}(s))^2||_1=||\nabla h^{(k-1)}(s)||_2^2\le
||\nabla h^{(k-1)}(s)||_1 ||\nabla h^{(k-1)}(s)||_{\infty}, \EEQ one obtains
\BEQ M_n^{(k+1)}\lesssim ||\nabla h_{T_n^*}||_1+C^{-1} M_n^{(k)} \EEQ
where $C^{-1}$ is proportional to the inverse of the constant $C$ in the definition of $T_1^*$. For $C$ large
enough, this yields $\sup_k M_n^{(k)}\le 2M_{n-1}$ and $M_n\le 2M_{n-1}$. Thus
\BEQ M_{n_0}\le ||\nabla h_0||_1 e^{C\lambda ||h_0||_{\infty}}\lesssim \frac{||\nabla h_0||_1}{||h_0||_1} I_{\infty}
\label{eq:2.32} \EEQ
with an appropriate definition of the constant in (\ref{eq:Iinfty}).


\item[(ii)] For $n\ge n_0$ one defines inductively $T_n^*$ by $T_{n+1}^*-T_n^*=C(\lambda||\nabla h_{T_n^*}||_{\infty})^{-2}$. Note that $T_{n_0}^*\approx \frac{||h_0||_{\infty}}{||\nabla h_0||_{\infty}^2}$; by the second
estimate (ii) on $||\nabla h_t||_{\infty}$, $T_{n+1}^*-T_n^*\gtrsim \frac{T_n^*}{\lambda ||h_0||_{\infty}}$,
so 
\BEQ T_n^*\ge \frac{||h_0||_{\infty}}{||\nabla h_0||_{\infty}^2}\left( 1+\frac{C}{\lambda||h_0||_{\infty}}\right)^{n-n_0},\qquad n\ge n_0.  \label{eq:1.32} \EEQ


Instead of  the bound $||\nabla e^{(t-T_n^*)\nu\Del} h_{T_n^*}||_1\le||\nabla h_{T_n^*}||_1$ used in (\ref{eq:M1}), it is more clever for $n$ and $t-T_n^*$ large enough to use the parabolic
estimate $||\nabla e^{(t-T_n^*)\nu\Del}h_{T_n^*}||_1\lesssim (t-T_n^*)^{-\half} I_{\infty}$ if the latter
expression is $\le ||\nabla h_{T_n^*}||_1$. Thus one gets the improved estimate
\BEQ M_n^{(k+1)}\lesssim\sup_{t\in[T_n^*,T_{n+1}^*]} \left( \inf(||\nabla h_{T_n^*}||_1,I_{\infty}
(t-T_n^*)^{-\half}) +\lambda||\nabla h_{T_n^*}||_{\infty} M_n^{(k)} (t-T_n^*)^{\half}\right). \label{eq:M2}\EEQ

If $||\nabla h_{T_n^*}||_1 \gtrsim I_{\infty} (T_{n+1}^*-T_n^*)^{-\half}\approx \lambda I_{\infty}
||\nabla h_{T_n^*}||_{\infty}$, the improved estimate (\ref{eq:M2})
is better than (\ref{eq:M1}) and yields
\BEA && M_n^{(k+1)}\lesssim \sup\left( ||\nabla h_{T_n^*}||_1+\lambda ||\nabla h_{T_n^*}||_{\infty} 
M_n^{(k)} \frac{||h_{T_n^*}||_1}{||\nabla h_{T_n^*}||_1}, \right.\nonumber\\
&&\qquad \qquad \left. \sup_{T_{n+1}^*-T_n^*\ge t-T_n^*\ge (||h_{T_n^*}||_1/||\nabla h_{T_n^*}||_1)^2}
\left( I_{\infty} (t-T_n^*)^{-\half}+\lambda ||\nabla h_{T_n^*}||_{\infty} M_n^{(k)} (t-T_n^*)^{\half} \right)\right)
\nonumber\\ \EEA
The function $x\mapsto \frac{a}{x}+bx$, here $x=\sqrt{t-T_n^*}$, is bounded on any interval of $\R_+$ by the max of its values at the two ends of the
interval. Hence
\BEQ M_n^{(k+1)}\lesssim \sup\left(  ||\nabla h_{T_n^*}||_1+\lambda ||\nabla h_{T_n^*}||_{\infty} 
M_n^{(k)} \frac{I_{\infty}}{||\nabla h_{T_n^*}||_1}, I_{\infty} (T_{n+1}^*-T_n^*)^{-\half}+C^{-1}M_n^{(k)}
\right)\EEQ
with $C^{-1}<1$. Iterating these affine inequalities yields either 
\BEQ M_n\lesssim I_{\infty}  (T_{n+1}^*-T_n^*)^{-\half} \approx \lambda I_{\infty} ||\nabla h_{T_n^*}||_{\infty}
\le I_{\infty} ||\nabla h_0||_{\infty}; \EEQ
or, assuming on the contrary that $||\nabla h_{T_n^*}||_1\gtrsim I_{\infty}(T_{n+1}^*-T_n^*)^{-\half}$,
\BEQ M_n\lesssim  \frac{||\nabla u_{T_n^*}||_1}{1-\lambda I_{\infty} \frac{||\nabla u_{T_n^*}||_{\infty}}{||\nabla
u_{T_n^*}||_1}}.\EEQ
Since the sequence $n\mapsto \lambda I_{\infty}||\nabla h_{T_n^*}||_{\infty}$ is exponentially decreasing
(as follows from the bound (ii) and the fact that the sequence $(T_n^*)$ is exponentially increasing, see
(\ref{eq:1.32})), the
recursive sequence
\BEQ x_{n+1}=\frac{x_n}{1-\lambda I_{\infty}\frac{ ||\nabla h_{T_n^*}||_{\infty}}{x_n}}\approx x_n+
\lambda I_{\infty} ||\nabla h_{T_n^*}||_{\infty}\EEQ
starting from $x_{n_1}\approx I_{\infty}(T_{n+1}^*-T_n^*)^{-\half}\lesssim \lambda||\nabla h_0||_{\infty} I_{\infty}$,
converges to 
\BEQ x_{\infty} \lesssim \lambda||\nabla h_0||_{\infty}I_{\infty} \left(1+\frac{1}{1-(1+\frac{C}{\lambda||h_0||_{\infty}})^{-1/2}}\right)\lesssim \lambda ||\nabla h_0||_{\infty} I_{\infty} (1+O(\lambda ||h_0||_{\infty})).\EEQ

This gives a global bound for $M_n,n\ge 0$,
\BEQ \sup_{n\ge 0}M_n\le \tilde{I}_{\infty}:=\sup\left(\frac{||\nabla h_0||_1}{||h_0||_1},
\lambda ||\nabla h_0||_{\infty} (1+O(\lambda ||h_0||_{\infty})) \right) 
I_{\infty},\EEQ
but no time decay yet in general.

\item[(iii)] For $t\gtrsim I_{\infty}^{2/d}$ we use the estimate (iii), $||\nabla h_t||_{\infty}\lesssim I_{\infty}
t^{-(d+1)/2}$ and prove the time decay in $O(t^{-\half})$. Let $\tilde{M}_n:=\sup_{t\in[2^{n-1},2^n]}
||\nabla h_t||_1$ for $n\ge n_2:=1+\log_2 I_{\infty}^{2/d}$. For all $t\in[2^n,2^{n+1})$,
\BEA ||\nabla h_t||_1 &\lesssim & 2^{-n/2} ||h_{t-2^{n-1}}||_1+\lambda \int_{t-2^{n-1}}^{t} ds (t-s)^{-\half}
||\nabla h_s||_1 ||\nabla h_s||_{\infty} \nonumber\\
&\lesssim& 2^{-n/2}I_{\infty}+\lambda 2^{n/2} (\tilde{M}_n+\tilde{M}_{n+1})I_{\infty} (2^{-n})^{(d+1)/2},\EEA
hence
\BEQ \tilde{M}_{n+1}\lesssim 2^{-n/2}I_{\infty}+\lambda I_{\infty} 2^{-nd/2} \tilde{M}_n+\lambda I_{\infty}
2^{-nd/2}\tilde{M}_{n+1}.\EEQ
For $n\ge n_2$ one has by definition $I_{\infty}2^{-nd/2}\le 1$, so
\BEQ \tilde{M}_{n+1}\lesssim (1+O(\lambda))(2^{-n/2}I_{\infty}+\lambda\tilde{M}_n),\EEQ
while $\tilde{M}_{n_2}\lesssim \tilde{I}_{\infty}$ by (ii),
implying by a straightforward induction 
\BEQ \tilde{M}_n\lesssim 2^{-n/2} I_{\infty}+\tilde{I}_{\infty} \lesssim  2^{-n/2}(I_{\infty}+ I_{\infty}^{1/d}\tilde{I}_{\infty}) \EEQ
and finally
\BEQ \sup_{t\ge I_{\infty}^{2/d}} \sqrt{t} ||\nabla h_t||_1\lesssim I_{\infty}+ I_{\infty}^{1/d} \tilde{I}_{\infty}.\EEQ
Finally,
\BEQ \sup_{t\le I_{\infty}^{2/d}} \sqrt{t} ||\nabla h_t||_1\lesssim I_{\infty}^{1/d} \tilde{I}_{\infty}.\EEQ
Hence the result.\hfill\eop
\end{itemize}

\smallskip

{\bf Remark.} If $V$ does not satisfy (\ref{eq:even-stronger-assumption}), then the beginning of the proof is modified as follows:
substituting to (ii) the bound $||\nabla h_t||_{\infty}\lesssim ||h_0||_{\infty} t^{-\half}$, see Proposition
\ref{prop:decay-gradient} (i), leads to 
$n_0$ defined such as to satisfy $||\nabla h_0||_{\infty}\approx ||h_0||_{\infty} (T_{n_0}^*)^{-\half}$, namely,
$n_0\approx (\lambda ||h_0||_{\infty})^2$, and (compare with (\ref{eq:2.32})) $M_{n_0}\lesssim ||\nabla h_0||_1
e^{C(\lambda ||h_0||_{\infty})^2}$. A bound comparable to (\ref{eq:decay-nabla-L1}) probably holds with
the quadratic exponential $e^{C(\lambda ||h_0||_{\infty})^2}$ substituting $e^{C\lambda ||h_0||_{\infty}}$, which
is clearly not optimal for the quadratic KPZ equation (see next paragraph). 

%%%%%%%%%%%%

\medskip
Our second result is valid under our general assumptions on $V$ stated in section 1.

\begin{Theorem}[bounds on higher derivatives] \label{th:bound-higher-derivatives}

Let $h$ be the solution of eq. (\ref{eq:KPZdet}) with initial condition $h_0\in {\cal W}^{3,\infty}$. Then:

\BEQ ||\nabla^2 h_t||_{\infty}\lesssim  P_1(||h_0||_{\infty}, ||\nabla h_0||_{\infty}, ||\nabla^2 h_0||_{\infty})
\frac{\ln (1+t)}{t}   \label{eq:nabla2}\EEQ
\BEQ ||\nabla^3 h_t||_{\infty}\lesssim P_2(||h_0||_{\infty}, ||\nabla h_0||_{\infty}, ||\nabla^2 h_0||_{\infty},
||\nabla^3 h_0||_{\infty}) \left(\frac{\ln^2 (1+t)}{t^{3/2}}\right) \label{eq:nabla3} \EEQ
where $P_1,P_2$ are polynomials.
\end{Theorem}

{\bf Proof.} 

We already know that $\sup_{[0,T_1^*]} ||\nabla^k h_t||_{\infty}\le 2||\nabla^k h_0||_{\infty}$, $k=2,3$ 
for $T_1^*\approx (\lambda||\nabla h_0||_{\infty})^{-2}.$
For $t\ge T_1^*$, $\nabla^2 h_t$ is the solution of an integral equation,
\BEQ \nabla^2 h_t=e^{t\nu\Del}\nabla^2 h_0+\lambda \left[ \int_{(1-\eps)t}^t ds (\nabla e^{(t-s)\nu\Del}) \nabla
(V(\nabla h_s)) + \int_0^{(1-\eps) t} ds (\nabla^2 e^{(t-s)\nu\Del}) V(\nabla h_s) \right].\EEQ
The idea is to commute  the gradient with the  heat operator $e^{(t-s)\nu\Del}$ in order to make the most
of  parabolic estimates; the implied
decay may be put to good use only for $t-s$ large enough, and we shall choose the parameter $\eps$ accordingly.
First, $||e^{t\nu\Del}\nabla^2 h_0||_{\infty}=||\nabla^2 e^{t\nu\Del}h_0||_{\infty}\lesssim ||h_0||_{\infty} t^{-1}$. Then, using $\nabla (V(\nabla h_s))=V'(\nabla h_s)\cdot \nabla^2 h_s$ and Proposition \ref{prop:decay-gradient} (i),
together with the inequality $V'(y)\lesssim y$, consequence of Assumption (\ref{assumptions}) (3), (4) (namely,
$(2y)^2\ge V(2y)\ge V(y)+yV'(y)$)
\BEQ   \lambda\left|\int_{(1-\eps)t}^t ds (\nabla e^{(t-s)\nu\Del}) \nabla
(V(\nabla h_s)) \right|  
 \lesssim \lambda \int_{(1-\eps)t}^t \frac{ds}{\sqrt{t-s}} ||\nabla h_s||_{\infty} \sup_{[(1-\eps)t,t]} ||\nabla^2
h_s||_{\infty} 
\lesssim\lambda  ||h_0||_{\infty} \sqrt{\eps} \sup_{[(1-\eps)t,t]} ||\nabla^2 h_s||_{\infty} \nonumber\\
\EEQ

provided $(1-\eps)t\gtrsim t$. To get a useful inequality we choose $\eps$ so that 
$\lambda||h_0||_{\infty}\sqrt{\eps}\le \frac{1}{4}$, namely, $\eps\approx \min(\frac{1}{4},\frac{1}{(\lambda^
||h_0||_{\infty})^2}) $. Finally (if $t\ge 1$) we split the second integral, $\int_0^{(1-\eps)t}$, into several pieces:

\BEQ \lambda \int_{t/2}^{(1-\eps)t} ds (\nabla^2 e^{(t-s)\nu\Del}) V(\nabla h_s) \lesssim
\lambda \int_{t/2}^{(1-\eps)t} \frac{ds}{t-s} \left( \frac{||h_0||_{\infty}}{\sqrt{t}} \right)^2 \lesssim \lambda
\ln(\eps^{-1}) ||h_0||_{\infty}^2 t^{-1};\EEQ

\BEQ \lambda \int_{1}^{t/2} ds (\nabla^2 e^{(t-s)\nu\Del}) V(\nabla h_s) \lesssim
\lambda t^{-1} \int_1^{t/2} \frac{ds}{s} ||h_0||_{\infty}^2 \lesssim \lambda\frac{\ln t}{t} ||h_0||^2_{\infty};
\label{eq:1.52}\EEQ

\BEQ \lambda \int_0^1 ds (\nabla^2 e^{(t-s)\nu\Del}) V(\nabla h_s) \lesssim
\lambda t^{-1}  \int_0^1 ds ||\nabla h_0||^2_{\infty}= \lambda ||\nabla h_0||^2_{\infty} t^{-1}. \label{eq:1.53}\EEQ

If $t<1$, we merge (\ref{eq:1.52},\ref{eq:1.53}) into
\BEQ \lambda \int_0^{t/2} ds (\nabla^2 e^{(t-s)\nu\Del}) V(\nabla h_s) \lesssim \lambda t^{-1}
\int_0^{t/2} ds ||\nabla h_0||_{\infty}^2=\frac{\lambda}{2} ||\nabla h_0||_{\infty}^2.\EEQ


We finish as in the proof of Theorem \ref{th:decay-nabla-L1} (iii), namely,  letting 
$M_0:=\sup_{[0,T_1^*]} ||\nabla^2 h_t||_{\infty}$ and $M_n:=\sup_{[2^{n-1} T_1^*,2^{n}T_1^*]} ||\nabla^2 h_t||_{\infty}$ ($n\ge 1$),
one has $M_0\le 2||\nabla^2 h_0||_{\infty}$ and 
\BEQ M_{n+1}\le C^{-1} M_n+Q_1(||h_0||_{\infty},||\nabla h_0||_{\infty})2^{-n} +\lambda ||\nabla h_0||_{\infty}^2 \EEQ
for $t<1$,
\BEQ M_{n+1}\le C^{-1} M_n+Q_2(||h_0||_{\infty}, ||\nabla h_0||_{\infty})2^{-n}+Q_3(||h_0||_{\infty},||\nabla
h_0||_{\infty})n2^{-n} \EEQ
for $t\ge 1$. 
 Hence (\ref{eq:nabla2}).

\medskip

This result is used as input to get similar a bound for $||\nabla^3 h_t||_{\infty}$. This time we must move around
three gradients in the best way; this gives three integrals rewritten as 
$$ \int_{(1-\eps)t}^{t} ds
(\nabla e^{(t-s)\nu\Del})\nabla^2 (V(\nabla h_s)), \quad  \int_{t/2}^{(1-\eps)t} ds \nabla^2 e^{(t-s)\Del} 
\nabla(V(\nabla h(s))), \quad \int_0^{t/2} ds  \nabla^3 e^{(t-s)\nu\Del} V(\nabla h_s).$$  
One has $\nabla^2 (V(\nabla h_s))=V'(\nabla h_s)\cdot\nabla^3 h_s+V''(\nabla h_s)(\nabla^2 h_s)^2$, yielding the same constraints on $\eps$, plus a supplementary
quadratic term in $\nabla^2 h_s$. The other terms are computed as before. Details are left to the reader.

\hfill\eop





%%%%%%%%%%%%%%%%%%%%

\subsection{An explicit example: decay of a 'bump' for the quadratic KPZ equation}

%%%%%%%%%%%%%%%%%%%%%%%%%


We consider here the time-decay (pointwise and with respect to various norms) of the solution of the
quadratic, homogeneous KPZ equation,
\BEQ \partial_t h=\Del h+|\nabla h|^2 \EEQ
with initial "bump" condition $h_0(x)=A {\bf 1}_{|x|\le L}$, where $A,L>0$. The coefficient $\lambda$ in front
of the nonlinearity has been disposed of by a simple rescaling. Note that, if $A=||h_0||_{\infty}\lesssim 1$ (i.e.
for a small initial condition), then the decay of the solution and of its derivatives in $L^p$-norms, $1\le p\le \infty$
follow the parabolic estimates as for the solutions of the  linear heat equation; thus we may assume that $A\gg 1$.
We want to compare the decays obtained by explicit computation to those obtained in much greater generality
in the previous paragraphs.

\medskip
 Through the exponential transformation, $w=\exp h$,
the equation becomes simply the heat equation,
\BEQ \partial_t w=\Del w,\qquad w_0=e^{h_0} \EEQ
so that
\BEA  h_t(x) & = & \ln\left( t^{-d/2} \int dy e^{-(x-y)^2/2t} w_0(y)\right)  \nonumber\\
&=& \ln\left (1+ (e^A-1)t^{-d/2}\int_{|y|\le L} dy e^{-(x-y)^2/2t} \right) \nonumber\\
&\approx& \ln\left (1+ e^A t^{-d/2}\int_{|y|\le L} dy e^{-(x-y)^2/2t} \right).
\EEA
Though the initial data is not in ${\cal W}^{1,\infty}$, this defines a solution. We are interested in its 
behaviour for $t\gtrsim L^2$, corresponding to the approximate amount of  time necessary for the solution to
smoothen up. Then
\BEQ h_t(x)\approx \ln\left(1+e^A \left(\frac{L^2}{t}\right)^{d/2} e^{-x^2/2t}\right).\EEQ There are two regimes:

\begin{itemize}
\item[(i)] (initial regime) Assume $L^2\lesssim t\lesssim L^2 e^{\frac{2}{d}A}$. Define $x_{max}(t)\in\R_+$ as the
solution of the equation $e^A \left(\frac{L^2}{t}\right)^{d/2} e^{-x^2_{max}(t)/2t}=1$; explicitly,
$x_{max}(t)=\sqrt{2t(A-\frac{d}{2}\log(\frac{t}{L^2}))}$. If $|x|\gtrsim x_{max}(t)$ then $h_t(x)\approx e^A
\left(\frac{L^2}{t}\right)^{d/2} e^{-x^2/2t}\lesssim 1$. On the other hand, if $|x|\lesssim x_{max}(t)$, then
$h_t(x)$ is still large, $h_t(x)\approx A-\frac{d}{2}\log\left(\frac{t}{L^2}\right)-\frac{x^2}{2t}$. In
particular, 
\BEQ ||h_t||_{\infty}\approx A-\frac{d}{2} \log\frac{t}{L^2}\gtrsim 1 \label{eq:htinfty}\EEQ
 and 
\BEQ  ||h_t||_{1}\approx ||h_t||_{\infty}
\cdot \Vol(B(0,x_{max}(t)))+e^A \left(\frac{L^2}{t}\right)^{d/2}\cdot t^{\half} x^{d-2}_{max}(t). \label{eq:htL1} \EEQ
The reader may easily check that dividing  (\ref{eq:htinfty}), resp. (\ref{eq:htL1}) by $t^{q/2}$ gives
bounds for $||\nabla^q h_t||_{\infty}$, resp. $||\nabla^q h_t||_1$ if $q\in\N^*$.

\item[(ii)]  (final regime) Assume $t\gtrsim L^2 e^{\frac{2}{d} A}$. Then $h_t(x)\approx e^A \left(\frac{L^2}{t}
\right)^{d/2} e^{-x^2/2t}\lesssim 1$; in other words, the bump has essentially disappeared. Furthermore,
\BEQ ||h_t||_{\infty}\approx e^A L^d t^{-d/2}\approx  ||h_0||_1 \frac{e^{||h_0||_{\infty}}}{||h_0||_{\infty}} t^{-d/2} \EEQ
and
\BEQ ||h_t||_1\approx e^A L^d\approx  ||h_0||_1 \frac{e^{||h_0||_{\infty}}}{||h_0||_{\infty}}.\EEQ

Again, bounds for $||\nabla^q h_t||_{\infty}$ or $||\nabla^q h_t||_1$ are obtained by dividing by $t^{q/2}$. 

\end{itemize}


The above computations make it clear that the general bounds for $||\nabla h_t||_{\infty}$, see eq. (\ref{eq:d+1/2}), 
$||\nabla h_t||_1$ and $||\nabla^q h_t||_{\infty}$, $q=2,3$ obtained in Theorems \ref{th:decay-nabla-L1} and \ref{th:bound-higher-derivatives}, are  essentially optimal. 


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%�
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\section{Bound for the homogeneous equation: the case of unbounded initial conditions}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


We now want to prove existence of and  bound  solutions of the homogeneous KPZ equation,
\BEQ \partial_t h=\nu\Del h+V(\nabla h) \label{eq:2.1}\EEQ
with {\em unbounded} initial condition $h_0$. We would typically like to consider a random initial condition
which is a smoothened white noise (see Appendix A). This raises various problems. First (1), one would like to identify
a functional space preserved by the linear heat equation, for which generalized parabolic estimates hold. Second (2),
one would like to extend the comparison principle to such a functional space, in such a way as to prove existence
of and bound the solution. Finally (3), one would like to identify the solution as the limit of solutions of 
(\ref{eq:2.1}) associated to a sequence of compactly supported (hence bounded) initial conditions converging to the original initial condition, so as to extend to the limit regularity results and estimates obtained in the previous
section. 

Let us now answer  questions (1), (2), (3).



%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\subsection{The functional spaces ${\cal H}^0_{\alpha}$}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%



For $f\in L^1_{loc}(\R^d)$ and $\alpha\ge 0$, $x\in\R^d$, one may define
\BEQ f^*_{\alpha}(x)=\sup_{\tau> 0} (1+\tau)^{\alpha} e^{\tau\Del}|f|(x)\in[0,+\infty]\EEQ
and in particular
\BEQ f^*(x):=f^*_0(x)=\sup_{\tau>0} e^{\tau\Del}|f|(x)\in[0,+\infty]. \EEQ
Note that $f^*\le f_{\alpha}^*\le f_{\beta}^*$ if $\alpha\le\beta$.
If $f$ is bounded, then $f^*(x)\le ||f||_{\infty}$. On the other hand, the kind of random initial conditions
we are interested in (see Appendix A) are a.s. unbounded, but satisfy a.s. $f^*(x)<\infty$ for every $x$
(see Lemma \ref{lem:LD-main} and discussion thereafter); compare with the standard parabolic estimates, $f^*_{d/2}(x)\lesssim ||f||_1$ for $f\in L^1(\R^d)$. 
Note that, if $\alpha=0$, the {\em pointwise} estimates
\BEQ |e^{t\Del}f(x)|\le f^*(x),\qquad t\ge 0\EEQ
and, better still,
\BEQ (e^{t\Del}f)^*(x)\le f^*(x),\qquad t\ge 0\EEQ 
(generalizing to 
\BEQ (e^{t\Del}f)^*_{\alpha}(x)\le f^*_{\alpha}(x),\qquad t\ge 0)\EEQ
are improvements on the {\em global} estimate $||e^{t\Del} f||_{\infty}\le ||f||_{\infty}$ which is useless
for unbounded functions. If $f^*_{\alpha}(x)<\infty$ for some $\alpha>0$, then 
\BEQ |e^{t\Del}f(x)|\le (1+t)^{-\alpha} f^*_{\alpha}(x)\EEQ
 decays in time. 


\begin{Definition}
Let, for $0\le\alpha\le d/2$,
\BEQ {\cal H}^0_{\alpha}:=\{ f\in L^1_{loc}(\R^d) \ |\ \forall x\in\R^d, f^*_{\alpha}(x)<\infty\}\EEQ
and
\BEQ {\cal H}^0:={\cal H}^0_0=\{ f\in L^1_{loc}(\R^d) \ |\ \forall x\in\R^d, f^*(x)<\infty\}.\EEQ
\end{Definition}

Since $\alpha\le d/2$, ${\cal H}^0_{\alpha}\supset L^1(\R^d)\cap L^{\infty}_{loc}(\R^d)$ (actually, it is easy to prove that ${\cal H}^0_{\alpha}=
\{0\}$ for $\alpha>d/2$). 

\medskip


Another closely related definition is by averaging: if $f\in L^1_{loc}(\R^d)$, and $r\ge 0$, $x\in\R^d$, one may define
\BEQ f^{\sharp}_{\alpha}(x)=\sup_{\rho> 0} (1+\rho^2)^{\alpha} 
\frac{\int_{B(x,\rho)} |f(y)|dy}{\Vol(B(x,\rho))}\EEQ 
(called  {\em maximal function} in real analysis for $\alpha=0$, see the classical book by  Stein \cite{Ste})
where $B(x,\rho)=\{y\in\R^d; |y-x|<\rho\}$ is the Euclidean ball and $\Vol(B(x,\rho))$ its volume.  Here also,
$f^{\#}_0\le f^{\#}_{\alpha}\le f^{\#}_{\beta}$ if $0\le\alpha\le\beta$. It is convenient
to denote averages by barred integrals, so that, by definition, $\fint_{\Omega} f=\frac{\int_{\Omega} f}{\Vol(\Omega)}$.
A simple result is the following:

\begin{Lemma} \label{eq:f*ftilde*}
There exists constants $c,C>0$ such that, for every  $f\in C(\R^d)$, $cf^*_{\alpha}(x)\le f^{\sharp}_{\alpha}(x)\le Cf^*_{\alpha}(x).$
\end{Lemma}

{\bf Proof.} We must prove two inequalities. First,
\BEA && \int dy \frac{e^{-(x-y)^2/2t}}{(2\pi t)^{d/2}} |f(y)| = \int dr \frac{e^{-r^2/2t}}{(2\pi t)^{d/2}}
\int_{\partial B(x,r)} dy |f(y)|  \nonumber\\
&& =\int dr \frac{r}{t} \frac{e^{-r^2/2t}}{(2\pi t)^{d/2}} \int_{B(x,r)} dy |f(y)| \nonumber\\
&&\le f^{\sharp}_0(x) \int dr \frac{r}{t} \frac{e^{-r^2/2t}}{(2\pi t)^{d/2}} \int_{B(x,r)} dy \nonumber\\
&&=f^{\sharp}_0(x) \int dy \frac{e^{-|y|^2/2t}}{(2\pi t)^{d/2}}=f^{\sharp}_0(x). \label{eq:H0H0}\EEA
Thus $f^*_0\le f^{\sharp}_0$. If $\alpha>0$ and $t\gtrsim 1$ then
\BEQ (1+t)^{\alpha} \int dy \frac{e^{-(x-y)^2/2t}}{(2\pi t)^{d/2}} |f(y)| \lesssim
 f^{\sharp}_{\alpha} (1+t)^{-1-\frac{d}{2}+\alpha} \int dr (1+r)^{1+d-2\alpha} e^{-r^2/2t} \lesssim f^{\sharp}_{\alpha}(x),\EEQ
 so $f^*_{\alpha}\lesssim f^{\#}_{\alpha}$.
 
Conversely, 
\BEQ e^{r^2\Del}|f|(x)=\int \frac{e^{-\half (|x-y|/r)^2}}{(2\pi)^{d/2}r^d} |f(y)|dy\ge C\fint_{B(x,r)} |f|.\EEQ
\hfill\eop

In particular, an equivalent definition for ${\cal H}^0_{\alpha}$ is:
\BEQ {\cal H}^0_{\alpha}=\{f\in L^{\infty}_{loc}(\R^d)\ |\ \forall x\in\R^d, f^{\#}_{\alpha}(x)<\infty\}.\EEQ

Note also that, if $f$ is lower semicontinuous (in particular, if $f$ is continuous),  $f^{\sharp}_{\alpha}(x)\ge \lim_{r\to 0} \fint_{B(x,r)} |f|\ge
|f(x)|$, and similarly
$f^*_{\alpha}(x)\ge |f(x)|$.
\medskip

\medskip




\begin{Lemma}
\begin{enumerate}
\item
Let $f\in {\cal H}^0_{\alpha}\cap C(\R^d)$. Then $f^{\sharp}_{\alpha}$ and $f^*_{\alpha}$ are continuous.
\item  Let furthermore 
\BEQ \beta:=\sup \{\gamma\in[0,1]\ |\ (x,y)\mapsto \frac{|f(x)-f(y)|}{|x-y|^{\gamma}}\ {\mathrm{in}} \ 
L^{\infty}_{loc} \}\in[0,1] \EEQ
be the maximum local H\"older exponent of $f$, and assume $\beta>0$.  Then $f^{\sharp}$ and $f^*$ are 
H\"older continuous, with H\"older exponent $\frac{\beta}{1+\beta}\in[0,\half]$.
\end{enumerate}
\end{Lemma}

In particular, $f^{\sharp}_{\alpha},f^*_{\alpha}$ are $\left(\half\right)$-H\"older continuous if $f\in C^1$.

\medskip



{\bf Proof.} For the sake of the proof we choose a bounded function $\phi:B(0,1)\to \R_+$ such that $\phi(u)=\phi(|u|)$ is
strictly increasing ,
$\phi(0)=0$, $\frac{\phi(u)}{u}>2$ and  $\frac{\phi(u)}{u}\to_{u\to 0}\infty$; we assume furthermore that $\phi(u)=
o_{u\to 0}(u^{1/3})$, so that the function $\chi(u)=\chi(|u|)=\phi(u)\sqrt{\frac{\phi(u)}{u}}$
satisfies the same properties but $\frac{\chi(u)}{\phi(u)}\to_{u\to 0}\infty$. The core of the proof
is a bound on the modulus of continuity of $f^*:=f^*_{\alpha}$,
$f^{\sharp}:=f^{\sharp}_{\alpha}$ given in terms of these two functions. We assume in the following lines that 
$|y-x|\le 1$.

\begin{itemize}
\item[(i)] Let us first prove that $f^{\sharp}$ is locally bounded (from which it follows by Lemma 
\ref{eq:f*ftilde*} that $f^*$ is also). Since $|y-x|\le 1$, then
\BEQ \fint_{B(y,r)} |f|\le \sup_{B(x,2)} |f|,\qquad r\le 1\EEQ
and
\BEQ \fint_{B(y,r)} |f|\le \left(\frac{r+1}{r}\right)^d \fint_{B(x,r+1)} |f|\le 2^d f^{\sharp}(x),\qquad r>1.\EEQ
So 
\BEQ \sup_{B(x,1)}f^{\sharp}\le 2^{\alpha} \max(\sup_{B(x,2)} |f|, 2^d f^{\sharp}(x)). \label{eq:fsharp-bounded} \EEQ

\item[(ii)] We now obtain a modulus of continuity for  $f^{\sharp}$. Fix $x\in\R^d$ and let $y$ vary in $B(x,1)$.
Consider first $r\le \phi(x-y)$. Then, letting $\tau_{x-y}f(z)=f(z-(x-y))$,
\BEA \left|\fint_{B(x,r)} |f|-\fint_{B(y,r)}|f|\right| &\le& \fint_{B(x,r)} |f-\tau_{x-y}f| \nonumber\\
&\le& \sup_{x'\in B(x,\phi(x-y)),y'\in B(y,\phi(x-y))} |f(x')-f(y')| \nonumber\\
&\le& osc_{B(x,2\phi(x-y))} (f). \EEA Since $f$ is continuous in an neighbourhood of $x$ this quantity goes to zero
when $y\to x$.

Consider now $r>\phi(x-y)$. Letting $r'=r+|x-y|$ so that $B(x,r')\supset B(y,r)$,
\BEQ \fint_{B(x,r')}|f|\ge \frac{\Vol(B(y,r))}{\Vol(B(x,r'))} \fint_{B(y,r)} |f|\EEQ
hence
\BEQ (1+r'^2)^{\alpha} \fint_{B(x,r')} |f|- (1+r^2)^{\alpha} \fint_{B(y,r)}|f|\ge \left[ \left(\frac{r}{r+|x-y|}\right)^d-1\right]f^{\sharp}(y)
\gtrsim -\left(\frac{|x-y|}{\phi(x-y)}\right)f^{\sharp}(y).\EEQ
Similarly, with $r''=r-|x-y|$ (note that $r''>|x-y|$ by hypothesis),
\BEQ (1+r^2)^{\alpha} \fint_{B(y,r)}|f|-(1+(r'')^2)^{\alpha} \fint_{B(x,r'')}|f|\gtrsim -\left(\frac{|x-y|}{\phi(x-y)}\right)f^{\sharp}(x).\EEQ
Thus, with $M=\sup_{B(x,1)}f^{\sharp}$, $M<\infty$ by (i),
\BEA && \sup_r (1+r^2)^{\alpha} \fint_{B(y,r)}|f|-\sup_r (1+r^2)^{\alpha} \fint_{B(x,r)}|f|\le \max\left( \sup_{r\le\phi(x-y)} (1+r^2)^{\alpha}
\left\{  \fint_{B(y,r)}|f|- \fint_{B(x,r)}|f|\right\},  \right.\nonumber\\
&& \qquad\qquad\qquad \left. \sup_{r>\phi(x-y)} \left\{ (1+r^2)^{\alpha} \fint_{B(y,r)}|f|-
(1+r'^2)^{\alpha} \fint_{B(x,r')}|f|
\right\} \right) \nonumber\\
&& \qquad \qquad \qquad \lesssim \max\left( \osc_{B(x,\phi(x-y))\cup B(y,\phi(x-y))}(f),M\frac{|x-y|}{\phi(x-y)}\right).\EEA
Exchanging $x$ and $y$ gives the same inequality. Hence we have shown that $f^{\sharp}$ is continuous, and obtained
more precisely that, for every function $\phi$ satisfying the above hypotheses,
\BEQ \osc_{B(x,u)}f^{\sharp} \lesssim \max(\osc_{B(x,2\phi(u))}(f),M\frac{u}{\phi(u)}),\qquad u\in(0,1). \label{eq:modulus}\EEQ
In particular, choosing $\phi(u)=2u^{1/(1+\beta)}$ if $\beta>0$ yields $\osc_{B(x,u)}f^{\sharp}\lesssim u^{\beta/
(1+\beta)}$, so $f^{\sharp}$ is $\frac{\beta}{1+\beta}$-H\"older continuous.



\item[(iii)] Let us finally obtain a modulus of continuity for  $f^*$. The proof is a slightly different from (ii) because
the  support of the heat kernel is the whole space; hence we must deal with the  queue of the exponential 
$e^{-u^2/t}$ for $u\gg \sqrt{t}$. Assume first $\sqrt{t}\le\phi(x-y)$. Then
\BEQ \left| e^{t\Del}|f|(x)-e^{t\Del}|f|(y)\right|\le I_1+I_2+I_3,\EEQ
where
\BEQ I_1=\int_{B(x,\chi(x-y))} \frac{e^{-|x-z|^2/2t}}{(2\pi t)^{d/2}} |f(z)-\tau_{x-y}f(z)| dz \le osc_{B(x,\chi(x-y))
\cup B(y,\chi(x-y))} (f)\EEQ
and
\BEA && I_2=\int_{\R^d\setminus B(x,\chi(x-y))}  \frac{e^{-|x-z|^2/2t}}{(2\pi t)^{d/2}} |f(z)|dz =
\int_{|u|>\chi(x-y)} \frac{e^{-|u|^2/2t}}{(2\pi t)^{d/2}} |f(x+u)|du \nonumber\\
&&\qquad \le 2^d \int \frac{e^{-|u|^2/8t}}{(8\pi t)^{d/2}} |f(x+u)|du \ \cdot\ e^{-3\chi^2(x-y)/8t}\nonumber\\
&&\qquad \lesssim  e^{-\frac{3}{8}\frac{\phi(x-y)}{|x-y|}} f^*(x),\EEA
while $I_3=\int_{\R^d\setminus B(y,\chi(x-y))}  \frac{e^{-|y-z|^2/2t}}{(2\pi t)^{d/2}} |f(z)|dz$ is similar to $I_2$.
The exponential factor in front of $f^*$ decreases to $0$ when $y\to x$. 



Assume now $\sqrt{t}>\phi(x-y)$. Then
$|x-z|^2\le |x-y|^2+|y-z|^2+2|x-y||y-z|\le (1+\eps)|y-z|^2+(1+\eps^{-1})|x-y|^2$ for every $z\in\R^d$ and $\eps>0$.
Choose $\eps=\frac{|x-y|}{\phi(x-y)}<\half$ so that $\frac{(1+\eps^{-1})|x-y|^2}{t}\lesssim\eps$. Letting 
$t'=t(1+\eps)$, one obtains
\BEA && (1+t')^{\alpha} \int \frac{e^{-|x-z|^2/2t'}}{(2\pi t')^{d/2}} |f(z)|dz - 
(1+t)^{\alpha} \int \frac{e^{-|y-z|^2/2t}}{(2\pi t)^{d/2}} |f(z)|dz \nonumber\\
&&\gtrsim -\eps (1+t)^{\alpha} \int \frac{e^{-|y-z|^2/2t}}{(2\pi t)^{d/2}} |f(z)|dz\ge -\eps f^*_{\alpha}(y).\EEA
Exchanging $x$ and $y$ gives a similar inequality, and one concludes to (\ref{eq:modulus}) as in (ii) by noting
that $e^{-\frac{3}{8}
\frac{\phi(u)}{u}}\lesssim \frac{u}{\phi(u)}$ for
$u<1$.
\end{itemize}

\hfill \eop




\medskip

A result in the same direction is

\begin{Lemma} \label{lem:tto0}
Let $f\in {\cal H}^0_{\alpha}\cap C(\R^d)$. Then, for every $t>0$ and $\alpha>0$,
\BEQ |e^{t\Del}f(x)-f(x)|\lesssim \osc_{B(x,t^{\alpha})} f+e^{-\frac{1}{4} t^{2\alpha-1}} f^*_0(x). \label{eq:tto0}\EEQ
Consequently, $e^{t\Del}f\to_{t\to 0}f$ uniformly on every compact.
\end{Lemma}

{\bf Proof.} (\ref{eq:tto0}) follows directly from the inequality
\BEQ |e^{t\Del}f(x)-f(x)|\le \int_{y\in B(x,\eps)} \frac{e^{-|x-y|^2/2t}}{(2\pi t)^{d/2}} |f(y)-f(x)| dy +
\int_{y\in B(x,\eps)^c} e^{-\eps^2/4t} \frac{e^{-|x-y|^2/4t}}{(2\pi t)^{d/2}} (|f(y)|+|f(x)|) dy.\EEQ
Taking $\eps=t^{\alpha}$ with $\alpha<\half$, using the local boundedness of $f^*_0$  (proved in the previous lemma) and letting $t\to 0$ yields the uniform
convergence on a compact set. \hfill\eop

\medskip


Finally, we shall later on need to approximate functions in ${\cal H}^0_{\alpha}$ by functions with compact support, and
use the following lemma:

\begin{Lemma} \label{lem:bump-alpha}
Let $\chi:\R^d\to\R_+$ be a smooth 'bump' scale 1 function, i.e. $\chi|_{B(0,1)}=1$, $\chi|_{\R^d\setminus B(0,2)}=0$.
Denote by $\chi_n(x)=\chi(\frac{x}{n})$ its dilatations for $n\in\N^*$. Then, if $f\in{\cal H}^0_{\alpha}$,
the functions $f_n:=f\cdot\chi_n$, $n\ge 1$ also belong to ${\cal H}^0_{\alpha}$,
 and $(f_n)^*_{\alpha}\to f^*_{\alpha}$, $(f_n)^{\sharp}_{\alpha}\to
f^{\sharp}_{\alpha}$ uniformly on every compact.
\end{Lemma}

Uniform convergence implies in particular that $f^*_{\alpha},f^{\#}_{\alpha}$ are continuous, but does not give any bound on
the modulus of continuity.

\medskip


{\bf Proof.} Let $K\subset\R^d$ compact containing $0$. We skip  $\alpha$ lower indices as in the previous lemma and
  prove that $(f_n)^*\to f^*$ uniformly on $K$. Let $B(0,r)$ a ball containing $K$,
and assume $n\gg r$. Then $\frac{|y-x|^2}{2t}\ge \frac{|y|^2}{4t}$ for all $t,x,y$ with $t>0$, $x\in K$, $|y|>n$. 
Hence
\BEQ 0\le e^{t\Del}|f|(x)-e^{t\Del}|f_n|(x)\le 2^{d/2} \int_{|y|>n} \frac{e^{-|y|^2/4t}}{(2\pi (2t))^{d/2}} |f(y)|dy
=2^{d/2} e^{2t\Del}(|f|-|f_n|)(0),\EEQ
from which  uniform convergence follows provided simple convergence holds at $0$. But
\BEQ f^*(0)=\sup_t (1+t)^{\alpha} e^{t\Del} (\sup_n |f_n|)(0)=\sup_{t,n} (1+t)^{\alpha} (e^{t\Del}|f_n|)(0)=\lim_n (f_n)_{\alpha}^*(0)\EEQ by monotone convergence.

The proof for $(f_n)^{\sharp}$ is similar: let us just state that
\BEQ \fint_{B(x,R)} (|f|-|f_n|)=0 \EEQ if  $B(x,R)\subset B(0,n)$, and
\BEQ \fint_{B(x,R)} (|f|-|f_n|)\le \left(\frac{R+r}{R}\right)^d \fint_{B(0,R+r)} (|f|-|f_n|)\EEQ
otherwise. Details are left to the reader.
\hfill\eop


\bigskip
We may now finally write down our {\em pointwise parabolic estimates}: 
\begin{Lemma}[pointwise parabolic estimates]  \label{lem:pointwise-parabolic-estimates}
Let $f\in{\cal H}^0_{\alpha}$.
For every $k\ge 0$,
\BEQ |\nabla^k e^{t\Del}f(x)|\lesssim t^{-\alpha-k/2} f^*_{\alpha}(x)\EEQ
and
\BEQ (\nabla^k e^{t\Del}f)^*_{\alpha}(x)\lesssim t^{-k/2}f^*_{\alpha}(x).\EEQ
\end{Lemma}

{\bf Proof.} 

By differentiating $k$ times the computations leading to (\ref{eq:H0H0}), one gets 
\BEA |\nabla^k e^{t\Del}f(x)| &\lesssim & \int dr \left(\frac{r}{t}\right)^{k+1} \frac{e^{-r^2/2t}}{(2\pi t)^{d/2}}
\int_{B(x,r)} dy |f(y)| \nonumber\\
&\lesssim & f^{\sharp}_{\alpha}(x) \int dr \left(\frac{r}{t}\right)^{k+1} \frac{e^{-r^2/2t}}{(2\pi t)^{d/2}}
(1+r^2)^{-\alpha} \Vol(B(x,r)) \nonumber\\
&\lesssim& t^{-\alpha-k/2} f^*_{\alpha}(x) \int dr \frac{r}{t} \frac{e^{-r^2/2t}}{(2\pi t)^{d/2}}
\Vol(B(x,r))=t^{-\alpha-k/2} f^*_{\alpha}(x) .\EEA
In particular, if $\alpha=0$, one gets 
\BEQ (\nabla^k e^{t\Del} f)^*_{\alpha}(x)\lesssim t^{-k/2} (f^*)^*_{\alpha}(x)
=t^{-k/2}\sup_{\tau,\sigma>0} (1+\tau)^{\alpha} e^{(\tau+\sigma)\Del}|f|(x)\le t^{-k/2}  f_{\alpha}^*(x).\EEQ
 \hfill\eop


\medskip
In the sequel we restrict for simplicity to the case $\alpha=0$. All results below are easily adapted to the
case $\alpha>0$ or to similar functional spaces with  pointwise bounds  of the form $||f||(x)=\sup_{\tau>0} 
F(\tau,e^{\tau \Del}|f|(x))$.  

%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%

\subsection{The comparison principle}


We now want to use as initial condition of (\ref{eq:2.1}) functions $h_0$ such that $h_0\in {\cal H}^{\lambda}\cap
C(\R^d)$, where
\BEQ {\cal H}^{\lambda}:=\{h_0\in L^1_{loc}(\R^d) \ |\ e^{\lambda |h_0|}\in{\cal H}^0\}. \EEQ
The  comparison to the linear heat equation (see subsection 2.1) actually suggests to consider initial
conditions in the unpleasant-looking space,
\BEQ \widetilde{\cal H}^{\lambda}:=\{h_0\in L^1_{loc}(\R^d) \ |\ e^{\lambda h_0^+},h_0^-\in{\cal H}^0\} \EEQ
However, by Jensen's inequality, $e^{\lambda (h_0^-)^*(x)}\le (e^{\lambda h_0^-})^*(x)$, so ${\cal H}^{\lambda}
\subset \widetilde{\cal H}^{\lambda}$.  
 Note that the definition is compatible with that of ${\cal H}^0$
in the previous paragraph, in the sense that $\frac{1}{\lambda}(e^{\lambda |h_0(x)|}-1)\to |h_0(x)|$ when
$\lambda\to 0$.  Also, by Jensen's inequality,  ${\cal H}^{\lambda}$ is  for $\lambda>0$ a convex subset (but not
a vector subspace) of ${\cal H}^0$, and 
\BEQ |||h_0|||_{{\cal H}^{\lambda}}(x):=\frac{1}{\lambda}\sup_{\tau>0} 
\ln\left( (e^{\tau\Del} e^{\lambda |h_0|})(x) \right)=\frac{1}{\lambda} \ln\left( (e^{\lambda|h_0|})^*(x)\right) \EEQ
defines a {\em family of pointwise "quasi-norms"}, in the sense that
\BEQ |||f|||_{{\cal H}^{\lambda}}(x)\le |||f|||_{{\cal H}^{\lambda'}}(x) \qquad (\lambda\le\lambda');\EEQ
\BEQ |||\mu f|||_{{\cal H}^{\lambda}}(x)\le |\mu| \  |||f|||_{{\cal H}^{|\mu|\lambda}} (x)\qquad (\mu\in\R)\EEQ
(the last inequality is actually an equality);
\BEQ |||f_1+f_2|||_{{\cal H}^{\lambda}}(x)\le \frac{1}{p_1}|||p_1 f_1|||_{{\cal H}^{\lambda}}(x)+
\frac{1}{p_2}|||p_2 f_2|||_{{\cal H}^{\lambda}}(x) \qquad (p_1,p_2\ge 1,\ \frac{1}{p_1}+\frac{1}{p_2}=1).\EEQ

  We then expect the solution of (\ref{eq:2.1}) to be "uniformly bounded
in ${\cal H}^{\lambda}$", at least locally in time (thus allowing for further generalizations to 
equations with time-dependent coefficients), and thus  to lie for all $T>0$ in the functional space
\BEQ {\cal H}^{\lambda}([0,T]):=\{h\in L^1_{loc}(\R\times\R^d)\ |\ \forall x\in\R^d, 
\sup_{t\in[0,T]} (e^{\lambda |h_t|})^*(x)<\infty\}.\EEQ




As mentioned previously, the comparison principle in its different forms usually requires as a cornerstone
assumption the boundedness of the solutions. However, various authors have proved ad hoc comparison principles for
PDE's with unbounded coefficients; the solution lies in functional spaces including functions growing at infinity.
The KPZ equation is a very particular class of Hamilton-Jacobi-Bellman equations for which a comparison principle
holds under {\em quadratic growth conditions}, see Ito \cite{Ito}, Da Lio-Ley \cite{DaLioLey,DaLioLey2}.  Lemma 2.1 and Theorem
2.1 in \cite{DaLioLey} state the following in our case:

\begin{Proposition}  \cite{DaLioLey}

Let $\underline{U}\in USC([0,T]\times\R^d)$ (resp. $\bar{U}\in LSC([0,T]\times\R^d)$) be a viscosity sub-solution (resp. super-solution)
of (\ref{eq:2.1}). Assume there exists $C>0$ such that $|\underline{U}(t,x)|,|\bar{U}(t,x)|\le C(1+|x|^2)$ for all $x\in\R^d,t\le T$.
Then $\underline{U}\le \bar{U}$ in $[0,T]\times\R^d$.

\end{Proposition}

A continuous function $h_0$ with quadratic growth at infinity, $|h_0(x)|\lesssim 1+|x|^2$, is
in general not in  ${\cal H}^{\lambda}$ for any $\alpha\ge 0$ or $\lambda\ge 0$. Conversely, a function in ${\cal H}^{\lambda}$, $\alpha,\lambda\ge 0$ may grow arbitrarily fast   in small
domains $\Omega_n$, $n\to\infty$ with $d(0,\Omega_n)\to_{n\to\infty} \infty$ provided the Lebesgue measure of $\Omega_n$
decreases to zero fast enough.  On the other hand, since the supremum of $n$ i.i.d. random variables grows like
$O(\sqrt{\log n})$,  one does expect random initial data $h_0$ to have a.s. quadratic growth at infinity.
Actually, if $\eps>0$, then a.s.  a random initial condition grows more slowly at infinity than $|x|^{\eps}$.
Thus the above comparison principle holds for such data, and the existence of a sub-solution and a super-solution in 
this class of functions entails by Perron's method the existence and unicity of a viscosity solution of (\ref{eq:2.1}).

It seems however much more natural  in our setting to  prove a comparison principle for functions in 
${\cal H}^{\lambda}$ since
the bounds one expects for the solution will depend on the pointwise maximal estimates $(h_0^-)^*$ and 
$(e^{\lambda h_0^+})^*$ (on the contrary, solutions are expected to have a finite explosion time for initial conditions
with quadratic growth, showing that this is in some sense too large a functional space). As it happens, 
we get such a comparison principle, but only
for solutions in spaces ${\cal H}^{\lambda'}$ with parameter  $\lambda'\ge 2\lambda$ (our proof does not hold for $\lambda'=\lambda$). In some sense ${\cal H}^0$ is the largest natural
functional space for globally defined solutions of parabolic PDE's. We conjecture that this extension of the viscosity solution theory to spaces modelled after ${\cal H}$ (like  ${\cal H}^{\lambda}$ in
the present case) is valid and of interest not only for the KPZ equation, but  probably much beyond for
many nonlinear parabolic PDE's.



Let us state our main theorem, following closely the strategy of Da Lio and Ley:

\begin{Theorem}[comparison principle] \label{th:comparison}

Let $\underline{U}\in USC([0,T]\times\R^d)\cap{\cal H}^{2\lambda}([0,T])$ (resp. $\bar{U}\in LSC([0,T]\times\R^d)\cap {\cal H}^{2\lambda}([0,T])$) be a viscosity sub-solution (resp. super-solution)
of (\ref{eq:2.1}) for some $\alpha\ge 0$. 
Then $\underline{U}\le \bar{U}$ in $[0,T]\times\R^d$.

\end{Theorem}

The proof is very similar to  \cite{DaLioLey}, section 2. The essential element is the following lemma.

\begin{Lemma} \label{lem:U-muV}
Let $\underline{U}\in USC([0,T]\times\R^d)\cap{\cal H}^{2\lambda}([0,T])$ be a sub-solution, and $\bar{U}\in
LSC([0,T]\times\R^d)\cap{\cal H}^{2\lambda}([0,T])$ be a super-solution of (\ref{eq:2.1}). Then
$\Psi_{\mu}:=\underline{U}-\mu \bar{U}$, $\mu\in(0,1)$ is  a sub-solution of the quadratic KPZ equation,
\BEQ \partial_t \psi=\Del\psi+\frac{\lambda}{1-\mu}|\nabla \psi|^2.\EEQ
\end{Lemma}

Note that $\left(\underline{U},\bar{U}\in {\cal H}^{2\lambda}([0,T]) \right)\Longrightarrow \Psi_{\mu}
\in{\cal H}^{\lambda}([0,T])$ (hence our choice of parameter, $\lambda'\ge 2\lambda$, see discussion above).

\medskip


{\bf Proof.}

If $U,V\in C^{1,2}$ then the proof is elementary. First,
\BEQ \partial_t \Psi_{\mu}\le \Del \Psi_{\mu}+\lambda(V(\nabla\underline{U})-\mu V(\nabla\bar{U})).\EEQ
Then, since $V$ is convex, 
\BEQ V(a)\le \mu V(b)+(1-\mu)V(\frac{a-\mu b}{1-\mu}),\qquad a,b\in\R^d. \label{eq:V-convex-mu} \EEQ
Finally, applying this inequality to $a=\nabla\underline{U}$, $b=\nabla\bar{U}$, and using Assumption \ref{assumptions}
(4), $V(y)\le y^2$, yields the result.

\medskip

Otherwise the proof is essentially a very particular case of \cite{DaLioLey}, Lemma 2.2. Let us reproduce the main
arguments for the sake of the reader. Let $\psi\in C^2([0,T]\times\R^d)$ and $(\bar{t},\bar{x})$ a strict local maximum
of $\Psi-\psi$; we must prove that $\partial_t \psi(\bar{t},\bar{x})\le \nu\Del\psi(\bar{t},\bar{x})+
\frac{\lambda}{1-\mu}|\nabla\psi(\bar{t},\bar{x})|^2$. This is done by the standard doubling of variables argument, namely, we let
$\Theta(t,x,y):=\psi(t,x)+\frac{|x-y|^2}{\eps^2}$, and $M_{\eps}=(\Psi-\Theta)(t_{\eps},x_{\eps})$ be the
maximum of $\Psi-\Theta$ in a small ball centered at $(\bar{t},\bar{x})$; it is known that $|x_{\eps}-y_{\eps}|=o(\eps)$
and $M_{\eps}\to_{\eps\to 0}\Psi(\bar{t},\bar{x})-\psi(\bar{t},\bar{x})$. By Theorem 8.3 in the User's guide, see
 in \cite{DaLioLey} for the details of computations, one finds, exploiting the hypotheses on $\underline{U}, \bar{U}$,
\BEQ \partial_t \psi(t_{\eps},x_{\eps})+H(t_{\eps},x_{\eps},\nabla\psi(t_{\eps},x_{\eps})+p_{\eps},X)-\mu 
H(t_{\eps},y_{\eps},\frac{p_{\eps}}{\mu},\frac{Y}{\mu})\le 0 \EEQ
where $p_{\eps}=2\frac{x_{\eps}-y_{\eps}}{\eps^2}$, $H(x,t,p,X):=-\lambda V(p)-\nu\Tr(X)$, and $X,Y$ are symmetric $d\times d$ matrices, depending on $\eps$ and on a parameter $\rho>0$, such that
$\Tr(X-Y)\le \Del\psi(t_{\eps},x_{\eps})+O(\rho/\eps^4).$
Hence
\BEQ \partial_t\psi(t_{\eps},x_{\eps})\le \nu\Del\psi(t_{\eps},x_{\eps})+ \lambda\left[
V(\nabla\psi(t_{\eps},x_{\eps})+p_{\eps})-\mu V(\frac{p_{\eps}}{\mu})\right]+O(\rho/\eps^4).\EEQ
Letting $\rho\to 0$ and using (\ref{eq:V-convex-mu}) as above yields
\BEQ \partial_t\psi(t_{\eps},x_{\eps})\le\nu\Del\psi(t_{\eps},x_{\eps})+\frac{\lambda}{1-\mu}|\nabla\psi(t_{\eps},
x_{\eps})|^2.\EEQ
Finally, letting $\eps\to 0$ gives the result. \hfill\eop

\medskip

We shall also need a non-standard comparison lemma for the linear heat equation:

\begin{Lemma} \label{lem:comparison}
Let $\underline{U}\in USC([0,T]\times\R^d)\cap{\cal H}^0([0,T])$ (resp. $\bar{U}\in LSC([0,T]\times\R^d)\cap
 {\cal H}^0([0,T])$) be a viscosity sub-solution (resp. super-solution)
of the linear heat equation for some $\alpha\ge 0$. 
Then $\underline{U}\le \bar{U}$ in $[0,T]\times\R^d$.

\end{Lemma}

\noindent In other words, Theorem \ref{th:comparison} holds for $\lambda=0$.
\smallskip

{\bf Proof.} Since the equation is linear, we may (by replacing $\underline{U}$ with 
$(\underline{U}-\bar{U})-e^{t\Del} (\underline{U}_0-\bar{U}_0)$) assume that $\bar{U}=0$ and $\underline{U}_0=0$.
Now, we have no bound at infinity available for $\underline{U}$, and the classical maximum principle does not hold.
Instead we choose a smooth function $\chi\ge 0$ with $\chi\big|_{(-\infty,0]}\equiv 1$, $\supp(\chi)\subset (-\infty,1]$, define
$\chi_n(x):=\chi(|x|-n)$ 
 and obtain the following inequality for $\underline{U}_n:=\underline{U}\chi_n$,
\BEQ (\partial_t-\Del)\underline{U}_n+\Del\chi_n \underline{U}+2\nabla\chi_n\cdot \nabla\underline{U}\le 0.
\label{eq:Un} \EEQ

Assume that $\underline{U}\in C^{1,2}$ is a classical sub-solution to begin with. Then, since $\underline{U}_n$,
$\Del\chi_n \underline{U}$ and $\nabla\chi_n\cdot \nabla\underline{U}$ are bounded, the classical comparison
principle entails
\BEA \underline{U}_n(t,x) & \lesssim & -\int_0^t ds e^{(t-s)\Del}(\Del\chi_n \underline{U}(s))-2
\int_0^t ds e^{(t-s)\Del}(\nabla\chi_n\cdot \nabla\underline{U}(s)) \nonumber\\
&=& \int_0^t ds e^{(t-s)\Del}(\Del\chi_n \underline{U}(s)) -2\int_0^t ds \nabla e^{(t-s)\Del}\cdot
(\nabla\chi_n \underline{U}(s))\nonumber\\
&\lesssim&
\int_0^t ds e^{(t-s)\Del} (|\tilde{\chi}_n\underline{U}(s)|) + 
\int_0^t ds \left| \nabla e^{(t-s)\Del} (\nabla\chi_n\underline{U}(s)) \right|
, \label{eq:3.41} \EEA


where $\tilde{\chi}_n=\max(|\nabla\chi_n|,|\Del\chi_n|)$. Now $\sum_n \tilde{\chi}_n\lesssim 1$, so (by the
pointwise parabolic estimates)
\BEQ \sum_n \int_0^t ds e^{(t-s)\Del} |\tilde{\chi}_n \underline{U}(s)|(x)\lesssim
\int_0^t ds e^{(t-s)\Del}|\underline{U}(s)|(x)\lesssim T \sup_{s\in[0,T]} (\underline{U}(s))^*(x). \label{eq:UUU}\EEQ
Hence (for $x$ fixed) $\int_0^t ds e^{(t-s)\Del}|\tilde{\chi}_n\underline{U}(s)|\to_{n\to\infty} 0.$
Lemma  \ref{lem:pointwise-parabolic-estimates} yields the same bound as (\ref{eq:UUU}), with $T$ replaced by $\sqrt{T}$,
for the term with the gradient.


\medskip
The above proof does not seem to extend to fonctions in $USC([0,T]\times\R^d)\cap{\cal H}^0([0,T])$ by a
density argument (in particular, if $\chi$ is a smooth, positive 'bump' function, then $\chi\ast \underline{U}$ is a
smooth subsolution in the classical sense if $\underline{U}$ is since $(\partial_t-\Del)(\chi\ast \underline{U})=\chi\ast
(\partial-\Del)\underline{U}\le 0$,  but not in the viscosity sense in general if $\underline{U}$ is only upper-semicontinuous). Instead we use another truncation argument, which could also have been used in the classical case.
We fix $x\in\R^d$ and let $n\to\infty$ as above.
Since $\underline{U}\in{\cal H}^0$, it is locally bounded, so the function $\underline{U}$ is a bounded sub-solution of the heat equation on $[0,T]\times B(0,n)$ with boundary value $\underline{U}\big|_{\partial B(0,n)}$. Thus the
classical maximum principle and Green's formula imply that
\BEQ \underline{U}(t,x)\le \int_0^t ds \int_{\partial B(0,n)} \nabla_{\vec{n}}G_n(t,x;s,y) \underline{U}(s,y),
\qquad t\le T, x\in B(0,n)\EEQ
where $\nabla_{\vec{n}}$ is the normal derivative and  $G_n$ is the Green function of the heat equation on $\R_+\times B(0,n)$. By standard estimates, $|\nabla G_n(t,x;s,y)|\lesssim (t-s)^{-\half} G(t,x;s,y)$ if $y\in \partial B(0,n)$, where
$G(t,x;s,y)$ is the usual heat kernel on $\R\times\R^d$. One has thus obtained an estimate very similar to
(\ref{eq:3.41}), and the end of  the proof is the same.

 \hfill\eop

\bigskip



{\bf Proof of Theorem \ref{th:comparison}.} 


By Lemma \ref{lem:U-muV}, $\partial_t \Psi_{\mu}\le \Del\Psi_{\mu}+\frac{\lambda}{1-\mu}|\nabla\Psi_{\mu}|^2$. Equivalently,
$(\partial_t-\Del)\left(e^{\frac{\lambda}{1-\mu}\Psi_{\mu}}\right)\le 0$. By Lemma \ref{lem:comparison}, 
\BEQ \Psi_{\mu}(t,x)\le \frac{1-\mu}{\lambda} \ln \left[ \int \frac{e^{-|x-y|^2/2t}}{(2\pi t)^{d/2}} e^{\lambda h_0(y)}
dy \right].\EEQ
Letting $\mu\to 1$, one finds $\Psi_{\mu}\le 0$. \hfill\eop

%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%

\subsection{Bounds for the solution}

%%%%%%%%%%%%%%%%%%%%%%%%


 Let $h_0\in{\cal H}^{2\lambda}\cap C(\R^d)$. Then $\underline{h}_t:=-e^{t\nu\Del}h_0^-$ is a sub-solution, and
$\bar{h}_t:=\frac{1}{\lambda}\ln(e^{t\nu\Del} e^{\lambda h_0^+})$  a super-solution of (\ref{eq:2.1}), and
the pointwise parabolic estimates, together with Jensen's inequality, imply that
$\underline{h},\bar{h}\in C([0,T]\times\R^d)\cap {\cal H}^{2\lambda}([0,T])$.
Perron's method (see User's guide \cite{CraIshLio}, Theorem 4.1), in combination with the comparison principle
of the previous paragraph, shows that 
\BEQ h(x):=\sup\{\tilde{h}(x)\ |\ \underline{h}\le \tilde{h}\le\bar{h}\ {\mathrm{and}}\ \tilde{h}\ 
{\mathrm{is\ a\ subsolution\ of}}\ (\ref{eq:KPZdet})\}  \EEQ
is the unique viscosity solution in $C([0,T]\times\R^d)\cap {\cal H}^{2\lambda}([0,T])$ of (\ref{eq:2.1}) for every $T>0$. We simply call $h$ the solution on $[0,T]$ of (\ref{eq:KPZdet}) with initial condition $h_0$.

The analogue of the space ${\cal W}^{1,\infty}$ in our setting is 
\BEQ {\cal W}^{1,\infty;2\lambda}_{loc}:=\left\{h_0\in {\cal H}^{2\lambda}\ |\  \forall x\in\R^d,
\sup_{\vec{\eps}\in B(0,1)\setminus\{0\}} \left(\exp \left\{2\lambda \frac{|h_0(\vec{\eps}+\cdot)-h_0(\cdot)|}{|\vec{\eps}|}
\right\} \right)^*(x)<\infty \right\}.\EEQ
Note that the choice of  $B(0,1)$ in the supremum in the above definition is largely irrelevant (any smaller neighbourhood
of the origin would do, larger ones would just require a rescaling). 
Just like  ${\cal H}^{2\lambda}$ before, ${\cal W}^{1,\infty;2\lambda}_{loc}$ is a convex subset
of $C(\R^d)$, and
\BEQ |||h_0|||_{{\cal W}^{1,\infty;2\lambda}_{loc}}(x):=|||h_0|||_{{\cal H}^{2\lambda}}(x)+ \sup_{\eps\in B(0,1)} ||| \frac{|h_0(\vec{\eps}+\cdot)
-h_0(\cdot)|}{|\vec{\eps}|} |||_{{\cal H}^{2\lambda}}(x) \EEQ
defines a family of pointwise "quasi-norms".

\medskip


A first easy result is:

\begin{Lemma} \label{lem:easy}
Let $h$ be the viscosity solution on [0,T] of (\ref{eq:2.1}) with initial condition $h_0\in{\cal H}^{2\lambda}\cap 
C(\R^d)$. Then,
for every $t\in[0,T]$,
\BEQ (e^{a\lambda |h_t|})^*(x)\le (e^{a\lambda |h_0|})^*(x),\qquad a\ge 1.\label{eq:ut*bound}\EEQ
In particular, 
\BEQ |h_t(x)|\le |||h_0|||_{{\cal H}^{\lambda}}(x).\EEQ
\end{Lemma}

{\bf Proof.} 
By the comparison principle, Theorem (\ref{th:comparison}), $|h|\le u$, where $u$ is the solution of the quadratic
KPZ equation $\partial_t u=\Del u+\lambda|\nabla u|^2$ with initial condition $|h_0|$. Then $e^{\lambda u}$
is a solution of the linear heat equation, hence  (by Jensen's theorem and pointwise parabolic estimates)
\BEQ \left(e^{a\lambda |h_t|} \right)^*(x)\le \left(e^{a\lambda u_t}\right)^*(x)=
\left( (e^{t\Del} e^{\lambda |h_0|})^a\right)^*(x)\le \left(e^{t\Del}e^{a\lambda |h_0|}\right)^*(x)
\le (e^{a\lambda |h_0|})^*(x) \label{eq:3.50} \EEQ   for $a\ge 1$. 
\hfill\eop

\smallskip

Thus $h$ extends to $t\in\R_+$ and satisfies (\ref{eq:ut*bound}) for arbitrary $t$.


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\subsection{Bounds for the gradient}

%%%%%%%%%%%%%%%%%%%%%%%


We now assume $h_0\in{\cal W}^{1,\infty;2\lambda}_{loc}$ or, better still, $h_0\in{\cal W}^{1,\infty;4\lambda}_{loc}$.

\begin{Lemma} \label{lem:classical}
\begin{itemize}
\item[(i)]
Assume  $h_0\in{\cal W}^{1,\infty;2\lambda}_{loc}\cap C(\R^d)$. Then the solution $h$ is classical for $t>0$.
Furthermore,
\BEQ |\nabla h(t,x)|\lesssim |||h_0|||_{{\cal W}_{loc}^{1,\infty;2\lambda}}(x).\EEQ
\item[(ii)]
Assume $h_0\in{\cal W}^{1,\infty;4\lambda}_{loc}\cap C(\R^d)$. Then $h_t\in {\cal W}_{loc}^{1,\infty;\lambda}$ and
\BEQ e^{\lambda|||h_t|||_{{\cal W}^{1,\infty;\lambda}_{loc}}(x)}\le 
\left[(e^{4\lambda |h_0|})^*(x)\right]^{1/2} \left[ (e^{4\lambda |||h_0|||_{{\cal W}^{1,\infty;\lambda}_{loc}}(x)}
\right]^{1/4}.\EEQ
\end{itemize}
\end{Lemma}

{\bf Proof.} 

\begin{itemize}
\item[(i)] Let $\vec{\eps}\in B(0,1)\setminus\{0\}$. By Lemma \ref{lem:U-muV},  $\Psi_{\vec{\eps}}(t,\cdot):=
h(t,\cdot+\vec{\eps})-(1-|\vec{\eps}|) h(t,\cdot)$ is a sub-solution of the KPZ equation $\partial_t \psi=\Del\psi+\frac{\lambda}{|\eps|}
|\nabla\psi|^2$, hence  $t\mapsto (e^{\frac{\lambda}{|\vec{\eps}|} \Psi_{\vec{\eps}}(t,\cdot)})^*(x)$ is decreasing and
\BEQ \Psi_{\vec{\eps}}(t,x)\le \frac{|\vec{\eps}|}{\lambda} \ln\left\{ e^{t\Del}\left(e^{\lambda \left[ |\frac{h_0(\cdot+
\vec{\eps})-h_0(\cdot)|}{|\vec{\eps}|} + |h_0(\cdot)| \right] }\right)\right\}(x) 
\le \frac{|\vec{\eps}|}{2\lambda} \left\{ \ln e^{t\Del}\left(e^{2\lambda \frac{|h_0(\cdot+
\vec{\eps})-h_0(\cdot)|}{|\vec{\eps}|}}\right)(x) + \ln e^{t\Del} (e^{2\lambda |h_0|})(x) \right\}.\EEQ
 Since $h_0\in {\cal W}^{1,\infty;2\lambda}_{loc}$, the
functions $||h_0||_{{\cal H}^{2\lambda}}(x)$ and  $\tilde{h}_0(x):=\left(\exp \left\{2\lambda
\frac{|h_0(\vec{\eps}+\cdot)-h_0(\cdot)|}{|\vec{\eps}|})
\right\} \right)^*(x)$ are locally bounded; exchanging the r\^oles of $x$ and $x+\vec{\eps}$ gives a two-sided
inequality for $\Psi_{\eps}$,  
\BEQ |\Psi_{\eps}(t,x)|\le |\vec{\eps}|\, \sup_{y\in B(x,1)}  ||h_0||_{{\cal H}_0^{2\lambda}}(y)+ \frac{|\vec{\eps}|}{2\lambda} \ln \sup_{y\in B(x,1)} \tilde{h}_0(y). \EEQ
 Thus
\BEQ \frac{|h(t,x+\vec{\eps})-h(t,x)|}{|\vec{\eps}|}\lesssim  \frac{\ln \sup_{y\in B(x,1)} \tilde{h}_0(y)}{2\lambda}+
\sup_{y\in B(x,1)} ||h_0||_{{\cal H}_0^{2\lambda}}(y)+ \sup_{y\in B(x,1)} |h(t,y)|.\EEQ
Letting $\vec{\eps}\to 0$, one shows that $h\in{\cal W}^{1,\infty}_{loc}$. Now, fix a compact $K\subset\R^d$
with smooth boundary. Solving the KPZ equation  
on $\R_+\times K$ with boundary condition $h(t,\cdot)\big|_{\partial K}$ as in \cite{AmoBen} and using the above a priori bound
shows that the solution is classical in $K$ for $t>0$.

\item[(ii)]
We introduce the following notations,
\BEQ \del f(\cdot):=\frac{f(\cdot+\vec{\eps})-f(\cdot)}{|\vec{\eps}|},\qquad 
\tilde{\del}f(\cdot):=\frac{f(\cdot+\vec{\eps})-(1-|\vec{\eps}|)f(\cdot)}{|\vec{\eps}|}.\EEQ
Note that $\tilde{\del}h_t(\cdot)=\frac{1}{|\vec{\eps}|} \Psi_{\eps}(t,\cdot)$ and $\tilde{\del}f=\del f+f$. 
Then we get for every $a\ge 1$
\BEQ \left(e^{\lambda a|\tilde{\del}h_t|}\right)^*(x)=\left(e^{a\frac{\lambda}{|\eps|}|\Psi_{\eps}(t)|}\right)^*(x)
\le \left(e^{a\frac{\lambda}{|\eps|}|\Psi_{\eps}(0)|}\right)^*(x)=\left(e^{\lambda a|\tilde{\del}h_0|}\right)^*(x).\EEQ
Now we use twice H\"older's inequality,
\BEQ (e^{\lambda |\del h_t|})^*(x)\le \left[ (e^{2\lambda |\tilde{\del} h_t|})^*(x) \right]^{1/2}
\left[ (e^{2\lambda |h_t|})^*(x) \right]^{1/2} \le 
\left[ (e^{2\lambda |\tilde{\del} h_0|})^*(x) \right]^{1/2}
\left[ (e^{2\lambda |h_0|})^*(x) \right]^{1/2}
\EEQ
and
\BEQ \left[(e^{2\lambda |\del h_0|})^*(x) \right]^{1/2} \le \left[ (e^{4\lambda |\del h_0|})^*(x) \right]^{1/4}
\left[ (e^{4\lambda |h_0|})^*(x) \right]^{1/4}. \EEQ 
Hence the result.
\end{itemize}
 \hfill \eop
 

\bigskip

We now consider a rate $V=V(y)$ satisfying assumption (\ref{eq:even-stronger-assumption}), i.e. behaving
like $y^2$ for $y$ small or large, and show how to generalize
the conclusions of Proposition \ref{prop:decay-gradient} (iii).


Let $\chi$ be a 'bump' function as in 
Lemma \ref{lem:bump-alpha}, and $\chi^{(n)}(x):=\chi(\frac{x}{n})$.



\begin{Lemma} \label{lem:h(n)}
Let $h^{(n)}$ be the solution of the homogeneous KPZ equation with initial condition $h^{(n)}_0=h_0(x)\chi^{(n)}(x)$.
 Then $h^{(n)}$ converges to $h$ uniformly on every compact.
\end{Lemma}

{\bf Proof.} It follows from Lemma \ref{lem:classical} that $h^{(n)}$ is locally uniformly differentiable. Thus,
by Ascoli's theorem and the classical diagonal extraction procedure, one may construct a subsequence $h^{(n_m)}$
converging locally uniformly. By the  stability principle for continuous viscosity solutions (see e.g.
\cite{Bar}, Theorem 3.1), the limit is a solution of the KPZ equation with initial condition $h_0$. Since the
solution is unique, we have shown that $h^{(n_m)}\to_{m\to\infty} h$ in $C(\R_+\times\R^d)$. Since the
sequence $(h^{(n)})_n$ is pre-compact and all subsequences converge to $h$, the sequence $(h^{(n)})$ itself converges
to $h$. \hfill\eop

\begin{Corollary} Let $V$ satisfy assumption (\ref{eq:even-stronger-assumption}), $yV'(y)-V(y)\ge Cy^2$. Then
\BEQ |\nabla h_t(x)|\lesssim \left(\frac{|||h_0|||_{{\cal H}^{\lambda}}(x)/\lambda}{t}\right)^{1/2}. \EEQ
\end{Corollary}

{\bf Proof.} By (\ref{eq:pointwise-decay-gradient}),
\BEQ |\nabla h^{(n)}_t(x)|\lesssim \left(\frac{|h^{(n)}_t(x)|/\lambda}{t}\right)^{1/2}, \qquad x\in\R^d.\EEQ
By Lemma \ref{lem:easy}, $|h^{(n)}_t(x)|\le |||h_0^{(n)}|||_{{\cal H}^{\lambda}}(x).$ Hence, for every 
$\vec{\eps}\in B(0,1)$, 
$\frac{1}{|\vec{\eps}|} |h_t^{(n)}(x+\vec{\eps})-h_t^{(n}(x)| \lesssim \sup_{B(x,1)}
\left(\frac{|||h_0^{(n)}|||_{{\cal H}^{\lambda}}(\cdot)/\lambda}{t}\right)^{1/2}$. 
 The corollary
follows from letting first $n\to\infty$ and then $\vec{\eps}\to 0$. \hfill \eop 



%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\section{Bounds for the infra-red cut-off inhomogeneous equation}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\subsection{General philosophy of scale decompositions}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%



In this section, we  start our study of the inhomogeneous KPZ equation,
\BEQ \partial_t \psi(t,x)=\nu\Del\psi(t,x)+\lambda V(\nabla\psi(t,x))+g(t,x) \label{eq:full-KPZ} \EEQ
where $g(t,x)$ is a $C^1$ force term. For the time being, we only consider an infra-red cut-off version of this
equation, see (\ref{eq:1.2}) or Definition \ref{def:KPZdowntoj} below. We  only require
here  good scale-dependent averaging properties for $g$ (see precise assumptions below). For the complete study (to be
developed in the further articles) we shall take for  $g$  a regularized white noise, denoted by $\eta$.

\medskip

The general motivation in the subsequent analysis is to get an effective {\em scale separation} mechanism. 
In other words, let $G$ be the Green kernel,
\BEQ G:g\mapsto (Gg)(t):=\int_0^{+\infty} e^{\nu s\Del}g(t-s)ds \EEQ
(called {\em propagator} in the physics
literature). Eq. (\ref{eq:full-KPZ}) is equivalent to the integral equation,
\BEQ \psi=G(\lambda V(\nabla\psi)+g) \label{eq:integral-equation}).\EEQ
Now we want to write $G$ as a sum $G=\sum_{j\ge 0} G^j$ over scales,  and similarly for the force term,
$g=\sum_{j\ge 0}g^j$, in such a way that
\begin{itemize}
\item[(1)] $\psi$ be well approximated by the sum $\sum_j \psi^j$, where $\psi^j$ is 
the solution  of the integral equation
\BEQ \psi^j=G^j(\lambda^j V(\nabla\psi^j)+g^j); \label{eq:4.3} \EEQ and
\item[(2)] $\psi^j$ by the solution $\phi^j$ of the linearized equation, $\phi^j=G^j g^j$, at least for
$\lambda$ small enough or $j$ large enough.
\end{itemize}

The approximations in (1) and (2) are responsible for the renormalization procedure in which $\lambda$ becomes
the scale-dependent parameter $\lambda^j$, and $G^j$, $g^j$ also receive correction terms (see third article
in the series, (3) {\em The multi-scale expansion in the small-field regime} for further explanations).

\medskip
At this point we are not interested in the renormalization procedure and would like in principle to consider a single-scale equation such
as (\ref{eq:4.3}),
\BEQ \psi^j=G^j(\lambda V(\nabla\psi^j)+g^j)\EEQ
or simply (since a priori $G^j g^j\simeq G^j g$)
\BEQ \psi^j=G^j(\lambda V(\nabla\psi^j)+g) \label{eq:one-scale}.\EEQ
The easiest way to select  fluctuations at time, resp. space distances of order $M^j$, resp. $M^{j/2}$ is to set
 \BEQ (G^j f)(t)=\int_{M^{j-1}}^{M^j} ds e^{s\nu\Del}f(t-s)\EEQ
(see Definition \ref{def:non-local-cut-off} for a better definition, in which  a smooth time cut-off is used instead). Coming back to $g=\eta$ to mimic the behaviour of the noisy KPZ
equation, we are led to set $\phi^j=G^j\eta$, $\eta^j=(\partial_t-\Del)\phi^j$. 
Recall
\BEQ d_{\phi}:=\half(\frac{d}{2}-1)\EEQ
is the  {\em scaling dimension} of the solution of the Ornstein-Uhlenbeck or of the KPZ equation, see Introduction.
 It is proved in Appendix A that 
\BEQ \esper[\phi^j(t,x)\phi^{j'}(t',x')]\lesssim M^{-|j-j'|}(M^{-\max(j,j')})^{2d_{\phi}} e^{-cM^{-\max(j,j')}|t-t'|-cM^{-\max(j,j')/2}|x-x'|} \label{eq:Cphi} \EEQ
\BEQ \esper[\eta^j(t,x)\eta^{j'}(t',x')]\lesssim M^{-2|j-j'|}(M^{-\max(j,j')})^{2+2d_{\phi}} e^{-cM^{-\max(j,j')}|t-t'|-cM^{-\max(j,j')/2}|x-x'|}
\label{eq:Ceta}\EEQ
for some constant $c>0$.  Consider first the diagonal covariance ($j=j'$): since $\phi^j$ and $\eta^j$  are 
Gaussian, (\ref{eq:Cphi}), (\ref{eq:Ceta}) essentially mean that the following {\em scalings} hold,
\BEQ \phi^j(t,x)=O(M^{-jd_{\phi}}), \qquad \eta^j(t,x)=O((M^{-j})^{1+d_{\phi}}), \label{eq:scaling}\EEQ
with random prefactors. The bounds in Appendix A also yield an order of magnitude of the gradients, with a
supplementary $M^{-j/2}$ factor,
\BEQ \nabla\phi^j(t,x)=O((M^{-j})^{\half+d_{\phi}}), \qquad \nabla\eta^j(t,x)=O((M^{-j})^{\frac{3}{2}+d_{\phi}})
\label{eq:scaling-gradient}.\EEQ
  For $j\not= j'$ one has
an extra decaying exponential factor in $M^{-|j-j'|}$ which lies at the root of the scale separation mechanism. 

\medskip

Our actual mechanism of separation of scales (see third article in the series) is actually different. The reason
is that the integral equation (\ref{eq:4.3}) is a delay, non-local equation which does not satisfy at all the
maximum principle, and we have no a priori bounds for its solutions. So we  introduce instead in the sequel
 a very simple {\em infra-red cut-off of scale $j$} for the propagator, namely, we replace $\nu\Del$ by $\nu\Del-M^{-j}$.  Denoting by $G^{j\to}$ the Green kernel of the operator
 $\nu\Del-M^{-j}$, one has the explicit formula
\BEQ G^{j\to}(t,x;t',x')={\bf 1}_{t>t'} e^{-M^{-j}(t-t')} \frac{e^{-|x-x'|^2/2\nu(t-t')}}{(2\pi\nu(t-t'))^{d/2}},
\label{eq:Gjfleche} \EEQ
which makes apparent an exponential decay in time and space: since $\inf_{s>0} (\frac{|x-x'|^2}{2\nu s}+sM^{-j})\approx
M^{-j/2} |x-x'|$, 
\BEQ G^{j\to}(t,x;t',x')\lesssim (t-t')^{-d/2} e^{-cM^{-j}(t-t')-cM^{-j/2}|x-x'|}. \label{eq:exp-decay} \EEQ
for some constant $c>0$.  The idea is that $G^{j\to}$ is a good substitute for the sum $\sum_{k\le j} G^k$. We also
replace the force term $g$ by $g^j$ such that $g^j(t,x)=O((M^{-j})^{1+d_{\phi}})$ as for $\eta^j$, see (\ref{eq:scaling}). 
Thus the new equation is the following.

\begin{Definition}[inhomogeneous KPZ equation with scale $j$ infra-red cut-off] \label{def:KPZdowntoj}
The inhomogeneous KPZ equation with scale $j$ infra-red cut-off is
\BEQ \partial_t\psi=(\nu \Del-M^{-j})\psi+\lambda V(\nabla\psi)+g^j. \label{eq:KPZdowntoj}\EEQ 
\end{Definition}

The integral form of this equation is
\BEQ \psi^j=G^{j\to}(V(\nabla\psi)+g^j).\EEQ
Note that the kernel $G^{j\to}$ has no {\em ultra-violet cut-off}, in the sense that it behaves like
the full Green kernel  $G$ for time separations $|t-t'|\ll M^j$. 
 It actually turns out that 
the solution $\psi^j$ of (\ref{eq:KPZdowntoj})
has the correct scaling, $\psi^j(t,x)=O(M^{-jd_{\phi}})$, see (\ref{eq:scaling}), under appropriate assumptions
on $g^j$ that 
we now proceed to write down. 


%%%%%%%%%%%%%%%%%
\subsection{Functional spaces of scale $j$}

%%%%%%%%%%%%%%%%%%%%%%%%%



\begin{Assumption} \label{assumption:etaj}
Let $g^j$ be a continuous function such that
\begin{itemize}
\item[(i)] for all $t\in[0,T]$ and $x\in\R^d$,
\BEQ |||g^j|||_{\lambda,j}([0,t],x):=\frac{1}{\lambda}\sup_{0<\del t\le M^j} M^{-j}\del t \sum_{p=0}^{\lfloor
t/\del t\rfloor} \left(e^{-M^{-j}\del t}\right)^p \ln\left[\left(e^{\lambda M^j \left|\fint^{t-p\del t}_{t-(p+1)\del t} g^j(s) \ ds \right| }
\right)^*(x)\right]<\infty ; \label{eq:etaj1}\EEQ
\item[(ii)] for all $t\in[0,T]$ and $x\in\R^d$,
\BEQ \frac{1}{\lambda}\sup_{0<\del t\le M^j,0<|\del x|\le 1} M^{-j}\del t\sum_{p=0}^{\lfloor t/\del t\rfloor} \left(e^{-M^{-j}\del t}\right)^p \ln\left[\left(e^{\lambda M^{3j/2} \left|\fint^{t-p\del t}_{t-(p+1)\del t} \frac{g^j(s,\cdot+\del x)-g^j(s,\cdot)}{|\del x|} \ ds \right| }
\right)^*(x)\right]<\infty  \label{eq:etaj2}\EEQ
\end{itemize}
where $\fint_{t-(p+1)\del t}^{t-p\del t} (\cdots) ds =\frac{1}{\del t}\int_{t-(p+1)\del t}^{t-p\del t} (\cdots) ds$
is the average over the corresponding subinterval.
\end{Assumption}

The conditions $0\le \del t\le M^j, 0<|\del x|\le 1$ are arbitrary (any smaller neighbourhood of the origin would do).
We denote by ${\cal W}_j^{1,\infty;\lambda}([0,T])\subset L^1_{loc}(\R\times\R^d)$ the convex subset of functions $g^j$ satisfying
(\ref{eq:etaj1}) and (\ref{eq:etaj2}) for all $t\in[0,T]$ and $x\in\R^d$. Note that the quantity in (\ref{eq:etaj2}) is $\sup_{0<|\del x|\le 1} ||| M^{j/2} \frac{g^j(\cdot,\cdot+\del x)-g^j(\cdot,
\cdot)}{|\del x|} |||_{\lambda,j}([0,t],x).$ Summing (\ref{eq:etaj1}) and (\ref{eq:etaj2})  yields a  pointwise "quasi-norm",
\BEQ |||g|||_{{\cal W}^{1,\infty;\lambda}_j([0,t])}(x):=|||g|||_{\lambda,j}([0,t],x)+\sup_{0<|\del x|\le 1} ||| M^{j/2} \frac{g(\cdot,\cdot+\del x)-g(\cdot,
\cdot)}{|\del x|} |||_{\lambda,j}([0,t],x). \EEQ
By Jensen's inequality, $||| M^{j/2} \frac{g(\cdot,\cdot+\del x)-g(\cdot,
\cdot)}{|\del x|} |||_{\lambda,j}([0,t],x)\ge M^{j/2} ||| \frac{g(\cdot,\cdot+\del x)-g(\cdot,
\cdot)}{|\del x|} |||_{\lambda,j}([0,t],x).$

\bigskip\medskip
Finally, we introduce a new, scale-dependent convex subspace ${\cal W}_j^{1,\infty;\lambda}\subset{\cal W}^{1,\infty;\lambda}_{loc}$.

\begin{Definition}[the space ${\cal W}_j^{1,\infty;\lambda}$]
Let
\BEQ {\cal W}_j^{1,\infty;\lambda}:=\{h_0\in {\cal H}^{\lambda};
\forall x\in\R^d, |||h_0||_{{\cal W}^{1,\infty;\lambda}_{j}}(x)<\infty\}, \EEQ
 where
\BEQ |||h_0|||_{{\cal W}^{1,\infty;\lambda}_{j}}(x):=|||h_0|||_{{\cal H}^{\lambda}}(x)+ \sup_{\eps\in B(0,1)} ||| 
M^{j/2} \frac{|h_0(\vec{\eps}+\cdot)
-h_0(\cdot)|}{|\vec{\eps}|} |||_{{\cal H}^{\lambda}}(x). \EEQ
\end{Definition}


\medskip



Let us now see (as promised just before eqs. (\ref{eq:etaj1}) and (\ref{eq:etaj2})) how to get a scaling
of order $O(M^{-jd_{\phi}})$ for $\psi^j(t,x)$. Very roughly, letting $C(s)$ be an order of magnitude of $g^j(s,\cdot)$, and $C$ a global order of magnitude of $g^j(\cdot,\cdot)$, 
 $|||g^j|||_{\lambda,j}([0,t],x)\approx \int_0^t ds e^{-M^{-j}s} C(t-s)\approx M^j C$. If 
 $|||g^j|||_{\lambda,j}([0,T],x)=O(M^{-jd_{\phi}})$, 
  then $C=O((M^{-j})^{1+d_{\phi}})$ as in (\ref{eq:scaling}). To get also the correct scaling 
  (\ref{eq:scaling-gradient}) of order $O((M^{-j})^{\half+
  d_{\phi}})$ for $\nabla\psi^j(t,x)$, we should therefore assume that
  \BEQ |||g^j|||_{{\cal W}^{1,\infty;\lambda}_j([0,T])}(x)=O(M^{-jd_{\phi}}). \label{eq:condition-on-gj} \EEQ
This condition is made precise in Corollary \ref{cor:4.6}.  It is essentially correct up to the replacement
of $\lambda$ by $4\lambda$.
 
 
%%%%%%%%%%%%%%%%%

\subsection{Bounds}

%%%%%%%%%%%%%%%%%%%%%

 
 

In order to take into account the right-hand side, we need the following Trotter-type formula. We use the
following notations in this paragraph. The homogeneous nonlinear semi-group generated by the homogeneous
KPZ equation (\ref{eq:2.1}) is denoted by $\Phi^{\lambda}(t)$, i.e.
$\Phi^{\lambda}(t)h_0$ is the solution at time $t$ of the homogeneous KPZ equation with initial condition
$h_0\in {\cal W}^{1,\infty}$. In the next two lemmas we assume $\psi_0\in {\cal W}^{1,\infty}$, let $g(t,x), \ (t,x)\in [0,T]\times\R^d$ be a function in $C([0,T],C^2(\R^d))$ and define
$\bar{g}_k(s):C(\R^d)\to C(\R^d), f\mapsto \bar{g}_k(s)f$ by $\bar{g}_k(s)f(x):=\int_{kt/n}^{u+kt/n} g(u,x) du+
f(x)$ ($0\le k\le n$). 

\begin{Lemma}[Trotter formula] \label{lem:Trotter}
 
  Let, for $k=0,\ldots,n$,
\BEQ \psi^{(n)}_{kt/n}(x):=\left(e^{-M^{-j}t/n} \Phi^{\lambda}(t/n) \tau_k(t/n) \right)
\left(e^{-M^{-j}t/n} \Phi^{\lambda}(t/n) \tau_{k-1}(t/n)\right)\cdots
\left(e^{-M^{-j}t/n} \Phi^{\lambda}(t/n) \tau_{0}(t/n) \right)
 \psi_0.\EEQ
Then  $\psi^{(n)}_t\to_{n\to\infty} \psi_t$ in $||\ \cdot ||_{\infty}$-norm.
\end{Lemma}

Trotter formulas have been shown with some generality for non-linear monotonous operators acting on Hilbert spaces
\cite{Bre}.
However here the natural spaces, $L^{\infty}$ or ${\cal W}^{1,\infty}$, are not  Hilbert spaces. To show this lemma we therefore follow 
instead the proof of convergence of
"viscous splitting" algorithms for the Navier-Stokes equation, as found in \cite{BerMaj}, \S 3.4, resting
on their {\em stability} and {\em consistency}. The proof
is however deferred after the following crucial bound in ${\cal H}^{\lambda}$ norm.






\begin{Lemma} \label{lem:bound-psin}
 The following bounds hold,
\BEQ (e^{\lambda |\psi^{(n)}_t|})^*(x)\le \left[(e^{\lambda|\psi_0|})^*(x)\right]^{e^{-M^{-j}t}} e^{\lambda |||g||_{\lambda,j}([0,t],x)} \label{eq:4.20}\EEQ
and 
\BEA &&  \left(e^{\lambda M^{j/2} \frac{|\psi^{(n)}_t(\cdot+\del x)-\psi^{(n)}_t(\cdot)|}{|\del x|}} \right)^*(x)\le
\left[(e^{4\lambda M^{j/2} \frac{|\psi_0(\cdot+\del x)-\psi_0(\cdot)|}{|\del x|}})^*(x)
(e^{4\lambda|\psi_0|})^*(x) \right]^{\frac{1}{4} e^{-M^{-j}t}} \left[ (e^{2\lambda|\psi_t|})^*(x)\right]^{1/2}   \quad .
\nonumber\\ && \qquad \qquad 
e^{2\lambda M^{j/2} |||\frac{g(\cdot,\cdot+\del x)-g(\cdot,\cdot)}{|\del x|}|||_{4\lambda,j}([0,t],x)} 
e^{2\lambda |||g|||_{4\lambda,j}([0,t],x)}, \label{eq:4.21}\EEA
where $|||\ .\ |||_{\lambda,j}(t,x)$ is the  family of pointwise "quasi-norms" appearing in (\ref{eq:etaj1}).
\end{Lemma}


{\bf Proof.}
We shall rely on the following two elementary bounds,
\BEQ (e^{\lambda a |\Phi^{\lambda}_s f|})^*(x)\le (e^{\lambda a |f|})^*(x),\qquad (f\in {\cal W}^{1,\infty},\ a\ge 1)
\label{eq:lambdaaPhis} \EEQ
(see (\ref{eq:3.50})) and H\"older's inequality
\BEQ (e^{\lambda |\tau_f \tilde{f}|})^*(x)\le \left[ (e^{\lambda \frac{p+1}{p} |\tilde{f}|})^*(x)\right]^{\frac{p}{p+1}}
\left[ (e^{\lambda (p+1)|f|})^*(x)\right]^{\frac{1}{p+1}}.
 \label{eq:Holder}\EEQ

\smallskip

Choose $p$ in (\ref{eq:Holder}) defined by $\frac{p}{p+1}=e^{-M^{-j}t/n}$. Note that $\frac{1}{p+1}\sim_{n\to\infty}
M^{-j} \frac{t}{n}.$  By (\ref{eq:lambdaaPhis}),
\BEQ \left(e^{\lambda \frac{p+1}{p} |\psi^{(n)}_{(k+1)t/n}|}\right)^*(x)=\left( e^{\lambda |\Phi^{\lambda}_{t/n}\circ
\tau_{\bar{g}_{(k+1)/n}}(\psi^{(n)}_{kt/n})|} \right)^*(x)\le \left[  (e^{\lambda \frac{p+1}{p}|\psi^{(n)}_{kt/n}|})^*(x)|
\right]^{\frac{p}{p+1}} \left[ (e^{\lambda (p+1)|\bar{g}_{(k+1)t/n}|})^*(x)\right]^{\frac{1}{p+1}}.\EEQ
By induction on $k$, this gives
\BEA \left(e^{\lambda |\psi^{(n)}_t|}\right)^*(x) &\le&  \left(e^{\lambda \frac{p+1}{p} |\psi^{(n)}_{t}|}\right)^*(x)\nonumber\\
&\le & \left[ (e^{\lambda \frac{p+1}{p} |\psi_0|})^*(x)\right]^{e^{-M^{-j} t}} e^{\lambda A},\EEA
where
\BEQ A:=\frac{1}{\lambda} \frac{1}{p+1} \sum_{k=0}^{n-1} \left(e^{-M^{-j}t/n}\right)^k \ln\left[\left(e^{\lambda (p+1) \left|\int^{t-kt/n}_{t-(k+1)t/n} g(s)ds \right| }
\right)^*(x)\right]\EEQ
is asymptotically bounded by the quantity appearing in (\ref{eq:etaj1}) with $g=g^j$.
Letting $n\to\infty$ yields the first bound (\ref{eq:4.20}). 



\medskip
The proof of (\ref{eq:4.21}) is similar but requires a further elaboration on the arguments developed in the course
of the proof of Lemma \ref{lem:classical}, to which we refer the reader. Let
\BEQ \del f(\cdot):=\frac{f(\cdot +\del x)-f(\cdot)}{|\del x|},\qquad
\tilde{\del}f(\cdot):=\frac{f(\cdot +\del x)-(1-M^{-j/2}|\del x|)f(\cdot)}{|\del x|} \EEQ
for $f\in C(\R^d)$. Note that $\tilde{\del}f(\cdot)=\del f(\cdot)+M^{-j/2}f(\cdot).$ Then 
\BEQ (e^{\lambda a M^{j/2} |\tilde{\del}\Phi^{\lambda}_{t/n}(h_0(\cdot))|})^*(x)=
(e^{\frac{\lambda a}{M^{-j/2}|\del x|}  \left[ \Phi^{\lambda}_{t/n}(h_0(\cdot+\del x))-(1-M^{-j/2}|\del x|)\Phi^{\lambda}_{t/n}(h_0(\cdot))\right] })^*(x)
\le (e^{\lambda a M^{j/2}|\tilde{\del}(h_0(\cdot))|})^*(x)\EEQ
for $a\ge 1$.
Hence

\BEA \left( e^{2\lambda M^{j/2} \left(\frac{p+1}{p}\right)^2 |\tilde{\del} \psi^{(n)}_{(k+1)t/n}|}\right)^*(x)
&\le & \left( e^{2\lambda M^{j/2} \frac{p+1}{p} |\tilde{\del} (\tau_{\bar{g}_{(k+1)t/n}}(\psi_{kt/n}^{(n)}))|} \right)^*(x)
\nonumber\\
&=& \left( e^{2\lambda M^{j/2} \frac{p+1}{p} |\tilde{\del} (\psi_{kt/n}^{(n)}) +
\tilde\del(\bar{g}_{(k+1)t/n}) |}\right)^*(x) \nonumber\\ 
&\le & \left[ \left(e^{2\lambda M^{j/2} \left(\frac{p+1}{p}\right)^2 |\tilde{\del}(\psi^{(n)}_{kt/n})|}\right)^*(x)\right]^{\frac{p}{p+1}}  
\left[ \left(e^{2\lambda M^{j/2} \frac{(p+1)^2}{p} |\tilde{\del}(\bar{g}_{(k+1)t/n})|}\right)^*(x)\right]^{\frac{1}{p+1}}.
\EEA

The end of the proof goes as before, up to the corrections necessitated by the difference between $\del$ and
$\tilde{\del}$, which simply consist in using  Holder's inequality twice (hence the use of "quasi-norms" with index $4\lambda$),
\BEQ \left(e^{2\lambda M^{j/2} \frac{(p+1)^2}{p} |\tilde{\del}(\bar{g}_{(k+1)t/n})|}\right)^*(x) \le 
\left[ \left(e^{4\lambda M^{j/2} \frac{(p+1)^2}{p} |\del(\bar{g}_{(k+1)t/n})|}\right)^*(x) \right]^{1/2}
\left[ \left(e^{4\lambda  \frac{(p+1)^2}{p} |\bar{g}_{(k+1)t/n})|}\right)^*(x)\right]^{1/2}; \EEQ

\BEQ \left( e^{\lambda M^{j/2} |\del\psi_t|}\right)^*(x)\le \left[ \left( e^{2\lambda M^{j/2} |\tilde{\del}\psi_t|}\right)^*(x)\right]^{\half}
\left[ \left( e^{2\lambda|\psi_t|}\right)^*(x) \right]^{\half} \EEQ
and
\BEQ \left[ \left( e^{2\lambda M^{j/2} |\tilde\del\psi_0|}\right)^*(x) \right]^{\half e^{-M^j t}} \le \left[ \left( e^{4\lambda M^{j/2} |\del\psi_0|}\right)^*(x)\right]^{\frac{1}{4} e^{-M^{-j}t}}
\left[ \left( e^{4\lambda|\psi_0|}\right)^*(x) \right]^{\frac{1}{4}e^{-M^{-j}t}}. \EEQ



\hfill \eop



\medskip

{\em Proof of Lemma  \ref{lem:Trotter}.}  Following \cite{BerMaj}, we split the proof into three points.

\begin{itemize}
\item[(i)] (stability) As follows from the bounds proved in the above lemma, 
$||\psi^{(n)}_{kt/n}||_{{\cal W}^{1,\infty}}\le C$, where $C$ is a constant depending on the initial condition
and on $g$, but independent from $n$ and $k=0,1,\ldots,n-1$.
\item[(ii)] (consistency)  For every $n\ge 1$, $k=0,1,\ldots,n-1$ and $s\in[0,t/n]$,  we let
$\psi_{s+kt/n}^{(n)}:=\tau_k(s)e^{-M^{-j}s} \Phi^{\lambda}(s) \psi_{kt/n}^{(n)}=\int_{kt/n}^{s+kt/n}
g(u)du+e^{-M^{-j}s} \Phi^{\lambda}(s) \psi_{kt/n}^{(n)}.$ Then
\BEA \partial_s \psi^{(n)}_{s+kt/n}&=& g_{s+kt/n}-M^{-j}e^{-M^{-j}s} \Phi^{\lambda}(s)\psi_{kt/n}^{(n)}
+e^{-M^{-j}s} \left( \nu \Del(\Phi^{\lambda}(s)\psi_{kt/n}^{(n)})+\lambda V(\nabla(\Phi^{\lambda}(s)\psi_{kt/n}^{(n)}))
\right) \nonumber\\
&=& g_{s+kt/n}-(\nu\Del-M^{-j})\left( \int_{kt/n}^{s+kt/n}g(u)du \right) +(\nu\Del-M^{-j})\psi_{s+kt/n}^{(n)}
+ \lambda V(\nabla(-\int_{kt/n}^{s+kt/n}g(u)du + \psi_{s+kt/n}^{(n)})) \nonumber\\
&=&  (\nu\Del-M^{-j})\psi_{s+kt/n}^{(n)}
+ \lambda V(\nabla \psi_{s+kt/n}^{(n)}) +g_{s+kt/n} -(\nu\Del-M^{-j})\left( \int_{kt/n}^{s+kt/n}g(u)du \right)\nonumber\\
&& \qquad \qquad \qquad \qquad \qquad + O(\lambda \sup_{y\lesssim C} |\nabla V(y)|)\nabla(\int_{kt/n}^{s+kt/n}g(u)du)) +   O(\lambda C \sup_{y\lesssim C} |\nabla V(y)| M^{-j} s), \nonumber\\ \EEA
viz.
\BEQ \partial_s\psi_{s+kt/n}=(\nu\Del-M^{-j})\psi_{s+kt/n}+\lambda V(\nabla \psi_{s+kt/n}) +g_{s+kt/n}.\EEQ

By the comparison principle, 
\BEQ ||\psi^{(n)}_{s+kt/n}-\psi_{s+kt/n}||_{\infty}\le ||\psi^{(n)}_{kt/n}-\psi_{kt/n}||+
 C'  s   \int_0^s \left(M^{-j}+M^{-j} ||g_{u+kt/n}||_{\infty} du +||\nabla g_{u+kt/n}||_{\infty}+
 \nu||\Del g_{u+kt/n}||_{\infty} \right) du\EEQ
 (where $C'$ depends like $C$ on the initial condition and on $g$). 
Hence
\BEQ || \psi^{(n)}_{s+kt/n} - \psi_{s+kt/n} ||_{\infty}
\lesssim  \ ||\psi_{kt/n}^{(n)} -\psi_{kt/n}||_{\infty} +O(s^2). \label{eq:consistency}\EEQ

\item[(iii)] We apply (\ref{eq:consistency}) with $s=t/n$.  By induction on $k$,
\BEQ ||\psi_{kt/n}^{(n)}-\psi_t||_{\infty}\lesssim \frac{t^2}{n} \to_{n\to \infty} 0.\EEQ
\end{itemize}
 \hfill \eop
 
 

\begin{Corollary} \label{cor:4.6}
Assume $\psi_0\in{\cal W}^{1,\infty;4\lambda}_j$ and $g\in {\cal W}^{1,\infty;4\lambda}_j([0,T]).$
Then the KPZ equation (\ref{eq:KPZdowntoj}) has a unique viscosity solution $\psi$ on $[0,T]$; the solution $\psi$  is classical,
is uniformly bounded in ${\cal W}_j^{1,\infty;\lambda}$, 
and satisfies the following estimates on $[0,T]\times\R^d$,
\BEQ (e^{\lambda|\psi_t|})^*(x)\le \left[(e^{\lambda\psi_0})^*(x)\right]^{e^{-M^{-j}t}} e^{\lambda
|||g||_{\lambda,j}([0,t],x)};\EEQ
\BEQ (e^{\lambda M^{j/2} |\nabla\psi_t|})^*(x)\le \left[ e^{2\lambda |||\psi_0|||_{{\cal W}^{1,\infty;4\lambda}_{j}}(x)}
\right]^{e^{-M^{-j}t}}  e^{3\lambda |||g|||_{{\cal W}^{1,\infty;4\lambda}_j([0,t])}(x)};\EEQ
\BEQ |||\psi_t|||_{{\cal W}_j^{1,\infty;\lambda}}(x)\lesssim |||\psi_0|||_{{\cal W}_j^{1,\infty;4\lambda}}(x)+
|||g|||_{{\cal W}^{1,\infty;4\lambda}([0,T])}(x).\EEQ
Therefore 
\BEQ \psi_t(x)=O( M^{-jd_{\phi}}),\qquad \nabla\psi_t(x) =O( (M^j)^{-\half-d_{\phi}}) \EEQ
if $|||\psi_0|||_{{\cal W}^{1,\infty;4\lambda}}(x)=O(M^{-jd_{\phi}})$ and  $|||g|||_{{\cal W}^{1,\infty;4\lambda}_j([0,T])}(x)=O(M^{-jd_{\phi}}).$
 \end{Corollary}

{\bf Proof.}
The comparison principle Theorem \ref{th:comparison} extends without any substantial modification to the inhomogeneous equation
(\ref{eq:KPZdowntoj}). By Lemma \ref{lem:Trotter}, the bounds proved in Lemma \ref{lem:bound-psin} for $\psi^{(n)}$ hold for $\psi$, as follows from  taking the limit $n\to\infty$. 
 Details are left
to the reader. \hfill \eop
 
 



%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\section{Appendix A. Scale decompositions}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


As a general motivation for this section, consider the Ornstein-Uhlenbeck process (\ref{eq:O.-U.}),
\BEQ \partial_t\phi=\nu\Del\phi+\eta \EEQ
where $\eta$ is a regularized white noise. Let $G=(\partial_t-\nu\Del)^{-1}$ be the  Green kernel of the
linear heat equation; formally, $\phi=G\eta$. Thus scale $j$ fluctuation fields  $\phi^j$ and $\eta^j$ should be
in direct link, namely, $\phi^j=G\eta^j$. A natural way to
accomplish this is to cut $G$ itself into scales, $G=\sum_j G^j$, and set $\phi^j=G^j\eta$, $\eta^j=(\partial_t-\nu
\Del)\phi^j$. 

\medskip

Before discussing various multi-scale decompositions of $G$, let us first introduce $\eta$.
Let $\tilde{\eta}(t,x)$ be the standard white noise on $\R\times\R^d$. We recall that $\tilde{\eta}$ is
the distribution-valued centered Gaussian process with integrated covariance 
\BEQ \esper\left[ \left(\int dt dx f_1(t,x)\tilde{\eta}(t,x)\right)\left(\int dt dx f_2(t,x)\tilde{\eta}(t,x)\right)
\right]=\int dt dx f_1(t,x)f_2(t,x)\EEQ
$(f_1,f_2\in C^{\infty}_c(\R\times\R^d))$. 

Let also $\chi=\chi(x)$ be a 'bump' function as in Lemma \ref{lem:bump-alpha}.

\begin{Definition}[regularized white noise]
Let $\eta(t,x)=\int dy \chi(x-y)\tilde{\eta}(t,y).$
\end{Definition}

Formally, $\esper[\eta(t,x)\eta(t',x')]=\chi^{*2}(x-x')\del(t-t')$, where $\chi^{*2}:=\chi\ast\chi$ is another 'bump' function.


\medskip


The {\em stationary} Ornstein-Uhlenbeck process,
\BEQ \phi(t,x)=\int_{-\infty}^t ds e^{(t-s)\nu\Del}\eta_s(x),  \label{eq:phi} \EEQ
 solution of (\ref{eq:O.-U.}),
has covariance kernel (assuming
 e.g. $t\ge t'$)
\BEA  \esper[\phi(t,x)\phi(t',x')] & =& \int_{-\infty}^t ds  \int_{-\infty}^{t'} ds' \int dy dy'  e^{\nu(t-s)\Del}(x-y)
e^{\nu(t'-s')\Del}(x'-y') \del(s-s') \chi^{*2}(y-y') \nonumber\\
&=& \int_0^{+\infty} du \left(\int dy  dy' e^{\nu (t-t'+u)\Del}(x-y) e^{\nu u\Del}(x'-y') \chi^{*2}(y-y') \right). \label{eq:stat-cov1}\EEA
The regularization has a measurable effect only around the diagonal $t=t',x=x',u=0$. Away from the diagonal
the last integral (\ref{eq:stat-cov1}) behaves like $\int^{+\infty} du \ e^{\nu (t-t'+2u)\Del}(x,x')=\int^{+\infty} du \frac{e^{-|x-x'|^2/2
\nu(t-t'+2u)}}{(2\pi\nu(t-t'+2u))^{d/2}}$, an integrable function at infinity since $d/2>1$. Thus
\BEQ \left|  \esper[\phi_t(x)\phi_{t'}(x')] \right|\lesssim \int_0^{+\infty} ds (t-t'+s)^{-d/2} \left(
1+O\left(\frac{|x-x'|}{\sqrt{t-t'+s}}\right)\right)^{-N},\qquad N\ge 1 \\ \EEQ
is bounded by a constant times $(t-t')^{1-d/2}$ if $|x-x'|\lesssim \sqrt{t-t'}$, and 
$\int_{|x-x'|^2}^{+\infty} s^{-d/2} ds=\frac{C}{|x-x'|^{d-2}}$ (the Green kernel of the Laplacian on $\R^d$)
in the contrary case.


\bigskip



We now want to cut $\phi$ into scales, i.e. understand how it behaves typically for time separations of order
$M^j$ ($j\ge 0$), or space separations of order $M^{j/2}$, where $M>1$ (the scale parameter) is an
arbitrary but fixed parameter.  The main task is to cut $G$ into $M$-adic scales, $G=\sum_{j\ge 0}G^j$; then (as discussed above) we
 define
\BEQ \phi^j=G^j\eta,\qquad \eta^j=(\partial_t-\nu\Del)\phi^j. \EEQ
With these definitions, $\sum_{j\ge 0}\phi^j=G\eta=\phi$ is the Ornstein-Uhlenbeck field, 
and $\sum_{j\ge 0}\eta^j=(\partial_t-\nu\Del)G\eta=\eta$.

We proceed as follows. Let $\bar{\chi}:\R_+\to\R_+$ be a smooth 'bump' function of scale $1$
supported away from the origin, say, $\bar{\chi}\big|_{[M^{-\half},M^{\half}]}\equiv 1$, $\bar{\chi}\big|_{\R_+\setminus
(M^{-1},M)}\equiv 0$, chosen in such a way that 
\BEQ \bar{\chi}^{0}(\cdot):=\sum_{n\ge 0} \bar{\chi}(M^n \cdot),\qquad \bar{\chi}^j(\cdot):=\bar{\chi}(M^{-j}\cdot)
\qquad (j\ge 1) \EEQ 
form a partition of unity, i.e. $\sum_{j\ge 0} \bar{\chi}^j\equiv 1$ on $\R_+$.


\begin{Definition}[cut-off] \label{def:non-local-cut-off}
Let $G^j$ be the operator
\BEQ (G^j g)(t):=\int \bar{\chi}^j(s) e^{s\nu\Del} g(t-s) ds,\qquad
j\ge 0\EEQ
and
\BEQ \phi^j=G^j\eta,\qquad
\eta^j=(\partial_t-\nu\Del)\phi^j.\EEQ
\end{Definition}
Clearly, $\sum_{j\ge 0}G^j=G$ and $\sum_{j\ge 0}\phi^j=\phi$ is the solution of the Ornstein-Uhlenbeck
equation (\ref{eq:O.-U.}).



Let $t\ge t'$. Assume $j\ge 1$. The diagonal covariance kernel $C^j_{\phi}(t,x;t',x')=\esper[\phi^j_{t}(x)\phi^j_{t'}(x')]$
 is non-zero only for $t-t'\le M^{j}(M-M^{-1})$, in which case (recall $d_{\phi}:=\half(\frac{d}{2}-1)$)
\BEA  C^j_{\phi}(t,x;t',x')&=&\int_{t-t'}^{M^{j}(M-M^{-1})} du \left(e^{\nu(M^{j-1}+u)\Del} e^{\nu(M^{j-1}+u-(t-t'))\Del}\right)\chi^{*2}(x-x') \nonumber\\
&\lesssim & M^j \frac{e^{-M^{-(j+1)}|x-x'|^2/4\nu}}{M^{jd/2}} \nonumber\\
&= & (M^{-j})^{2d_{\phi}}
e^{ -M^{-(j+1)}|x-x'|^2/4\nu}\lesssim  (M^{-j})^{2d_{\phi}} e^{-c M^{-j/2} |x-x'|}\EEA
for some constant $c>0$. A similar formula holds for $j=0$: if $t-t'\le M$,
\BEQ C^0_{\phi}(t,x;t',x')=\int_{t-t'}^{M} du \left(e^{\nu u\Del} e^{\nu(u-(t-t'))\Del}\right)\chi^{*2}(x-x')\lesssim 
e^{- M^{-1}|x-x'|^2/4\nu}\lesssim e^{-c|x-x'|} \EEQ
since $\chi^{*2}$ is smooth and compactly supported.

Then the off-diagonal covariances 
\BEQ C^{j,j'}_{\phi}(t,x;t',x')=\esper[\phi^j_{t}(x)\phi^{j'}_{t'}(x')]\EEQ
are similarly shown to satisfy for $j\ge j'$ the estimate
\BEQ |C^{j,j'}_{\phi}(t,x;t',x')|\lesssim M^{j'} \frac{e^{-M^{-(j+1)}|x-x'|^2/4\nu}}{M^{jd/2}} \lesssim M^{-|j-j'|} (M^{-j})^{2d_{\phi}}
e^{-cM^{-j/2} |x-x'|}.\EEQ
Since $C_{\phi}^{j,j'}(t,\cdot;t',\cdot)=0$ for $|t-t'|\gg M^j$, one may clearly also write
\BEQ |C^{j,j'}_{\phi}(t,x;t',x')|\lesssim M^{-|j-j'|} (M^{-j})^{2d_{\phi}}
e^{-cM^{-j}|t-t'|-cM^{-j/2} |x-x'|}.\EEQ

\medskip

Finally gradients, resp. time-derivatives applied to the heat kernel produce  small factors
of order $O(M^{-\max(j,j')/2})$, resp. $O(M^{-\max(j,j')})$. Let us recapitulate.

\begin{Lemma}[covariance kernel estimates] \label{lem:cov-ker-est}
Let
\BEQ C^{j,j'}_{\phi}(t,x;t',x')=\esper[\phi^j_{t}(x)\phi^{j'}_{t'}(x')], C^{j,j'}_{\eta}(t,x;t',x')=\esper[\eta^j(t,x)\eta^{j'}(t',x')]\EEQ 
and
\BEQ C^{j}_{\phi}:=C^{j,j}_{\phi},\qquad C^j_{\eta}:=C^{j,j}_{\eta}.\EEQ

Then, for $j\ge j'$, 
\BEQ \left|\nabla_x^{p}\partial_t^q \nabla_{x'}^{p'}\partial_{t'}^{q'} C^{j,j'}_{\phi}(t,x;t',x')\right|\lesssim
M^{-|j-j'|} M^{-\frac{j}{2}(p+p')
-j(q+q')} (M^{-j})^{2d_{\phi}}
e^{-cM^{-j}|t-t'|-cM^{-j/2} |x-x'|} \\ \EEQ
and
\BEQ \left|\nabla_x^{p}\partial_t^q \nabla_{x'}^{p'}\partial_{t'}^{q'} C^{j,j'}_{\eta}(t,x;t',x')\right|\lesssim
M^{-|j-j'|} M^{-\frac{j}{2}(p+p')
-j(q+q')} (M^{-j})^{2+2d_{\phi}}
e^{-cM^{-j}|t-t'|-cM^{-j/2} |x-x'|}. \\ \EEQ
Furthermore, if $j\ge 0$,
\BEQ \esper[(\eta^j_t(x)-\eta^j_t(y))^2]\lesssim (M^{-j})^{3+2d_{\phi}}|x-y|^2 \label{eq:eta-distance} \EEQ
and
\BEQ \esper[(\eta^j_t(x)-\eta^j_s(x))^2]\lesssim (M^{-j})^{4+2d_{\phi}}|t-s|^2 \label{eq:5.20bis}.\EEQ
\end{Lemma}


The  last two estimates (\ref{eq:eta-distance}), (\ref{eq:5.20bis})  follows immediately from Taylor's formula: letting $\vec{v}:=\frac{y-x}{|y-x|}$,
\BEQ \esper[(\eta^j_t(x)-\eta^j_t(y))^2]\le \int_0^{|y-x|} dz \int_0^{|y-x|} dz' 
\left|\nabla_{\vec{v}}\nabla'_{\vec{v}} C^j_{\eta}(t,x+z\vec{v};t,x+z'\vec{v}) \right|\EEQ
and similarly for $\esper[(\eta^j_t(x)-\eta^j_s(x))^2]$.

\medskip


One has thus obtained a very elaborate version of the scalings (\ref{eq:scaling}), $\phi^j(t,x)=O(M^{-jd_{\phi}}),
\eta^j(t,x)=O((M^{-j})^{1+d_{\phi}})$, together with a first indication of the scale-separation mechanism: the prefactors
in powers of $M^{-|j-j'|}$ show clearly that fields of widely separated scales are effectively independent.









%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\section{Appendix B. Large deviations estimates for the single-scale noisy equation}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\subsection{Introduction}

%%%%%%%%%%%%%%%%%%%%%



We consider here the noisy KPZ equation with scale $j$ infra-red cut-off,
\BEQ \partial_t \psi=(\nu\Del-M^{-j})\psi+\lambda V(\nabla\psi)+\eta^j\EEQ
with right-hand side $\eta^j=G^j \eta$ defined as in Appendix A, subsection 2.
Recall the outcome of the computations leading to Corollary \ref{cor:4.6}: if $|||\eta^j|||_{{\cal W}^{1,\infty;4\lambda}}([0,t],x)=O(M^{-jd_{\phi}})$, then $|\psi(t,x)|=O(M^{-jd_{\phi}}), |\nabla\psi(t,x)|=O((M^{-j})^{\half+d_{\phi}})$. 

We  show in this section that $|||\eta^j|||_{{\cal W}^{1,\infty;4\lambda}_j}([0,\infty),x)$ is a.s. bounded, and
prove large deviation estimates for this quantity when it is much larger than $O(M^{-jd_{\phi}})$.

The random variables appearing in the definition of the pointwise "quasi-norms" associated with  ${\cal W}^{1,\infty;4\lambda}$ are essentially time- and space-averages of a large number of independent {\em log-normal variables}, such as
$e^{4\lambda M^j|\eta^j(t,x)|}$. Log-normal variables  have large tails in $e^{-a(\ln z)^2}$ and thus no exponential moment,
hence standard large-deviation theory (notably Cram\'er's theorem) does not give any valuable information on the
probability that such averages become large. Some authors have been considering this problem, notably Russians, starting from the 60es; one may cite Linnik \cite{Lin},
Nagaev \cite{Nag1,Nag2}, Rozovski \cite{Roz}, see also e.g. Kl\"uppelberg and Mikosch \cite{KluMik} for a renewal of
the theory with a view to applications in insurance. The theory is not easily accessible, partly because written
originally in Russian journals in the 60es and 70es (in particular in {\em Teoriya Veroyatnostei i ee Primeneniya}, later translated to English as
{\em Theory of Probability and its  Applications}), partly for the lack of a theory as general and satisfactory as the standard large-deviation
theory. 

\medskip
Let us just point out the difficulties (this very short abstract is taken from an inspiring review in \cite{MikNag}).
Choose a random variable $X$ with finite first and second moments; by translation and rescaling we may assume
that $\esper[X]=0,\esper[X^2]=1$. Let $S_n:=X_1+\ldots+X_n$, $M_n:=
\max(X_1,\ldots,X_n)$, where $X_1,\ldots,X_n$ are independent copies of $X$. Let finally 
$\bar{F}^X(x):=\proba[X>x],\ \bar{F}_n^X(x):=\proba[S_n>x]$ and $\Errfc(x):=\int_x^{+\infty}
 \frac{e^{-y^2/2}}{\sqrt{2\pi}} \, dy$ be resp. the queues of $X$, of $S_n$ and of a standard Gaussian variable. By the central limit theorem,
one expects 
\BEQ \bar{F}_n^X(x)\approx \Errfc(x/\sqrt{n}), \label{eq:LD1}\EEQ
at least if $x\approx \sqrt{n}$. On the other hand, one clarly has if $X\ge 0$
\BEQ \bar{F}_n^X(x)\ge \proba[M_n>x]\sim_{x\to\infty} n\bar{F}^X(x). \label{eq:LD2}\EEQ
{\em Subexponential distributions} (including log-normal distributions) are precisely defined by the asymptotic relation
$\bar{F}_n^X(x)\sim_{x\to\infty}  n\bar{F}^X(x)$, implying a heavy queue. For distribution with lighter queues
 (such as e.g. Gaussian distributions), the inequality in (\ref{eq:LD2}) is very rough, in the sense that typically $n\bar{F}^X(x)\ll \bar{F}_n^X(x)$ for every $x\ge x_0$,
with $x_0$ independent from $n$. 

Thus, one expects, specifically for subexponential distributions, a {\em central limit theorem behaviour} as in (\ref{eq:LD1}) for
$x\ll c_n$, with $c_n$ defined by $\Errfc(c_n/\sqrt{n})\approx \bar{F}_n^X(c_n)$, and an {\em extreme-value regime},
\BEQ \bar{F}^X_n(x)\sim n\bar{F}^X(x),\qquad x\gg d_n\EEQ
with $d_n\ge c_n$, in which $n\bar{F}^X(x)\gg \Errfc(x/\sqrt{n})$.
Optimal sequences $c_n,d_n$ have been identified for various types of subexponential distributions; for a
standard log-normal variable $X=e^Z$, $Z\sim {\cal N}(0,1)$, one finds $c_n,d_n\approx n^{\half}\ln n$. One major
drawback of this picture is that it doesn't say anything about the behaviour of $\bar{F}_n^X(x)$ in the window
$c_n\lesssim x\lesssim d_n$ (in our case, for $x\approx c_n$ since $c_n=d_n$), which is expected to be a mixture
of (\ref{eq:LD1}) and (\ref{eq:LD2}). The complicated asymptotics, valid on the whole real line, proved by 
Rozovski \cite{Roz} -- a veritable tour de force -- give a more complete answer. 

\medskip
This being said, our problem does not fit exactly into this frame, since (1) we are only interested in {\em 
upper bounds} for $\bar{F}_n^X$, moreover in the {\em extreme-value regime}, with $x\gtrsim n$; on the other
hand (2) the variables $X_1,\ldots,X_i,\ldots,X_j,\ldots,X_n$  (chosen as local space or space-time averages of
the noise) are not independent, but have {\em correlations which decrease exponentially}
with the scaled distance $d^j$ (see below) or equivalently with $|j-i|$; (3) we need {\em scale-dependent estimates} for $\bar{F}^X_n$ since $X\approx e^{M^{-jd_{\phi}}|Z|}$,
$Z\sim {\cal N}(0,1)$ is strongly $j$-dependent. However all the previous results are strongly dependent on the
particular form of the distribution, in particular on the first and second moments, and it is often difficult
to retrace the $j$-dependence of the constants in the bounds.



\bigskip

Our main  result is the following. 

\begin{Theorem} \label{th:LD1}
Let $j\in\N$ and $\lambda>0$. Then the function $x\mapsto |||\eta^j|||_{{\cal W}^{1,\infty;\lambda}_j(\R_+)}(x)$
is a.s. everywhere defined (i.e. finite). Furthermore, the following large deviation estimates holds for every
$x\in\R^d$,
\BEQ \proba[\sup_{B(x,M^{j/2})} |||\eta^j|||_{{\cal W}^{1,\infty;\lambda}_j(\R_+)}(x)>AM^{-jd_{\phi}}]\lesssim 
A^{-c\ln(A)}, \qquad A\ge 1\EEQ
where $c>0$ is some constant.
\end{Theorem}

As follows from Corollary \ref{cor:4.6}, this implies (up to the replacement of $\lambda$ by $4\lambda$) that the solution $\psi$ of the full KPZ equation with scale $j$ infra-red cut-off (\ref{eq:KPZdowntoj}) is defined a.s. for all positive times $t\ge 0$ and sits in 
the space ${\cal W}^{1,\infty;\lambda}_j(\R_+)$, with $|||\psi|||_{{\cal W}^{1,\infty;\lambda}_j(\R_+)}(x)=O(M^{-jd_{\phi}})$ for every $x\in\R^d$, with a random multiplicative prefactor $A(x)$ whose queue is bounded {\em locally
in $x$} by that of
a log-normal distribution. (Note that the prefactor $A(x)$ is {\em not globally} bounded!)

\bigskip

The proof  includes both Gaussian inequalities taken from the monograph \cite{Adl}, and
an adaptation to weakly correlated variables of a result about large deviations for subexponential distributions
\cite{Nag2}. We shall need quite a few preliminary results before the proof, given at the very end of the present
section. 

We finish this introductory paragraph with the tiny bit of stochastic domination and Gaussian inequalities used
in the sequel, and a little bit of geometry.

\medskip


\begin{Definition}
Let $X:\Omega\to\R,Y:\Omega'\to\R$ be two real-valued random variables, defined a priori on two different
probability spaces. Then $X$ is {\em stochastically dominated} by $Y$ if
\BEQ \forall x\in\R, \proba[X>x]\le \proba[Y>x]. \EEQ
We then write $X\lele Y$.
\end{Definition}
By Strassen's theorem \cite{Lind}, if $X\lele Y$, there exists a coupling between $X$ and $Y$, i.e. random variables $X',Y':\Omega''\to\R$ defined on the same probability space, with $X'\stackrel{(d)}{=}X,\ Y'\stackrel{(d)}{=} Y$, 
and such that $X'\le Y'$. 


\medskip



\begin{Proposition} (see \cite{Adl}) \label{prop:Slepian}
Let $(Z_1,\ldots,Z_n)$ be a centered Gaussian vector, and $\phi:\R^n\to\R$ be a convex function with
polynomial growth at infinity. Then $\esper[\phi(Z_1,\ldots,Z_n)]$ is an increasing function of the coefficients
$c_{ij}=\esper[Z_i Z_j]$, $i,j=1,\ldots,n$.
\end{Proposition}
This technical lemma, due to Slepian (whose short proof relies on a Gaussian integration by parts) is one of the main tools for
Gaussian inequalities. It extends to Gaussian fields $(Z_x)_{x\in\R^d}$ and convex functionals $\phi$ under adequate regularity assumptions.

\medskip



\begin{Proposition}[Borell-Tsirelson-Ibragimov-Sudakov or BTIS inequality] (see \cite{Adl}) \label{prop:BTIS}\\
Let $(Y_x)_{x\in {\cal D}}$, ${\cal D}\subset B(0,1)$ be a centered Gaussian process, such that 
$\sigma^2_{{\cal D}}=\sup_{x\in{\cal D}} \esper[Y^2_x]<\infty$, and
$\del(x,y):=\sqrt{\esper[ (Y_x-Y_y)^2]}\lesssim |x-y|.$ Let $||Y||_{\infty}:=\max_{x\in{\cal D}} |Y_{x}|$. Then a.s.
$||Y||_{\infty}<\infty$, $\esper[||Y||_{\infty}]\lesssim 1$ and
\BEQ \proba[||Y||_{\infty}-\esper[||Y||_{\infty}]>u]\le e^{-u^2/2\sigma_{{\cal D}}^2}.\EEQ
\end{Proposition}
This is actually a particular case of the BTIS inequality. For a Gaussian process $Y$ indexed by an abstract set $\cal D$, $\esper[||Y||_{\infty}]$ is bounded by the integral of the square-root of the entropy $\log N(\eps)$, $\esper[||Y||_{\infty}]
\lesssim \int_0^{+\infty} \sqrt{\ln N(\eps)}d\eps$, where $N(\eps)$ is the minimum number of balls of diameter
$\le\eps$ (with respect to the metric induced by $\del(\cdot,\cdot)$) covering $\cal D$. In our proposition, 
$\ln N(\eps)=0$ for $\eps\gg 1$ since $\sup_{x,y\in{\cal D}} \del(x,y)\lesssim 1$, and $N(\eps)=O(\eps^d)$
otherwise by hypothesis, hence the result.


\medskip
The above proposition applies for fixed $t_0,x_0$ to $Y_x:=M^{j(1+d_{\phi})}\eta^j(t_0,x_0+M^{j/2}x)$, 
with ${\cal D}=B(0,1)$. It follows from  Lemma \ref{lem:cov-ker-est} in Appendix A that $\sigma^2_{{\cal D}}\approx 1$ and $d(x,y)\lesssim |x-y|$. Thus 
\BEQ \esper \left[ \sup_{B(x_0,M^{j/2})} |\eta^j_{t_0}| \right] \lesssim M^{-j(1+d_{\phi})} \EEQ
and there exists a constant $C\lesssim 1$ such that
\BEQ \proba[M^{j(1+d_{\phi})} \sup_{B(x_0,M^{j/2})} |\eta^j_{t_0}|>u+C]\le e^{-u^2/2C}.\EEQ
One easily deduces that 
\BEQ M^{j(1+d_{\phi})} \sup_{B(x_0,M^{j/2})} |\eta^j_{t_0}|\lele C' (|Z|+1) \label{eq:suplele}\EEQ
if $Z\sim {\cal N}(0,1)$.

\bigskip

Recall from section 3.1 that $f^*\le f^{\sharp}$ $(f\in C(\R^d,\R))$  -- note, and this is very important, that  the inequality is {\em exact}, with 
a coefficient one --, where $f^{\#}(x)=\sup_{\rho>0} \fint_{B(x,\rho)}
|f|.$ We cannot bound directly a supremum over a continuous parameter (here $\rho$), so it is natural to start
by rewriting $(\eta^j_{t_0})^*$ in terms of its local averages or suprema on balls of radius $M^{j/2}$, over which
we have a good control. However, we cannot obviously cover $\R^d$ (nor $B(x,\rho)$) by disjoint balls of fixed
radius, and taking into account error terms due to overlaps or boundary effects would cost a multiplicative coefficient,
which we cannot afford to do. Hence we first transform balls centered at $x$ into cubes by letting
\BEQ \Phi:\R^d\to\R^d,\quad y\mapsto \Phi(y)=x+\frac{|y-x|}{||y-x||_{\infty}}(y-x)\EEQ
where $||y-x||_{\infty}:=\sup(|y_1-x_1|,\ldots,|y_d-x_d|)$ is the supremum norm. The Euclidean norm $|\ \cdot\ |$ and
the supremum norm $||\ \cdot\ ||_{\infty}$ are equivalent, hence (the easy proof is left to the reader)
$\Phi$ and $\Phi^{-1}$ are uniformly Lipschitz.  Thus $\sup_{B(x,\rho)} |\eta^j_{t_0}|=\sup_{\tilde{B}(x,\rho)}
|\eta^j_{t_0}\circ\Phi^{-1}|$, where $\tilde{B}(x,\rho)=\{y\in\R^d\ |\ ||y-x||_{\infty}=\rho\}$ is a cube.
The field $\eta^j_{t_0}\circ\Phi^{-1}$ has the same general properties as $\eta^j_{t_0}$ (scaling, exponentially
decreasing covariance) as stated in Lemma \ref{lem:cov-ker-est}, so (by abuse of notation) we simply denote
$\eta^j_{t_0}\circ\phi^{-1}$ by $\eta^j_{t_0}$ in the sequel.

\begin{Definition}[scale $j$ cubes]
Let
$\D^j$ be the set of all {\em scale $j$ cubes}, i.e. of all primitive cells $[k_1 M^{j/2},(k_1+1)M^{j/2}]\times\ldots\times[k_d M^{j/2},(k_d+1)M^{j/2}]$,
$k_1,\ldots,k_d\in\Z$  
of the square lattice $M^{j/2}\Z^d$.


We denote by $x_{\Del}=(x_{\Del,1},\ldots,x_{\Del,d})$ the center of a cube $\Del\in\D^j$.
\end{Definition}

We now show how to bound an average  $\fint_{\tilde{B}(x,\rho)} |f|$, $f\in C(\R^d,\R)$ over a cube of arbitrary radius in terms of the 
local suprema $f_{\Del}:=\sup_{\Del} |f|,\ \Del\in \D^j$. We give the proof in dimension 2 to simplify notations
(in general, we would need the whole cellular decomposition of a cube). Let, for $\rho>0$,
\BEQ \tilde{B}^j(x,\rho):=\cup\{\Del\in\D^j\ |\ \Del\subset \tilde{B}(x,M^{j/2}\rho)\},\qquad
\partial \tilde{B}^j(x,\rho):=\cup\{\Del\in\D^j\ |\ \Del\cap \tilde{B}(x,M^{j/2}\rho)\not=\emptyset\}\setminus
\tilde{B}^j(x,\rho),  \label{eq:Bjxrho1} \EEQ
and $n:=\tilde{B}^j(x,\rho)$.
The boundary $\partial\tilde{B}^j(x,\rho)$ decomposes into 8 disjoint subsets,
\BEQ \partial\tilde{B}^j_{{\mathrm{right}}}(x,\rho):=\cup\{\Del=[x_{\Del,\min},x_{\Del,\max}]\times
[y_{\Del,min},y_{\Del,max}]\ | \ a<x_{\Del,min}<b<x_{\Del,max},\ c\le y_{\Del,min}<y_{\Del,max}\le d\} \EEQ
and similary $ \partial\tilde{B}^j_{{\mathrm{left}}}(x,\rho), \partial\tilde{B}^j_{{\mathrm{up}}}(x,\rho), \partial\tilde{B}^j_{{\mathrm{down}}}(x,\rho)$ for the sides of the square;
\BEQ  \partial\tilde{B}^j_{{\mathrm{up,right}}}(x,\rho)=\cup\{\Del=[x_{\Del,\min},x_{\Del,\max}]\times
[y_{\Del,min},y_{\Del,max}]\ |\ a<x_{\Del,min}<b<x_{\Del,max},\ c< y_{\Del,min}<d<y_{\Del,max}\} \EEQ
and similary for the three other corners. We let $c_{{\mathrm{right}}}:=\frac{\Vol(\partial\tilde{B}^j_{{\mathrm{right}}}
(x,\rho)\cap \tilde{B}(x,\rho))}{\Vol((\partial\tilde{B}^j_{{\mathrm{right}}}
(x,\rho))}$, and similarly $c_{{\mathrm{left}}},\ldots$ be the corresponding volume ratios. Let
\BEQ F(c_{{\mathrm{right}}},c_{{\mathrm{left}}},\ldots):=\frac{\sum_{\Del\in \tilde{B}^j(x,\rho)} f_{\Del}+
c_{{\mathrm{right}}} \sum_{\Del'\in\partial\tilde{B}^j_{{\mathrm{right}}}(x,\rho)} f_{\Del'}+\ldots}{n+
c_{{\mathrm{right}}} \sharp \partial\tilde{B}^j_{{\mathrm{right}}}(x,\rho)+\ldots};\EEQ
note that $\fint_{\tilde{B}(x,\rho)} f\le F(c_{{\mathrm{right}}},\ldots)$ since $c_{{\mathrm{right}}}$ is the
uniform volume ratio $\frac{\Vol(\Del)\cap \tilde{B}^j(x,\rho)}{\Vol(\Del)}$ of all scale $j$ cubes at the right
border, as follows from the fact that the border is straight. Then trivially $F(c_{{\mathrm{right}}},c_{{\mathrm{left}}},\ldots)\le \max\left(  F(0,c_{{\mathrm{left}}},\ldots), F(0,c_{{\mathrm{right}}},\ldots) \right)$; this same elementary
remark may be repeated for the eight $c$ coefficients. Thus we have proved that
\BEQ \fint_{\tilde{B}(x,\rho)} f\le \max_{B^j} \frac{\sum_{\Del\in B^j} f_{\Del}}{\sharp B^j},\EEQ
where the $B^j$ range among
$2^8$ subsets of squares, and by definition  $\tilde{B}^j(x,\rho)\subset B^j\subset \tilde{B}^j(x,\rho)\cup \partial\tilde{B}^j(x,\rho)$. 




%%%%%%%%%%%%%%%%%%%%%%%

\subsection{A first preliminary result: large deviations for the noise}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

We prove in this paragraph the following result.



\begin{Lemma} \label{lem:LD-main}
Let $j\in\N$ and $t_0\in\R_+$. Then the function $x\mapsto  (\eta^j)^*(t_0,x)$ is a.s. everywhere defined
(i.e. finite). Furthermore, the following large deviation estimates holds,
\BEQ \proba[\sup_{B(x,M^{j/2})} (\eta^j)^*(t_0)>A M^{-j(1+d_{\phi})}]\le e^{-c(A-C)_+^2} \label{eq:6.14} \EEQ 
for some constants $c,C>0$, where $(A-C)_+^2=(A-C)^2 {\bf 1}_{A>C}.$
\end{Lemma}

It is actually reasonable to expect, on account of the central limit theorem, that $|\eta^j(t_0,x)|-\esper[|\eta^j(t_0,x)|]\in {\cal H}^0_{\alpha}$
for every $\alpha<d/4$,  and that the norm in ${\cal H}^0_{\alpha}$ satisfies large deviation estimates
as in (\ref{eq:6.14}), but we do not prove this. 
The above result, however natural it may be, is not really needed anywhere in the article, but the proof of Theorem
\ref{th:LD} is based on the arguments developed for the proof of the lemma. \medskip

{\bf Proof.} In the sequel $c,c',C>0$ are constants possibly varying from line to line (contrary to $c_0,m_0$,
see below, which are fixed once and for all). As already recalled,  $(\eta^j)^*(t_0,x)\le(\eta^j)^{\sharp}(t_0,x)=
\sup_{\rho>0}  \fint_{B(x,M^{j/2} \rho)} dy |\eta^j(t_0,y)|.$  Also, from the results of Appendix A,
the correlations of the field $(\eta^j_{t_0}(x))_{x\in\R^d}$ decay exponentially with the
{\em scaled distance} $d^j(x,x'):=M^{j/2}|x-x'|$, in the sense that, for a certain constant $c_0$, 
\BEQ |\esper[\eta^j_{t_0}(x)\eta^j_{t_0}(x')]|\lesssim M^{-2j(1+d_{\phi})} e^{-c_0 d^j(x,x')}.  \label{eq:c0} \EEQ

\medskip We split the proof into several points.

\begin{itemize}
\item[(i)] In order to use the exponential decay, we first choose $m_0\ge 2$ large enough (depending on further
considerations), and partition $\D^j$ into $m_0^d$ disjoint susets $\D^j_{\mu}$, $\mu\in\{1,\ldots,m_0\}^d$, 
with $\Del=[k_1 M^{j/2},(k_1+1)M^{j/2}]\times\ldots\times[k_d M^{j/2},(k_d+1)M^{j/2}]\in\D^j_{\mu}\Leftrightarrow
k_i\equiv \mu_i\mod m_0$. Two points $x,x'$ located in disjoint cubes $\Del\not=\Del'$ in the same sublattice
$\D^j_{\mu}$ are thus at distance $d^j(x,x')\gtrsim m_0$, which amounts (up to rescaling) to replacing $c_0$ by $m_0 c_0$ in (\ref{eq:c0}); in the sequel, we may thus assume that $c_0$ is large enough. By abuse of notation, we also denote by $\D^j_{\mu}$ the subset $\cup \{\Del;\Del\in\D^j_{\mu}\}
\subset\R^d$.
Clearly, 
\BEQ \fint_{B(x,M^{j/2}\rho)} dy |\eta^j(t_0,y)|\le \sup_{\mu} \fint_{B(x,M^{j/2}\rho)\cap\D^j_{\mu}} dy |\eta^j(t_0,y)|.
\label{eq:6.16} \EEQ
If $\psi:\R\to\R$ is increasing, then
\BEQ \esper\left[\psi\left(\fint_{B(x,M^{j/2}\rho)} dy |\eta^j(t_0,y)|\right)\right]\le \sum_{\mu}
\esper\left[\psi\left(\fint_{B(x,M^{j/2}\rho)\cap\D^j_{\mu}} dy |\eta^j(t_0,y)|\right)\right] \label{eq:mu}.\EEQ


\item[(ii)] Next, we want to bound the average $\fint_{B(x,M^{j/2}\rho)\cap\D^j_{\mu}} dy |\eta^j(t_0,y)|$
over some fixed sublattice by the average of a finite number of variables representing the supremum of $|\eta^j|$ on
each cube. For that (note that the following construction is $\mu$-dependent, which we do not always specify)
we introduce i.i.d. copies $(\eta^j_{\Del})_{\Del\in\D'_{\mu}}$ of the field $\eta^j_{t_0}\big|_{[-\half M^{j/2},
\half M^{j/2}]^d}$ restricted to some reference cube, and define a new random field $\tilde{\eta}^j$ on $\D^j_{\mu}$,
 \BEQ \tilde{\eta}^j(x):=\sum_{\Del'\in\tilde{B}^j(x,\rho)} e^{-c_0 d^j(\Del,\Del')} \eta^j_{\Del'}(x-x_{\Del}),\qquad x\in\Del \label{eq:new-random-field} \EEQ
 separately on each cube $\Del\in\D^j_{\mu}$, where
\BEQ d^j(\Del,\Del'):=M^{-j/2}\sup_{x\in\Del}\inf_{y\in\Del'} |x-y| \EEQ
is the set distance measured in scaled units, and 
\BEQ \tilde{B}^j(x,\rho):=\{\Del\in\D^j_{\mu} \ |\ 
\Del\subset \tilde{B}(x,M^{j/2}\rho)\} \label{eq:Bjxrho2}\EEQ
(compare with the previous definition, (\ref{eq:Bjxrho1})). 
By a simple computation, one finds 
\BEQ \esper[\tilde{\eta}^j(x)\tilde{\eta}^j(x')]\approx (1+d^j(\Del,\Del'))^d e^{-c_0 d^j(\Del,\Del')} 
\esper[\eta^j_{t_0}(x-x_{\Del})\eta^j_{t_0}(x'-x_{\Del'})]\gtrsim \esper[\eta^j_{t_0}(x)\eta^j_{t_0}(x')] \EEQ
if $x\in\Del,\ x'\in\Del'$ and 
\BEQ \Del,\Del'\in \tilde{B}^j(x,\rho)\cup\partial \tilde{B}^j(x,\rho):=\{\Del\in\D^j_{\mu} \ |\ 
\Del\cap \tilde{B}(x,M^{j/2}\rho)\not=\emptyset\}.\EEQ
Applying Proposition \ref{prop:Slepian} with $\phi(\eta^j_{t_0})=\psi\left( M^{j(1+d_{\phi})}
\fint_{B(x,M^{j/2}\rho)\cap\D'_{\mu}}dy |\eta^j_{t_0}(y)| \right)$ where $\psi$ is any convex, increasing function
\footnote{Observe that $\psi_1\circ\psi_2$ is convex if $\psi_1:\R_+\to\R$ is convex and increasing and 
$\psi_2:\R^n\to\R_+$ is convex, since $\nabla^2(\psi_1\circ\psi_2)=\psi''_1\circ\psi_2 \ \cdot\ \nabla\psi_2\otimes
\nabla \psi_2 + \psi'_1\circ\psi_2 \ \cdot\ \nabla^2 \psi_2$.} on $\R_+$,
\BEQ \esper\left[\psi\left( M^{j(1+d_{\phi})}
\fint_{B(x,M^{j/2}\rho)\cap\D'_{\mu}}dy |\eta^j_{t_0}(y)| \right)\right] \le 
\esper\left[\psi\left( M^{j(1+d_{\phi})}
\fint_{B(x,M^{j/2}\rho)\cap\D'_{\mu}}dy |\tilde{\eta}^j(y)| \right)\right]. \label{eq:6.21} \EEQ

As follows from the discussion in the previous paragraph,
\BEQ \fint_{B(x,M^{j/2}\rho)\cap \D^j_{\mu}} dy |\tilde{\eta}^j(y)|\le \max_{B^j} \frac{\sum_{\Del\in B^j} \tilde{Y}_{\Del}}{\sharp B^j},\EEQ
where
\BEQ \tilde{Y}_{\Del}:=\sup_{\Del}|\tilde{\eta}^j|\EEQ
and the $B^j$ are a finite number (depending only on $d$) of subsets of cubes such that 
$\tilde{B}^j(x,\rho)\subset B^j\subset \tilde{B}^j(x,\rho)\cup\partial\tilde{B}^j(x,\rho).$

\item[(iii)]
By construction, see (\ref{eq:new-random-field}),
\BEQ \tilde{Y}_{\Del}\le \sum_{\Del'\in\tilde{B}^j(x,\rho)} e^{-c_0 d^j(\Del,\Del')} \sup|\eta^j_{\Del'}|. \label{eq:6.27}\EEQ
We have seen in (\ref{eq:suplele}) that $M^{j(1+d_{\phi})} \sup|\eta^j_{\Del'}|\lele C(|Z_{\Del'}|+1)$ if
$Z_{\Del'}\sim {\cal N}(0,1)$. Since the fields $(\eta^j_{\Del})_{\Del}$ are independent, we may by the above
cited Strassen theorem define a coupling of the field $\eta^j\big|_{\D^j_{\mu}}$ with i.i.d. standard Gaussian
variables $(Z_{\Del})_{\Del\in\D^j_{\mu}}$ in such a way that 
\BEQ M^{j(1+d_{\phi})} \sup|\eta^j_{\Del}|\le C(|Z_{\Del}|+1). \label{eq:coupling} \EEQ
Hence 
\BEQ M^{j(1+d_{\phi})} \sum_{\Del\in B^j} \tilde{Y}_{\Del}\le C\sum_{\Delta\in \tilde{B}^j(x,\rho)}(|Z_{\Del}|+1) \EEQ
-- note that the  bound in the right hand side does not depend on the choice of $B^j$ --  and
\BEQ \esper\left[ \psi\left( M^{j(1+d_{\phi})}
\fint_{B(x,M^{j/2}\rho)\cap\D'_{\mu}}dy |\tilde{\eta}^j(y)| \right)\right] \le  \esper 
\left[  \psi\left( \frac{C}{n} \sum_{\Del\in\tilde{B}^j(x,\rho)} (|Z_{\Del}|+1) \right)
 \right].
        \label{eq:6.29} \EEQ


We rewrite the expectation as an integral by integration by parts,

\BEQ \esper 
\left[  \psi\left( \frac{C}{n} \sum_{\Del\in B^j} (|Z_{\Del}|+1) \right)
 \right] = \int_0^{+\infty} dA\, \psi'(A)\,  \proba\left[\frac{C}{n} \sum_{\Del\in B^j} (|Z_{\Del}|+1) >A\right]
 +\psi(0).
\EEQ

 

 
 

 Finally, $\sum_{\Del\in{B}^j} |Z_{\Del}|$ is
a sum of $n$ independent copies of $|Z|$, where $Z\sim {\cal N}(0,1)$, to
which
we may apply standard large deviation arguments in a trivial setting, 
\BEA  \proba[\sum_{\Del\in{B}^j} |Z_{\Del}|>nA] & \le & \min\left( 1, \min_{t\ge 0} e^{-tnA} \esper[e^{t\sum_{\Del} |Z_{\Del}|}] \right) \le 
\min\left( 1, 2^n \min_{t\ge 0} e^{-tnA+nt^2/2} \right) \nonumber\\
&=& \min\left( 1, 2^n e^{-nA^2/2} \right)\le C e^{-n(A-C)_+^2/2}.  \label{eq:standard-LD} \EEA
Thus we may choose $\psi(A)=e^{cn(A-C)_+^2} {\bf 1}_{A>C}+{\bf 1}_{A\le C}$ so that 
 $\esper\left[\psi\left( \frac{C}{n}\sum_{\Del\in B^j} (|Z_{\Del}|+1) \right)\right]\lesssim 1$.
Collecting  (\ref{eq:mu}), (\ref{eq:6.21}) and (\ref{eq:6.29}), one obtains
by Markov's inequality 
\BEQ \proba[M^{j(1+d_{\phi})} \fint_{B(x,M^{j/2}\rho)} dy |\eta^j_{t_0}(y)|>A]\lesssim \frac{1}{\psi(A)}\lesssim 
e^{-cn(A-C)^2},\qquad 
A\ge C.\EEQ 

For each fixed $n\ge 1$, the set $\{B^j(x,\rho),\rho\ge 0 \ |\ \sharp B^j(x,\rho)=n\}\cup\{
B^j(x,\rho)\cup\partial B^j(x,\rho),\rho\ge 0 \ |\ \sharp B^j(x,\rho)\cup\partial B^j(x,\rho)=n\}$
consists of $0,1$ or $2$ elements.
Thus, using (\ref{eq:mu}),
\BEQ \proba[(\eta^j)^*(t_0,x)>AM^{-j(1+d_{\phi})}] \lesssim \min\left(1, \sum_{n\ge 1}  e^{-cn(A-C)_+^2} \right)
\lesssim
 e^{-c(A-C')_+^2}.\EEQ


Finally, we use a scaled version of (\ref{eq:fsharp-bounded}), 
\BEQ \sup_{B(x,M^{j/2})}(\eta^j_{t_0})^{\sharp}\lesssim \sup_{B(x,2M^{j/2})} |\eta^j_{t_0}|+(\eta^j_{t_0})^{\sharp}(x),\EEQ
from which we conclude that
\BEQ \proba[\sup_{B(x,M^{j/2})} (\eta^j_{t_0})^*>AM^{-j(1+d_{\phi})}]\le e^{-c(A-C)_+^2}. \EEQ 
In particular,
\BEQ \proba[\exists x\in\R^d\ |\ (\eta^j_{t_0})^*(x)=+\infty]\le \sum_{\Del\in\D^j} \proba[\sup_{\Del} (\eta^j_{t_0})^*=
+\infty]=0.\EEQ
 
 
 
\end{itemize}

\hfill \eop

%%%%%%%%%%%%%%%%%%%%%%%�

\subsection{Large deviations for the exponential of the noise}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%




We now turn to large deviation estimates for $(e^{\lambda M^j |\eta^j_{t_0}|})^*(x)$ and prove
the following result.


\begin{Theorem} \label{th:LD}
Let $j\in\N$, $\lambda>0$ and $t_0\in\R_+$. Then the function $x\mapsto (e^{\lambda M^j |\eta^j_{t_0}|})^*(x)$
is a.s. everywhere defined (i.e. finite). Furthermore, the following large deviation estimates holds for every
$x\in\R^d$,
\BEQ \proba[ \sup_{B(x,M^{j/2})} \ln (e^{\lambda M^j |\eta^j_{t_0}|})^*(x)>\eps A]\lesssim 
A^{-c\ln(A)}, \qquad A\ge 1\EEQ
where $\eps=\lambda M^{-jd_{\phi}}$ and $c>0$ is some constant.
\end{Theorem}



The proof is essentially similar to that of Lemma \ref{lem:LD-main}, except that it is based on large deviation
estimates for log-normal variables. We cite a result by Nagaev,  show how to apply it in our context, and prove
a few technical lemmas before
turning to the proof of Theorem \ref{th:LD}.



%%%%%%%%%%%%%%%%%%%%%%%%%%%

\subsubsection{Log-normal large deviations}

%%%%%%%%%%%%%%%%%%%%%%%%





\begin{Proposition} (see \cite{Nag2}, Corollary 1.8)
Let $X$ be a real-valued random variable such that $\esper[X]=0$ and $\esper[|X|^t]<\infty$ for some $t\ge 2$, and
$X_1,\ldots,X_n$ n i.i.d. copies of $X$, $S_n:=X_1+\ldots+X_n$. Then
\BEQ \proba[S_n> A]\lesssim n\esper[X^t {\bf 1}_{X>0}] A^{-t}+e^{-2(t+2)^{-2} e^{-t} A^2/n\esper[X^2]}. \label{eq:Nag}\EEQ
\end{Proposition}
Note that this general bound mixes the two regimes (\ref{eq:LD1}) and (\ref{eq:LD2}).



\begin{Corollary} \label{cor:Nagaev0}
Let $(Z_i)_{i=1,\ldots,n}$, $n\ge 1$ be i.i.d. standard Gaussian variables, and
let 
\BEQ S_n:=\sum_{i=1}^n \left( e^{\eps|Z_i|}-\esper[e^{\eps|Z_i|}] \right),\qquad \tilde{S}_n:=
\sum_{i=1}^n \left( e^{\eps|Z_i|}-1\right)\EEQ
where $0<\eps\ll 1$. Let finally $A\gg n\eps$ and $B\gg \ln(n)$. Then there exists a constant $c>0$ such that
\begin{itemize}
\item[(i)]
\BEQ \proba[S_n>A]\lesssim (A/\eps)^{-c\ln(A/\eps)} \label{eq:cor:Nag01}\EEQ
or equivalently
\item[(ii)]
\BEQ \proba[\ln\left(\frac{S_n}{\eps}\right)>B]\lesssim e^{-cB^2}; \label{eq:cor:Nag02}\EEQ
\item[(iii)]
\BEQ \proba[\ln S_n>A]\lesssim (A/\eps)^{-c\ln(A/\eps)}. \label{eq:cor:Nag03}\EEQ 
\end{itemize}
Furthermore, the same estimates (\ref{eq:cor:Nag01}), (\ref{eq:cor:Nag02}), (\ref{eq:cor:Nag03}) still hold
if one replaces $S_n$ by $\tilde{S}_n$. 
\end{Corollary}

{\bf Proof.}

Note that (ii) is equivalent to (i), and (iii) follows directly from (ii) since $e^A\gg n\eps$ and
$\frac{e^A}{\eps}\gg \frac{A}{\eps}$ if $A\gg n\eps$. Also, since $\esper[e^{\eps |Z_i|}]=1+O(\eps)$,
$\tilde{S}_n-S_n=O(n\eps)\ll A$, so the same estimates hold indifferently for $S_n$ or $\tilde{S}_n$ (up to
the choice of $c$).

Therefore we need only prove (i) for $S_n$. 
We apply the above Proposition with $X=e^{\eps|Z|}-\esper[e^{ \eps|Z|}]$,
where $Z$ is any of the variables $Z_i$. 
 One finds  
$\esper X^2=\esper[e^{2\eps|Z|}]-(\esper[e^{\eps|Z|}])^2\approx \eps^2$ and
$\esper[X^t {\bf 1}_{X>0}]<\esper[e^{ t \eps |Z|}]\le 2e^{t^2 \eps^2/2}.$ The bound (\ref{eq:Nag})
is close to optimal if one chooses $t=\half \ln(A/\eps)\gg 1$; we then find (using $\ln^2(A/\eps)\ll (A/\eps)^{\kappa}$
for all $\kappa>0$)
\BEQ \proba[S_n> A]\lesssim ne^{\frac{\eps^2}{8} \ln^2(A/\eps)} A^{-\half \ln(A/\eps)}+
e^{-\frac{c}{n} (A/\eps)^{\frac{3}{2}-\kappa}}. \label{eq:Nag02}\EEQ

\medskip

\begin{itemize}
\item[(i)]
 Assume first that $A\gtrsim e^{1/\eps}\gg 1$. 
The second term in the right-hand side of (\ref{eq:Nag02}) is then the smaller one since (for $\kappa<\half$)
\BEQ e^{-\frac{c}{n} (A/\eps)^{\frac{3}{2}-\kappa}}\le e^{-\frac{c}{n} (A/\eps)  (A/\eps)^{\half-\kappa}}
 \le  e^{-\half  \ln(A/\eps)^2}= (A/\eps)^{-\half \ln(A/\eps)} \le A^{-\half\ln(A/\eps)}.  \label{eq:6.25} \EEQ
As for the first term, it is bounded by $A^{-c\ln(A/\eps)}$ since (using $A\gg 1$)
\BEA e^{\frac{\eps^2}{8} \ln^2(A/\eps)} &\le & e^{\frac{\eps}{8} \ln^2(A/\eps)}\le e^{\frac{1}{8}
(\eps \ln(A)+1) \ln(A/\eps)} \nonumber\\
&\le & e^{\frac{1}{3} \ln(A)\ln(A/\eps)}=A^{\frac{1}{3}\ln(A/\eps)} \label{eq:6.36} \EEA
and 
\BEQ n\ll A/\eps=e^{\ln(A/\eps)} \lesssim A^{\frac{1}{12}\ln(A/\eps)}. \EEQ
 All together one has obtained
\BEQ \proba[S_n>A]\lesssim A^{-c\ln(A/\eps)}\lesssim (A/\eps)^{-c'\ln(A/\eps)},\qquad A\gtrsim e^{1/\eps}.
\label{eq:A>>}\EEQ 

 

\item[(ii)] We now assume that $A\lesssim  e^{1/\eps}$, implying that
$t=\half \ln(A/\eps)\lesssim \frac{1}{\eps}$. Then $A^{t}$ is not
necessarily small, so we must first improve our bound on $\esper[X^t {\bf 1}_{X>0}]$:
\BEA \esper[X^t {\bf 1}_{X>0}] &\le & 2\esper[(e^{\eps Z}-1)^t {\bf 1}_{Z>0}] \nonumber\\
&\lesssim & \int_0^{1/\eps} dz (e^{\eps z}-1)^t e^{-z^2/2} + \int_{1/\eps}^{+\infty} dz (e^{\eps z}-1)^t e^{-z^2/2} 
\nonumber\\
&\lesssim & \int_0^{+\infty} dz (\eps z)^t e^{-z^2/2} + \esper[e^{t\eps Z}] e^{-1/2\eps^2} \nonumber\\
&=& 2^{-\half} (\eps\sqrt{2})^t \Gamma(\frac{t}{2}+1) + e^{\half(\eps^2 t^2-\frac{1}{\eps^2})}. \label{eq:6.39} \EEA
Since $t\lesssim \frac{1}{\eps}$, we find 
$e^{\half(\eps^2 t^2-\frac{1}{\eps^2})}\lesssim e^{-c/\eps^2}\ll e^{-\frac{1}{\eps}|\ln\eps|}\lesssim e^{-t|\ln\eps|}
=\eps^t.$ Hence
\BEQ n\esper[X^t {\bf 1}_{X>0}] A^{-t}\lesssim n t^t (A/\eps)^{-c\ln(A/\eps)}\lesssim (A/\eps)^{-c'\ln(A/\eps)}.\EEQ
As for the second term, clearly $e^{-\frac{c}{n} (A/\eps)^{5/4}}\le (A/\eps)^{-c'\ln(A/\eps)}$ (see (\ref{eq:6.25})).  All together,
\BEQ \proba[S_n>A]\lesssim (A/\eps)^{-c\ln(A/\eps)},\qquad A\lesssim e^{1/\eps}.  \label{eq:A>} \EEQ 


\end{itemize}
\hfill \eop

{\bf Remark.} The above results are actually valid as soon as $A\gg n^{\kappa}\eps$ with $\kappa>\half$, as the
reader may easily check (choose $t=c\ln(A/\eps)$ with $c$ small enough and see how (\ref{eq:Nag02}) and (\ref{eq:6.25})
are modified). The condition $A\gg n^{\kappa}\eps$ may certainly be further improved with some extra effort.

\medskip

Corollary \ref{cor:Nagaev0} has the following generalization.

\begin{Corollary}[block large deviation estimates] \label{cor:Nagaev}
Let $Z:=\sum_{i'=1}^{n'} |\tilde{Z}_{i'}|$, where $(\tilde{Z}_{i'})_{i'=1,\ldots,n'}$, $n'\gg 1$ are i.i.d. standard
Gaussian variables; $n\in\N^*$ a multiple of $n'$, $Z_i,\ i=1,\ldots,n/n'$ i.i.d. copies of $Z$;  
\BEQ S_n:=\sum_{i=1}^{n/n'} \left( e^{\eps Z_i}-\esper[e^{\eps Z_i}] \right),\qquad \tilde{S}_n:=
\sum_{i=1}^{n/n'} \left( e^{\eps Z_i}-1\right)\EEQ
where $0<\eps\ll 1$ and $\eps n'\ll 1$. Let finally $A\gg n\eps$ and $B\gg \ln(n)$. Then there exists a constant $c>0$ such that
\begin{itemize}
\item[(i)]
\BEQ \proba[S_n>A]\lesssim (A/n'\eps)^{-c\ln(A/n'\eps)} \label{eq:cor:Nag1}\EEQ
or equivalently
\item[(ii)]
\BEQ \proba[\ln\left(\frac{S_n}{n'\eps}\right)>B]\lesssim e^{-cB^2}; \label{eq:cor:Nag2}\EEQ
\item[(iii)]
\BEQ \proba[\ln S_n>A]\lesssim (A/n'\eps)^{-c\ln(A/n'\eps)}. \label{eq:cor:Nag3}\EEQ 
\end{itemize}
Furthermore, the same estimates (\ref{eq:cor:Nag1}), (\ref{eq:cor:Nag2}), (\ref{eq:cor:Nag3}) still hold
if one replaces $S_n$ by $\tilde{S}_n$. 
\end{Corollary}

\medskip


{\bf Proof.} The result is exactly the one stated in Corollary \ref{cor:Nagaev0} if $n'=1$. We want to prove
the same kind of result for blocks of size $n'$. Standard large deviation arguments apply to $Z$, yielding
(see (\ref{eq:standard-LD})) $\proba[Z>A]\le ce^{-\frac{1}{2n'} (A-cn')_+^2}$, hence (letting as before
$X:=e^{\eps Z}-\esper[e^{\eps Z}]$), $\esper[e^{\eps Z}]=1+O(n'\eps)$,
$\esper[X^2]\lesssim \eps^2 \Var(Z)=O(n'\eps^2)$, and
\BEA \esper[X^t {\bf 1}_{X>0}]\le \esper[e^{t\eps Z}] & \lesssim & t\eps \int_0^{+\infty} e^{t\eps z} e^{-\frac{1}{2n'}(z-cn')_+^2} dz\nonumber\\
&\le & \int_0^{2cn'}  t\eps e^{t\eps z} dz +\int_{-\infty}^{+\infty} e^{t\eps z-\frac{1}{8n'}z^2} dz \nonumber\\
&\lesssim & e^{2Cn't\eps}+e^{Cn'(t\eps)^2/2}.\EEA
We set $t:=\half \ln(A/n'\sqrt{\eps})$ and distinguish two regimes according to whether $A\gtrless e^{1/(\eps\sqrt{n'})}$,
corresponding to $t\gtrless \frac{1}{\eps\sqrt{n'}}$. Thus
\BEQ e^{2Cn't\eps}=e^{Cn'\eps \ln(A/\eps\sqrt{n'})}\le c^{\ln(A/\eps\sqrt{n'})}\ll A^{\frac{1}{3}\ln(A/\eps\sqrt{n'})} \EEQ
instead of (\ref{eq:6.36}), and 
\BEA \esper[X^t {\bf 1}_{X>0}] &\le & \esper[(e^{\eps Z}-1)^t] \nonumber\\
&\lesssim & \int_0^{1/\eps} dz (e^{\eps z}-1)^t e^{-\frac{1}{2n'}(z-cn')^2_+} + \int_{1/\eps}^{+\infty}
dz (e^{\eps z}-1)^t e^{-\frac{1}{2n'}(z-cn')^2_+} \nonumber\\
&\lesssim & \int_0^{2cn'} dz (\eps z)^t + \int_0^{+\infty} dz (\eps z)^t e^{-z^2/8n'} +
\int_{1/\eps}^{+\infty} dz\, e^{t\eps z} \,  e^{-\frac{z^2}{8n'}-\frac{c}{n'\eps^2}} \nonumber\\
&\lesssim & (2\eps Cn')^{t+1}+ (Cn')^{\half(t+1)}\eps^t \Gamma(\frac{t}{2}+1) + e^{c(n'\eps^2 t^2-\frac{1}{n'\eps^2})}
\EEA
instead of (\ref{eq:6.39}).  Hence all estimates contained in the proof of Corollary \ref{cor:Nagaev0} hold if one
replaces $A/\eps$ by $A/n'\eps$ or $A/\sqrt{n'}\eps$. 


\hfill\eop


%%%%%%%%%%%%%%%%%%%%%%%%

\subsubsection{Mayer expansion}

%%%%%%%%%%%%%%%%%%





We also  need in the course of the proof of Theorem \ref{th:LD} a technical result which we choose to
state separately for the sake of clarity. In the sequel, $B^j$ is one of the
$\mu$-dependent subsets of cubes with $\tilde{B}^j(x,\rho)\subset B^j\subset \tilde{B}^j(x,\rho)\cup \partial\tilde{B}^j(x,\rho)$ introduced in section 6.2.

\begin{Lemma}["Mayer expansion"]  \label{lem:Mayer0}
Let $(z_{\Del})_{\Del\in \tilde{B}_j(x,\rho)}\in\R_+$ and $c>0$. Define
\BEQ y_{\Del}:=\sum_{\Del'\in \tilde{B}_j(x,\rho)} e^{-c_0 d^j(\Del,\Del')} z_{\Del'} \EEQ
(see eq. (\ref{eq:6.27})).
\begin{itemize}
\item[(i)] ("Mayer expansion")
Let
\BEQ S_{\del}((z_{\Del})):=\sum_{\Del\in \tilde{B}_j(x,\rho)}  \left( e^{e^{-c_0\del}\eps z_{\Del}} -1\right), \qquad
\del\ge 0 \label{eq:6.60} \EEQ
and 
\BEQ S_0((y_{\Del})):=\sum_{\Del\in B_j} \left(e^{\eps y_{\Del}}-1\right).\EEQ
Then
\BEQ 0\le  S_0((y_{\Del}))\le \sum_{m\ge 1} \frac{1}{(m-1)!} \sum_{\del_1<\ldots<\del_m} \prod_{p=1}^m S_{\del_p}
((z_{\Del})) \label{eq:Mayer01}\EEQ
where $\del_i,\ i=1,2,\ldots$ range among the set $\{d^j(\Del,\Del'),\Del,\Del'\in B^j\}$.


More generally, if $\del_{max}\in\R_+$, then
\BEA &&  S_0((y_{\Del}))\le \sum_{m\ge 1} \frac{1}{(m-1)!} \sum_{\del_{max}<\del_1<\ldots<\del_m} \prod_{p=1}^m S_{\del_p}
((z_{\Del})) \nonumber\\
&& \qquad +\sum_{m\ge 1} \sum_{m'=1}^{m-1} \frac{1}{(m-1-m')!} \sum_{ \del_1<\ldots<\del_{m'}\le\del_{max}< \del_{m'+1}<\ldots<\del_m}  S_0
((z_{\Del}))^{1+O(e^{-c_0})} \prod_{p=m'+1}^{m} S_{\del_p}((z_{\Del})) \nonumber\\
&& \qquad +\sum_{m\ge 1}  \sum_{\del_1<\ldots<\del_m\le\del_{max}}  S_0
((z_{\Del}))^{O(e^{-c\del_1})}. \label{eq:Mayer0-generalized} \EEA






\item[(ii)]
Let
\BEQ T((z_{\Del})):=\sum_{\Del\in \tilde{B}_j(x,\rho)}   e^{\eps z_{\Del}}\EEQ
and similarly
\BEQ T((y_{\Del})):=\sum_{\Del\in B_j}   e^{\eps y_{\Del}}.\EEQ
Then 
\BEQ T((y_{\Del}))\le \left(T((z_{\Del}))\right)^{1+O(e^{-c})}. \label{eq:Mayer2} \EEQ
\end{itemize}
\end{Lemma}

\medskip

Note that (by invariance by translation) $\del_i\in \{d^j(\Del,\Del'),\Del'\in\D^j\}=\{0<d_1<d_2<\ldots\}$
where $\Del$ is some arbitrary fixed cube, and $d_i\approx_{i\to\infty} i^{1/d}$. The indexation is easier if we
 choose a distorted distance instead of the Euclidean distance, i.e., if we define e.g. $|x-y|=\sup_{i=1,\ldots,d} |({\cal R}(x-y))_i|$
where ${\cal R}$ is a generic rotation, so that the set $\{\Del'\in \D^j\ |\ d^j(\Del,\Del')
=\del\}$ contains at most one element for $\Del$, $\del$ fixed, which we denote by $\Del(\del)$. The nickname
"Mayer expansion" refers to a common expansion of the free energy in equilibrium statistical physics where $e^{\beta{\cal H}}$,
${\cal H}$ being the local energy density, is expanded
into $\left(e^{\beta{\cal H}}-1\right)+1$, which is exactly what we do in (i).


\medskip
{\bf Proof.}

\begin{itemize}
\item[(i)] 
Let $a_{\Del}(\del):=e^{e^{-c_0\del}\eps z_{\Del}}-1$ $(\Del\in\tilde{B}^j(x,\rho))$    and $a_{\Del}:=a_{\Del}(0)=e^{\eps z_{\Del}}-1$.
Rewriting $S_0((y_{\Del}))$ in terms of the $z$-variables and expanding  the product of terms of the form $a_{\Del(\del)}(\del)+1$ yields
\BEA  &&S_0((y_{\Del})) =\sum_{\Del\in B^j}  \left[ \prod_{\del} (a_{\Del(\del)}(\del)+1) \right]-1= \sum_{\Del\in B^j} \sum_{m=1}^n \sum_{\del_1<\del_2<\ldots 
< \del_m} \prod_{p=1}^m a_{\Del(\del_p)}(\del_p) \nonumber\\
 && \le \sum_{m=1}^n \frac{1}{(m-1)!} \sum_{\del_1<\del_2<\ldots<\del_m} \sum_{\Del\in B^j}
a_{\Del(\del_1)}(\del_1) \nonumber\\ && \qquad \qquad   \left( \sum_{\Del_2\in \{\Del(\del_2),\ldots,\Del(\del_m)\}} a_{\Del_2}(\del_2) \left( 
\sum_{\Del_3\in \{\Del(\del_2),\ldots,\Del(\del_m)\}\setminus \Del_2 } a_{\Del_3}(\del_3) \left(
\cdots \right)\right)
\right)  \label{eq:Mayer-entretemps} \\
&\le &  \sum_{m=1}^n \frac{1}{(m-1)!} \sum_{\del_1<\del_2<\ldots<\del_m} \left( \sum_{\Del\in B^j}
a_{\Del}(\del_1) \right) \left( \sum_{\Del_2\in B^j} a_{\Del_2}(\del_2) \left( \cdots \right)
\right) \nonumber\\
&=& \sum_{m\ge 1} \frac{1}{(m-1)!} \sum_{\del_1<\ldots<\del_m} \prod_{p=1}^m S_{\del_p}((z_{\Del})).
\label{eq:6.52} \EEA

For the proof of (\ref{eq:Mayer0-generalized}), we fix $\del_{max}\ge 0$, start from (\ref{eq:Mayer-entretemps}) and
pick its $p$-th factor,
$A_p=\sum_{\Del_p\in\{\Del(\del_2),\ldots,\Del(\del_m)\}\setminus\{\Del_2,\ldots,\Del_{p-1}\} }
a_{\Del_p}(\del_p)$. If $\del_p>\del_{max}$ we bound $A_p$ by $S_{\del_p}((z_{\Del}))$ as before. Otherwise we use the
identity $(x-1)^{\kappa}\le x^{\kappa}-1$ ($x\ge 1,\kappa\ge 1$) and
H\"older's inequality to get  $a_{\Del_p}(\del_p)\le \left(e^{\eps z_{\Del_p}}-1\right)^{e^{-c_0\del_p}}=a_{\Del_p}^{e^{-c_0\Del_p}}$ and
\BEQ A_p\le (m-p+1) \left(\sum_{\Del\in B^j} a_{\Del}\right)^{e^{-c\del_p}}=(m-p+1)S_0((z_{\Del}))^{e^{-c_0\del_p}}.\EEQ
Finally, $\sum_p e^{-c_0\del_p}\le 1+e^{-c_0 d_1}+e^{-c_0 d_2}+\ldots=1+O(e^{-c_0}).$



\item[(ii)]

One finds (all sums or supremums in the next expressions range over subsets of $\tilde{B}^j(x,\rho)$, unless
otherwise stated)
\BEA \sum_{\Del\in B^j} e^{\eps y_{\Del}} & \le &\sum_{\Del\in B^j}
\prod_{\Del'} e^{e^{-c_0 d^j(\Del,\Del')} \eps z_{\Del'}}  \nonumber\\
&\le & \frac{1}{(n-1)!} \sum_{\Del_1} e^{\eps z_{\Del_1}} \left( \sum_{\Del_2\not=\Del_1} e^{e^{-c_0 d_1}\eps z_{\Del_2}}
\left( \sum_{\Del_3\not=\Del_1,\Del_2} e^{e^{-c_0 d_2} \eps z_{\Del_3}} \left(\cdots\right)\right)\right) \nonumber\\
&\le & \frac{1}{(n-1)!}  \left( \sum_{\Del_1} e^{\eps z_{\Del_1}}\right) \sup_{\Del_1} \left( \sum_{\Del_2\not=
\Del_1} e^{e^{-c_0 d_1}\eps z_{\Del_2}}\right) \cdots \nonumber\\
&\le & \frac{1}{(n-1)!}  \left( \sum_{\Del_1} e^{\eps z_{\Del_1}}\right) \cdot \left[ (n-1)
\left(\sum_{\Del_2} e^{\eps z_{\Del_1}}\right)^{e^{-c_0 d_1}} \right] \cdots
\EEA
(H\"older's inequality was used in the last line). 
The product of the prefactors in the last expression, $(n-1)(n-2)\cdots$ is exactly compensated by the factorial $\frac{1}{(n-1)!}$,
and there remains 
\BEQ \sum_{\Del\in B^j} e^{\eps y_{\Del}} \le \left(\sum_{\Del\in \tilde{B}^j(x,\rho)}
e^{\eps z_{\Del}} \right)^{1+O(e^{-c_0})}. \EEQ

\end{itemize}
\hfill \eop


\medskip

Again, this lemma has a block generalization. Roughly speaking, we  want to group together all cubes $\Del'$ at
distance $\del\approx 3^k$ of a given cube $\Del$ and sum over $k$, instead of summing over the $\del_i$'s which (as a
detailed computation proves)
increase too slowly with $i$ to give a converging series. Actually, we bother to do so only for
$\del>\del_{max}$, in a region where the exponential decay governs essentially the estimates; the value of 
$\del_{max}$ is fixed later in the text. In order to avoid blocks with "holes" and overlaps between
blocks, we introduce the following definitions. Let $\D^{j,k}$, $k\ge \log_3 \del_{max}$ be the set of blocks ${\bf \Del}^k=[3^k M^{j/2}k_1,3^k M^{j/2}(k_1+1)]\times\ldots
\times[3^k M^{j/2}k_d,3^k M^{j/2}(k_d+1)]$ of size $3^k$ included in $\tilde{B}^j(x,\rho)$. The $3^d-1$ blocks of size $3^k$, $[x_{\Del,1}+\eps_1 3^k M^{j/2}(k_1-\half),
x_{\Del,1}+\eps_1 3^k M^{j/2}(k_1+\half)]\times\ldots  [x_{\Del,d}+\eps_d 3^k M^{j/2}(k_d-\half) x_{\Del,d}+\eps_d 3^k M^{j/2}(k_d+\half) ]$, where $\vec{\eps}=(\eps_1,\ldots,\eps_d)\in\{-1,0,1\}^d\setminus\{0,\ldots,0\}$, are all situated at
a scaled distance  $\ge\del=3^k$ of $\Del$. We denote them by $\vec{\Del}(\vec{\del})$, where  $\vec{\del}:=(\del,\vec{\eps})$
is a composite index including both  the distance $\del$ and a discrete index $\vec{\eps}$ ranging in a fixed finite set.
Then, for smaller distances $\del<3^k$, we set $\vec{\Del}(\del)=\Del(\del)$ as in the previous lemma, and 
$\vec{\del}=\del$ simply.
All together the blocks $(\vec{\Del}(\vec{\del}))_{\vec{\del}}$, $\vec{\del}=(\del,\vec{\eps})$ ($\del\ge 3^k$) or
$\del$ ($\del<3^k$) define for every fixed cube $\Del$ a partition of $\R^d$.
We choose in the sequel some arbitrary total ordering $<$ of the  indices $\vec{\del}$ such that 
$(\vec{\del}=(\del,\vec{\eps})$  or $\del, \ \vec{\del}'=(\del',\vec{\eps}')$ or $\del',\ \del<\del')\Rightarrow 
\vec{\del}<\vec{\del'}.$


\begin{Lemma}[block "Mayer expansion"]  \label{lem:Mayer}
Let $(z_{\Del})_{\Del\in B_j}\in\R_+$ and $c>0$. Define as in the previous lemma
\BEQ y_{\Del}:=\sum_{\Del'\in \tilde{B}^j(x,\rho)} e^{-c_0 d^j(\Del,\Del')} z_{\Del'},\quad S_0((y_{\Del})):=
\sum_{\Del\in B^j} \left(e^{\eps y_{\Del}}-1 \right),\quad S_0((z_{\Del})):=
\sum_{\Del\in \tilde{B}^j(x,\rho)} \left(e^{\eps z_{\Del}}-1 \right) \EEQ and
let, for $k\in\N$,
\BEQ \vec{S}_{3^k}((z_{\Del})):=\sum_{\Del^k\in\D^{j,k}}  \left( e^{e^{-c_0 3^k}\eps \sum_{\Del\in\Del^k\cap B^j} z_{\Del}} -1\right), \qquad
k \ge 0, \EEQ
a block version of (\ref{eq:6.60}) distinguished by the boldface letter. Choose some value of  $\del_{max}$ and order the 
indices $\vec{\del}$ as indicated above.
Then

\BEA &&  S_0((y_{\Del}))\le \sum_{m\ge 1} \frac{1}{(m-1)!} \sum_{\del_{max}<\vec{\del}_1<\ldots<\vec{\del}_m} \prod_{p=1}^m \vec{S}_{\del_p}
((z_{\Del})) \nonumber\\
&& \qquad +\sum_{m\ge 1} \sum_{k=1}^{m-1} \sum_{ \del_1<\ldots<\del_{m'}\le\del_{max}< \vec{\del}_{m'+1}<\ldots<
\vec{\del}_m}  S_0
((z_{\Del}))^{1+O(e^{-c_0})} \prod_{p=1}^k \vec{S}_{\del_p}((z_{\Del})) \nonumber\\
&& \qquad +\sum_{m\ge 1}  \sum_{\del_1<\ldots<\del_m\le \del_{max}}  S_0
((z_{\Del}))^{1+O(e^{-c_0})}. \label{eq:Mayer-generalized} \EEA

\end{Lemma}


{\bf Proof.} If $\vec{\Del}$ is a block of size $3^k$ for some $k\ge 0 $, we let $a_{\vec{\Del}}(\del):=
e^{e^{-c\del}\eps \sum_{\Del\in \vec{\Del}} z_{\Del}}-1$. Thus
\BEA  S_0((y_{\Del})) & \le &\sum_{\Del\in B^j} \left[ \prod_{\vec{\del}} a_{\vec{\Del}(\vec{\del})}(\del)+1\right]-1 \nonumber\\
&=& \sum_{\Del\in B^j} \sum_{m\ge 1} \sum_{\vec{\del}_1>\vec{\del}_2>\ldots>\vec{\del}_m} \prod_{p=1}^m
a_{\vec{\Del}(\vec{\del}_p)}(\del_p).\EEA
We then expand as in (\ref{eq:Mayer-entretemps}) and (\ref{eq:6.52}), and forget the unnecessary factorials
in the denominator (which would require a short discussion in any case since there is no symmetry factor for terms
belonging to blocks with different sizes). \hfill \eop

\hfill \eop

% Note that  (\ref{eq:etaYtildeY}) may be modified as follows: letting
% $\tilde{B}^j_m(x,\rho)$ be a decimation of $B^j$ of order $m$, namely, the union of all 
% $\Del=[k_1 M^{j/2},(k_1+1)M^{j/2}]\times\ldots\times[k_d M^{j/2},(k_d+1)M^{j/2}]\in B^j$
% such that $k_1,\ldots,k_d$ are multiples of $m\in\N^*$, we get
%\BEQ \proba[\fint_{B(y,M^{j/2}\rho)} dy e^{\lambda M^j |\eta^j(t_0,y)|}>A ] \lesssim m^d
% \proba[\sum_{\Del\in\tilde{B}^j_m(x,\rho)} e^{\eps |\tilde{Y}_{\Del}|} >c'_{\rho} n' A] \label{eq:decimation} \EEQ
% with $n'=\sharp \tilde{B}^j_m(x,\rho)$, $c'_{\rho}=1+O(\frac{1}{n'})$ as for $c_{\rho}$.
%Thus one  may actually assume that the constant $c'$ in (\ref{eq:c'}) is large enough (later on we actually
% need $e^{c'}\gg 4$), at the price of having to change some numerical constants.
 


%%%%%%%%%%%%%%%%%%%%%%%%

\subsubsection{Proof of Theorem \ref{th:LD}}

%%%%%%%%%%%%%%%%%%%%%%%%%%%



We shall  use several times the following elementary lemma.

\begin{Lemma} \label{lem:elementary}
Let $X_1,\ldots,X_n$, $n\ge 1$ be real-valued random variables, and $\lambda_1,\ldots,\lambda_n\in\R_+$ such
that $\lambda_1+\ldots+\lambda_n=1$. Then
\BEQ \proba[\lambda_1 X_1+\ldots+\lambda_n X_n>A]\le \proba[\sup_{p=1,\ldots,n} X_p>A]\le \sum_{p=1}^n \proba[X_p>A].\EEQ
\end{Lemma}




{\bf Proof of Theorem \ref{th:LD}.} 
The general scheme of the proof is the same as that of  Lemma \ref{lem:LD-main}. 
Applying Proposition \ref{prop:Slepian} with $\phi(\eta^j_{t_0})=\psi\left( \frac{1}{\eps}
\fint_{B(x,M^{j/2}\rho)\cap\D^j_{\mu}}dy \left(e^{\lambda M^j |\eta^j_{t_0}(y)|}-1\right) \right)$ where $\psi:\R_+\to\R$ is any convex, increasing function yields instead of (\ref{eq:6.21})
\BEQ \esper\left[\psi\left( \frac{1}{\eps}
\fint_{B(x,M^{j/2}\rho)\cap\D^j_{\mu}}dy \left( e^{\lambda M^j |\eta^j_{t_0}(y)|}-1\right) \right)\right] \le 
\esper\left[\psi\left( \frac{1}{\eps}
\fint_{B(x,M^{j/2}\rho)\cap\D^j_{\mu}}dy  \left(e^{\lambda M^j |\tilde{\eta}^j(y)|}-1\right) \right)\right].
\nonumber\\
  \EEQ

Then we bound the last integral by sums of local suprema $\tilde{Y}_{\Del}=\sup_{\Del}|\tilde{\eta}^j(y)|$,
\BEQ \fint_{B(x,M^{j/2}\rho)\cap \D^j_{\mu}} dy  e^{\lambda M^j |\tilde{\eta}^j(y)|} \le  \max_{B^j}  \frac{\sum_{\Del\in B^j} e^{\lambda M^j \tilde{Y}_{\Del}} }{\sharp B_j} 
 \label{eq:6.25bis}
\EEQ
where $B^j$ is a union of cubes ranging over a finite set as in subsections 6.1 and 6.2, and $\sharp B_j\le
\sharp \tilde{B}^j(x,\rho)=n$. 
Eq. (\ref{eq:6.27}), (\ref{eq:coupling}) imply 
\BEQ \lambda M^j \tilde{Y}_{\Del}\le C\eps \sum_{\Del'\in\tilde{B}^j(x,\rho)} e^{-c_0 d^j(\Del,\Del')} (|Z_{\Del'}|+1)
\le C'\eps + C'\eps \sum_{\Del'\in \tilde{B}^j(x,\rho)} e^{-c_0 d^j(\Del,\Del')} |Z_{\Del'}|. \label{eq:6.79} \EEQ


Finally, we use the formula
\BEA  \esper\left[\psi\left( \frac{1}{n\eps} \sum_{\Del\in B^j} \left(e^{\lambda M^j \tilde{Y}_{\Del}}-1\right)\right)\right]
& =& \int_0^{+\infty} dA \,  \psi'(A) \, \proba[\sum_{\Del\in B^j} e^{\lambda M^j \tilde{Y}_{\Del}}>n(1+\eps A)] +\psi(0) \EEA
Clearly, $e^{-C'\eps}(1+\eps A)\ge (1-C'\eps)(1+\eps A)\ge 1+c\eps A$ if $A\gg 1$. Hence, assuming
$\supp\psi'\in [C,+\infty]$ with $C$ large enough, one finds
\BEQ \esper\left[\psi\left( \frac{1}{n\eps} \sum_{\Del\in B^j} \left(e^{\lambda M^j \tilde{Y}_{\Del}}-1\right)\right)\right]
\le  \int_0^{+\infty} dA \, \psi'(A) \,  \proba[\sum_{\Del\in B^j} e^{c\eps Y_{\Del}}>n(1+\eps A)]+\psi(0) \EEQ
with
\BEQ Y_{\Del}:=\sum_{\Del'\in \tilde{B}^j(x,\rho)} e^{-c_0 d^j(\Del,\Del')} |Z_{\Del'}|\EEQ
(see (\ref{eq:coupling}) and (\ref{eq:6.79})).


Below we prove that 
\BEQ \proba[ \sum_{\Del\in B^j} e^{\eps Y_{\Del}}
 >n(1+\eps A)] \le e^{-c\ln^2(nA)}, \qquad A\gg 1. \label{eq:below}\EEQ
 Hence $\proba[ \sum_{\Del\in B^j} e^{\eps Y_{\Del}}
 >n(1+\eps A)] \lesssim \psi^{-2}(A)$ with
 \BEQ \psi(A):=e^{c\ln^2(n(A-C))} {\bf 1}_{A>C(1+\frac{1}{n})}+e^{c\ln^2(C)} {\bf 1}_{A<C(1+\frac{1}{n})}.\EEQ
 Therefore,
\BEQ\esper\left[\psi\left( \frac{1}{n\eps}\sum_{\Del\in B^j} \left( e^{\lambda M^j \tilde{Y}_{\Del}} - 1 \right) \right)\right] 
\lesssim [-\frac{1}{\psi(A)}]_{0}^{+\infty}+\psi(0)=O(1)\EEQ
and, by Markov's inequality,
\BEQ \proba[\frac{1}{n} \sum_{\Del\in B^j} e^{\lambda M^j \tilde{Y}_{\Del}}>1+\eps A] \lesssim \frac{1}{\psi(A)} \lesssim e^{-c\ln^2(nA)}, \qquad A\gg 1 \EEQ
so
\BEQ \proba\left[ \ln\left(\frac{1}{n} \sum_{\Del\in B^j} e^{\lambda M^j \tilde{Y}_{\Del}} \right)>\eps A\right]=
\proba\left[ \frac{1}{n} \sum_{\Del\in B^j} e^{\lambda M^j \tilde{Y}_{\Del}}>e^{\eps A} \right]\lesssim e^{-c\ln^2(nA)}.\EEQ
(Note that,  if $\eps A\gg 1$, one obtains in fact a Gaussian queue distribution,
\BEQ \proba\left[ \ln\left(\frac{1}{n} \sum_{\Del\in B^j} e^{\lambda M^j \tilde{Y}_{\Del}} \right)>\eps A\right]\lesssim
e^{-c(\eps A+\ln \frac{n}{\eps})^2}\le e^{-c(\eps A)^2}, \EEQ
with a very bad coefficient $\eps$ in front of $A$ however.)
Thus
\BEQ \proba\left[\ln (e^{\lambda M^j |\eta^j_{t_0}|})^*(x)>\eps A\right] \lesssim \sum_{n\ge 1}  e^{-
c\ln^2(nA)}
\lesssim A^{-c\ln(A)}.\EEQ



\vskip 2cm

It remains to prove the key estimate (\ref{eq:below}). Recall $Y_{\Del}=\sum_{\Del'\in \tilde{B}^j(x,\rho)} e^{c_0 d^j(\Del,\Del')} |Z_{\Del'}|$,
where $(Z_{\Del})_{\Del}$ are i.i.d. standard Gaussian variables.
It turns out that there are four different large deviation regimes, according to the value of
$S_0((|Z_{\Del}|)_{\Del})=\sum_{\Del\in B^j} \left(e^{\eps |Z_{\Del}|}-1\right)$,
written as $\tilde{S}_n$ in Corollary \ref{cor:Nagaev},  or
$S_0((Y_{\Del})_{\Del})=\sum_{\Del\in B^j} \left(e^{\eps Y_{\Del}}-1\right)$
(see Lemma \ref{lem:Mayer}). By assumption 
\BEQ \sum_{\Del\in B^j} e^{\eps Y_{\Del}} >   n(1+\eps A) \EEQ
 with $A\gg 1$; in other terms,
 \BEQ S_0((Y_{\Del})_{\Del})> n\eps A.\EEQ

\begin{itemize}
\item[(i)]
(Gaussian regime) Assume $S_0((|Z_{\Del}|)_{\Del})=\tilde{S}_n\le 2$. Then $\eps |Z_{\Del}|=O(1)$ for all $\Del$, hence $\eps Y_{\Del}=O(1)$
too,  so $ S_0((Y_{\Del})_{\Del})
\lesssim S_0((|Z_{\Del}|)_{\Del})\approx \eps \sum_{\Del\in\partial B^j} |Z_{\Del}|$. Therefore
\BEQ   \proba[\sum_{\Del\in B^j} e^{\eps Y_{\Del}} > n(1+\eps A)] \le 
\proba[S_0((Y_{\Del})_{\Del})>n\eps A]\le \proba[\sum_{\Del\in\tilde{B}^j(x,\rho)} |Z_{\Del}|>cnA]\lesssim
e^{-c'n(A-C)^2}. \EEQ
Clearly this last quantity is bounded by $e^{-c''\ln^2(nA)}$ for $A$ large enough.

\item[(ii)] (very large deviation regime) Assume $S_0(Y_{\Del})_{\Del})\gg n^{1+O(e^{-c_0})}$, or equivalently
$A\eps\gg n^{O(e^{-c_0})}$. Then  the "Mayer expansion"
(see Lemma \ref{lem:Mayer0} (i)) is not needed; we use Lemma \ref{lem:Mayer0} (ii) and find successively
$S_0((Y_{\Del})_{\Del})\approx T((Y_{\Del})_{\Del})\le \left( T((|Z_{\Del}|)_{\Del}) \right)^{1+O(e^{-c_0})}$, so $T((|Z_{\Del}|)_{\Del}))\gg n$, hence again $T((|Z_{\Del}|)_{\Del})) \approx
S_0((|Z_{\Del}|)_{\Del}))$. All together we have found $S_0((Y_{\Del})_{\Del}) \lesssim
\left( S_0((|Z_{\Del}|)_{\Del}) \right)^{1+O(e^{-c_0})}$.

So we may apply Corollary \ref{cor:Nagaev0}, to the result that
\BEQ \proba[\sum_{\Del\in B^j} e^{\eps Y_{\Del}} > n (1+\eps A)] \le
\proba[\tilde{S}_n>(n(1+\eps A))^{1/(1+O(e^{-c_0}))}]\lesssim e^{-c''\ln^2(nA)}. \EEQ


\item[(iii)] Assume $n\eps\lesssim 1$ and $A\eps\lesssim n^{O(e^{-c_0})}$. Then we use the generalized block "Mayer expansion" (\ref{eq:Mayer-generalized}) with $\del_{max}=0$:
\BEQ \proba[ S_0((Y_{\Del})_{\Del})> n\eps A] \le \proba\left[\sum_{m\ge 1}  \sum_{ \vec{\del}_1>\ldots>\vec{\del}_m} \prod_{p=1}^m \vec{S}_{\del_p}
((|Z_{\Del}|)_{\Del}) >n\eps A\right].\EEQ
Since $\sum_{m\ge 1}\sum_{\vec{\del}_1>\ldots>\vec{\del}_m} e^{-\frac{c_0}{2}(\del_1+\ldots+\del_m)}
\approx \sum_{m\ge 1} \frac{1}{m!} \sum_{\vec{\del_1},\ldots,\vec{\del}_m} e^{-\frac{c_0}{2}(\del_1+\ldots+\del_m)}
\approx 1$, we get by Lemma \ref{lem:elementary}
\BEA  \proba[ S_0((Y_{\Del})_{\Del})> n\eps A]  &\le &  \sum_{m\ge 1}  \sum_{\vec{\del}_1<\ldots<
\vec{\del}_m}
\proba\left[\prod_{p=1}^m \frac{\vec{S}_{\del_p}((|Z_{\Del}|)_{\Del})}{n\eps e^{-c_0\del_p}}> \prod_{p=1}^m
\left(e^{\frac{c_0}{2}\bar{\del}} (n\eps)^{\frac{1}{m}-1} A^{1/m} \right) \right] \nonumber\\
&\le & \sum_{m\ge 1}\sum_{\vec{\del}_1<\ldots<\vec{\del}_m} \sum_{p=1}^m 
\proba\left[\frac{\vec{S}_{\del_p}((|Z_{\Del}|)_{\Del})}{n\eps e^{-c_0\del_p}}> 
e^{\frac{c_0}{2}\bar{\del}} (n\eps)^{\frac{1}{m}-1}  A^{1/m}\right]
 \nonumber\\ \EEA
 where $\bar{\del}:=\frac{1}{m}(\del_1+\ldots+\del_m)$. Note that the expression $\vec{S}_{\del_p}$ is a sum
 over blocks of size $n'\approx \del_p^d\ll (\eps e^{-c_0\del_p})^{-1}$, and $e^{\frac{c_0}{2}\bar{\del}} (n\eps)^{\frac{1}{m}-1}  A^{1/m} \gg 1$, hence we are in the large deviation regime studied in Corollary \ref{cor:Nagaev}. Also,
 \BEQ n e^{\frac{c_0}{2}\bar{\del}} (n\eps)^{\frac{1}{m}-1}  A^{1/m}\ge ne^{\frac{c_0}{2}\bar{\del}}(n\eps)^{-\half}
 \ge e^{\frac{c_0}{2}\bar{\del}} A^{\half} n^{\half-O(e^{-c_0})}  \label{eq:6.94} \EEQ
 if $m\ge 2$. For $m=1$ the estimates of Corollary \ref{cor:Nagaev} give directly a log-normal queue, so we
 sum over $m\ge 2$. By construction $\del_m\ge 3^{m/(3^d-1)}$, so $\bar{\del}>\frac{\del_m}{m}\gtrsim \sqrt{\del_m}$
 and $n'\approx \del_p^d\le \del_m^d\ll e^{\frac{c_0}{2}\bar{\del}}$. Hence 
 \BEQ \proba\left[\frac{\vec{S}_{\del_p}((|Z_{\Del}|)_{\Del})}{\eps e^{-c_0\del_p}}> 
n e^{\frac{c_0}{2}\bar{\del}} (n\eps)^{\frac{1}{m}-1}  A^{1/m}\right] \lesssim e^{-c\ln^2 \left(\frac{n}{n'} e^{c_0\bar{\del}}
A \right)} \lesssim e^{-c\ln^2 \left(n e^{\frac{c_0}{2}\bar{\del}}
A \right)} \lesssim e^{-c''\bar{\del}^2} (nA)^{-c''\ln(nA)}.\EEQ

Let $V_m(r):=\sharp\{(\del_1,\ldots,\del_m)\ |\ \bar{\del}<r\}=\sharp\{(\del_1,\ldots,\del_m)\ |\ \sum_{p=1}^m \del_p<mr\}$: clearly $V_m(r)\lesssim \sharp\{(i_1,\ldots,i_m)\in \N^m\ |\ i_1+\ldots+i_m=mr\}$, hence  
\BEQ  V_m(r) \lesssim  \int_0^{+\infty} dx_1\ldots dx_m {\bf 1}_{\sum_p x_p<mr} =\frac{(mr)^m}{m!}\lesssim (Cr)^m.\EEQ

Thus  (with an extra factor $\frac{1}{m!}$ due to the ordering $\vec{\del}_1>\ldots>\vec{\del}_m$)
\BEA  \sum_{m\ge 2} \sum_{\vec{\del}_1>\ldots>\vec{\del}_m} \proba\left[ \frac{\vec{S}_{\del_p}((|Z_{\Del}|)_{\Del}) }{n\eps e^{-c_0\del_p}}> 
e^{\frac{c_0}{2}\bar{\del}} (n\eps)^{\frac{1}{m}-1}  A^{1/m}\right] & \lesssim&  \sum_m \frac{1}{m!}  (nA)^{-c\ln(nA)}
\sum_{r=0}^{+\infty} (V_m(r)-V_m(r-1))e^{-cr^2} \nonumber\\ &\le&   (nA)^{-c\ln(nA)} 
\sum_{r=0}^{+\infty} e^{-cr^2} \sum_m \frac{V_m(r)}{m!} \nonumber\\
&\lesssim & (nA)^{-c\ln(nA)}  \EEA



\item[(iv)] Finally, assume $n\eps\gtrsim 1$ and $A\eps\lesssim n^{0(e^{-c_0})}$, and apply the generalized "Mayer expansion"
with $\del_{max}=\frac{8}{c_0}\ln(n\eps)$ defined in such a way that $e^{\frac{c_0}{8}\del}\ge n\eps$
for $\del\ge\del_{max}$.
Since 
\BEQ  \sum_{m\ge 1}\sum_{m'=0}^m  \sum_{\del_1<\ldots<\del_{m'}< \del_{max}\le \vec{\del}_{m'+1}<\ldots<\vec{\del}_m} e^{-\frac{c_0}{2}(\del_{m'+1}+\ldots+\del_m)} 
 \lesssim \sharp{\cal P}(\{1,\ldots,\del_{max}\})=2^{\del_{max}}=(n\eps)^{2\ln(2)/c_0}, \label{eq:part} \EEQ
 the generalized "Mayer expansion", together with Lemma \ref{lem:elementary}, yield
\BEA &&  \proba[ S_0((Y_{\Del})_{\Del})> n\eps A]  \le
  \sum_{m\ge 1} \sum_{m'=0}^m    \sum_{\del_1<\ldots<\del_{m'}\le \del_{max}< \vec{\del}_{m'+1}<\ldots<\vec{\del}_m}
  \nonumber\\ && \qquad \qquad  
\proba\left[ S_0((|Z_{\Del}|)_{\Del})^{1+O(e^{-c_0})} \prod_{p=m'+1}^{m}
 \frac{\vec{S}_{\del_p}((|Z_{\Del}|)_{\Del})}{n\eps e^{-c_0\del_p}} > (n\eps)^{1-\frac{2\ln(2)}{c_0}}  A \prod_{p=m'+1}^m \frac{e^{\frac{c_0}{2}\bar{\del}}}{n\eps} \right] \nonumber\\
&&\le \sum_{m\ge 1} \left\{ \sum_{\del_1<\ldots<\del_m\le\del_{max}} \proba\left[\frac{S_0((|Z_{\Del}|)_{\Del})^{1+O(e^{-c_0})}}{\eps}>\eps^{-2\ln(2)/c_0} n^{1-\frac{2\ln(2)}{c_0}} A \right]  \right. \label{eq:6.100}\\
&&\left.
+\sum_{m'=0}^{m-1} \sum_{\del_1<\ldots<\del_{m'}\le\del_{max}< \vec{\del}_{m'+1}<\ldots<\vec{\del}_m}
\left(\proba\left[\frac{S_0((|Z_{\Del}|)_{\Del})^{1+O(e^{-c_0})}}{\eps}>\eps^{-2\ln(2)/c_0} n^{1-\frac{2\ln(2)}{c_0}} e^{\frac{c_0}{4}\bar{\del}} A \right]   \right.\right.\nonumber\\ && \left.\left. \qquad \qquad \qquad\qquad +
\sum_{p=m'+1}^m  \proba\left[  \frac{\vec{S}_{\del_p}((|Z_{\Del}|)_{\Del})}{\eps e^{-c_0\del_p}}  >  \frac{e^{\frac{c_0}{4}\bar{\del}}}{\eps} \right] \right)\right\} \label{eq:6.101}
 \EEA
 where $\bar{\del}:=\frac{1}{m-m'} (\del_{m'+1}+\ldots+\del_m)$. 

 
For $c_0$ large enough (recall $c_0$ has been multiplied by $m_0$, thus it suffices to choose $m_0$ large enough)
\BEQ n^{1-\frac{2\ln(2)}{c_0}} \eps^{-2\ln(2)/c_0} e^{\frac{c_0}{4}\bar{\del}} A\gtrsim e^{\frac{c_0}{4}\bar{\del}}
An^{\kappa} \gtrsim n^{\kappa} \EEQ
with $\kappa>\half$, and
\BEQ \frac{e^{\frac{c_0}{4} \bar{\del}}}{\eps}\ge \max\left(n, \frac{e^{\frac{c_0}{6} \bar{\del}} (n\eps)^{2/3}}{\eps}
\right) \gtrsim \max\left(n, e^{\frac{c_0}{6}\bar{\del}} A^{\frac{1}{3}} n^{\frac{2}{3}-O(e^{-c_0})} \right).\EEQ
Thus we are in the large deviation regime (see remark after Corollary \ref{cor:Nagaev0}, and Corollary \ref{cor:Nagaev}),
and the lower bounds are as in (\ref{eq:6.94}), yielding a bound $O((nA)^{-c\ln(nA)})$ for the sum (\ref{eq:6.101})
over $m'$ and $\del_{m'+1},\ldots,\del_m$. As for the first sum  over $\del_1<\ldots<\del_m\le\del_{max}$ in
(\ref{eq:6.100}), or $\del_1<\ldots<\del_{m'}$ in (\ref{eq:6.101}), it produces as in (\ref{eq:part}) a supplementary
multiplicative factor  of order $(n\eps)^{2\ln(2)/c_0}$, of no  incidence on the result since
$(n\eps)^{2\ln(2)/c_0} (nA)^{-c\ln(nA)}\le n^{2\ln(2)/c_0} (nA)^{-c\ln(nA)}
\lesssim (nA)^{-c'\ln(nA)}.$
 
 
\end{itemize}

\hfill\eop



%%%%%%%%%%%%%%%%%%%%%%%

\subsection{Proof of Theorem \ref{th:LD1}}

%%%%%%%%%%%%%%%%%%%%%%%%��




We may now finally prove Theorem \ref{th:LD1}. 
Let 
\BEQ F(\del t,[0,t]):= M^{-j}\del t \sum_{p=0}^{\lfloor t/\del t\rfloor} (e^{-M^{-j} \del t})^p \ln \left[
\left(e^{\lambda M^{j} \fint_{t-p\del t}^{t-(p-1)\del t} |\eta^j(s)| ds} \right)^*(x)\right],\EEQ
and 
\BEQ c_{\del t}:=\left[ M^{-j}\del t \sum_{p=0}^{\lfloor t/\del t\rfloor} (e^{-M^{-j} \del t})^p \right]^{-1}
\gtrsim 1.\EEQ

Clearly,
\BEQ F(\del t,[0,t])\le M^{-j}\del t \sum_{p=0}^{\lfloor t/\del t\rfloor} (e^{-M^{-j} \del t})^p \ln \left[
\left(e^{\lambda M^{j} ||\tilde{\eta}^j||_{\infty, [t-p\del t,t-(p-1)\del t]} }  \right)^*(x)\right]\EEQ
where $||\tilde{\eta}^j||_{\infty, [t-p\del t,t-(p-1)\del t]}(x):=\sup_{s\in [t-p\del t,t-(p-1)\del t]}
|\eta^j(s,x)|.$ Thus
\BEA   c_{\del t} F(\del t) & \le & \sup_p (e^{-M^{-j} \del t})^p \ln \left[
\left(e^{\lambda M^{j} ||\tilde{\eta}^j||_{\infty, [t-p\del t,t-(p-1)\del t]} }  \right)^*(x)\right] \nonumber\\
& \le & \sup_{q\in\N} e^{-q} \ln \left[
\left(e^{\lambda M^{j}  ||\tilde{\eta}^j||_{\infty, [t-q M^j,t-(q-1) M^j]} }  \right)^*(x)\right]
.\EEA

   The estimates we developed for $|\eta^j_{t_0}|$ in Theorem \ref{th:LD}  extend to 
$||\tilde{\eta}^j||_{\infty, [t-qM^j,t-(q-1)M^j]}(x)$ by using the BTIS inequality once again.
Hence 
\BEA \proba[F(\del t,[0,t])>\eps A] & \lesssim& \sum_q \proba\left [
\ln \left[
\left(e^{\lambda M^{j} ||\tilde{\eta}^j||_{\infty, [t-qM^j,t-(q-1)M^j]} }  \right)^*(x)\right] >c\eps
e^q A\right] \nonumber\\
&\lesssim & \sum_{q=0}^{+\infty} \left( e^q A \right)^{-c' \ln \left( e^q A \right)}
\lesssim A^{-c'' \ln A}\EEA
by Theorem \ref{th:LD}.


Now, by H\"older's inequality,
\BEA  F(\del t,[0,t]) & \le &  \half M^{-j}\del t \sum_{p=0}^{\lfloor t/\del t\rfloor} (e^{-M^j \del t})^p  \left\{\ln \left[
\left(e^{\lambda M^j \fint_{t-p\del t}^{t-(p-1/2)\del t} |\eta^j(s)| ds} \right)^*(x)\right]+ \ln \left[
\left(e^{\lambda M^j \fint_{t-(p-1/2)\del t}^{t-(p-1)\del t} |\eta^j(s)| ds} \right)^*(x)\right] \right\} \nonumber\\
& \le &  e^{M^j \del t/2} F(\del t/2,[0,t])\EEA
so $|||\eta^j|||_{\lambda,j}(\R_+,x)\le \limsup_{\del t\to 0,t\to +\infty} \frac{1}{\lambda} F(\del t,[0,t])$
by (\ref{eq:etaj1}), and by monotone convergence
we get $\proba[|||\eta^j|||_{\lambda,j}(\R_+,x)> A M^{-jd_{\phi}}]\le \limsup_{\del t\to 0,t\to +\infty} \proba[F(\del t,[0,t])>
\eps A]\lesssim A^{-c\ln A}.$ 

\medskip


The estimates for $||| M^{j/2} \frac{\eta^j(.,.+\del x)-\eta^j(.,.)}{|\del x|} |||_{\lambda,j}(\R_+,x)$ are proved in the same way since $M^{j/2}\frac{\eta^j(.,.+\del x)-\eta^j(.,.)}{|\del x|}=O((M^{-j})^{1+d_{\phi}})$ scales like 
$\eta^j_t(x)$ (see \ref{eq:5.20bis}).

\hfill \eop



%Here we shall be helped by the fact that $M^j\eta^j(t,x)=O(M^{-jd_{\phi}})$ is extremely small for $j$ large,
%hence $e^{4\lambda M^j\eta^j(t,x)}\simeq 1+4\lambda M^j\eta^j(t,x)$ is essentially Gaussian.




\bigskip
{\bf Acknowledgements.} We wish to express our thanks to S. Benachour and G. Barles for useful discussions.



%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

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\end{document}





























































1307.1980.extracted_bib
\begin{thebibliography}{99}




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