Local time of a diffusion in a stable L�vy environment
Roland Diel
Consider a one-dimensional diffusion in a stable L�vy environment. In this article we prove that the normalized local time process recentered at the bottom of the standard valley with height $\log t$, $(L_X(t,\mathfrak m_{\log t}+x)/t,x\in \R)$, converges in law to a functional of two independent L�vy processes, which are conditioned to stay positive. In the proof of the main result, we derive that the law of the standard valley is close to a two-sided L�vy process conditioned to stay positive. Moreover, we compute the limit law of the supremum of the normalized local time. In the case of a Brownian environment, similar result to the ones proved here have been obtained by Andreoletti and Diel \cite{ad:llltbd}.
0909.2929.tex
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\begin{document}





\title{Local time of a diffusion in a stable L�vy environment}
\subjclass[2000]{60G51 - 60J55 - 60J60}
\keywords{  Diffusion process, Local time, L�vy process, Random environment }
\author{Roland Diel} 

\address{
Roland Diel,
MAPMO, CNRS UMR 6628,
F\'ed\'eration Denis Poisson FR 2964,
Universit� d'Orl�ans,
B.P. 6759,
45067 Orl�ans cedex 2
FRANCE.
}
  
\email{roland.diel@univ-orleans.fr} 

\author{Guillaume Voisin}

\address{
Guillaume Voisin,
MAPMO CNRS UMR 6628, 
F\'ed\'eration Denis Poisson FR 2964,
Universit� d'Orl�ans,
B.P. 6759,
45067 Orl�ans cedex 2
FRANCE.}

\email{guillaume.voisin@univ-orleans.fr}


\begin{abstract}
Consider a one-dimensional diffusion in a stable L�vy environment. In this article we prove that the normalized local time process recentered at the bottom of the standard valley with height $\log t$, $(L_X(t,\mathfrak m_{\log t}+x)/t,x\in \R)$, converges in law to a functional of two independent L�vy processes, which are conditioned to stay positive. In the proof of the main result, we derive that the law of the standard valley is close to a two-sided L�vy process conditioned to stay positive. Moreover, we compute the limit law of the supremum of the normalized local time. In the case of a Brownian environment, similar result to the ones proved here have been obtained by Andreoletti and Diel \cite{ad:llltbd}.
\end{abstract}








\maketitle












\section{Introduction.}
Let $(V(x),x\in\R)$ be a real-valued c�d-l�g stochastic process such that $V(0)=0$. A \emph{diffusion in the environment} $V$ is a process $\left(X(V,t), t\geq0\right)$ (or $\left(X(t), t\geq0\right)$ if there is no ambiguity on the environment) whose generator given $V$
 is 
\begin{displaymath}
	\frac{1}{2}e^{V(x)}\frac{d}{dx}\left( e^{-V(x)}\frac{d}{dx} \right).
\end{displaymath}
Remark that if $V$ is almost surely differentiable and $V'$ is bounded, then $(X(t),t\geq0)$ is the solution of the following SDE:
\begin{displaymath}
\left\{ \begin{array}{ll}
\ud X(t)=\ud \beta(t)-\frac{1}{2}V'(X(t))\ud t,\\
X(0)=0%\qquad ,x\in\mathbb{R}
\end{array} \right.
\end{displaymath}
where $\beta$ is a standard Brownian motion independent of $V$. Historically, this model has been introduced by Schumacher  \cite{schumacher}. It was mainly studied when the environment $V$ is a Brownian motion (Brox \cite{b:oddpwm} and Kawazu and Tanaka \cite{kt:omdpdbe} ), a drifted Brownian motion (Kawazu and Tanaka \cite{kt:adpbewd}, Hu, Shi and Yor \cite{hsy:rcddbp}, Devulder \cite{d:assdpdbp} and Talet \cite{t:atebmdbp}  ) or a process which has the same asymptotic behavior as a Brownian motion(Hu and Shi \cite{hs:ltsrwre,hs:lssrwre}), this last case allows to study the discrete analogue of the diffusion in a Brownian environment, the so-called Sinai's random walk. More recently, general L�vy environments have been studied by Carmona \cite{c:mvbmrlp}, Tanaka \cite{t:lpcspdre}, Cheliotis \cite{c:oddare} and Singh \cite{s:rctdsnlp}.

The local time process $(L_X(t,x),t\geq 0,x\in\R)$ is defined by the occupation time formula, i.e., it is the unique process, almost surely continuous in the variable $t$, such that for all bounded Borel function $f$ and for all $t\geq0$,
\begin{equation}\label{toccu}
	\int_0^t f(X_s)\ud s=\int_\R f(x)L_X(t,x)\ud x.
\end{equation}
The existence of such a process can be easily proved, see e.g. Shi \cite{s:ltcre}. In the Brownian case, the local time process was largely studied see e.g. Hu and Shi \cite{hs:ltsrwre,hs:pmvsre}, Shi \cite{s:ltcre}, Devulder \cite{d:mltdpdbp} and Andreoletti and Diel \cite{ad:llltbd}. In this paper, we generalize the result of \cite{ad:llltbd} to the L�vy environment case. However several difficulties arise: contrary to the Brownian motion, the law of a L�vy process is not stable by time inversion; moreover the processes $(V(x)-\inf_{0\leq y \leq x}V(y),x\geq0)$, $(\sup_{0\leq y\leq x}V(y)-V(x),x\geq0)$ and $|V|$ do not have the same law.

In what follows, we focus on the local time of the diffusion when the environment $V$ is an $\alpha$-stable L�vy process on $\R$. We say that $V$ is $\alpha$-stable if there exists $\alpha>0$, such that for all $c>0$
$$ (c^{-1} V(c^\alpha x),x\in \R ) \stackrel{\cl}{=} (V(x),x\in \R) $$ 
where $ \stackrel{\cl}{=} $ is the equality in law. And $V$ is an $\alpha$-stable L�vy process if on top of that it is a L�vy process on $\R$.


One of the main results of this paper is the convergence in law of the supremum of the local time
\begin{displaymath}
\forall t\geq0,\quad L_X^*(t)=\sup_{x\in\R}L_X(t,x)
\end{displaymath}
which measures the time passed by the diffusion in the most visited point.

We next recall the notion of \emph{valley} introduced in the Brownian case by Brox \cite{b:oddpwm} (see also \cite{NevPit}). Denote by $\cv$ the space of c�d-l�g functions $\omega:\R \rightarrow \R$, endowed with the Skorohod topology (the space $\cv$ is then Polish). Let $c>0$, \label{defvalley} $\omega\in\mathcal{V}$ has a $c$-\textit{minimum} in the point $x_0$ if there exists $\xi$ and $\zeta$ such that $\xi<x_0<\zeta$ and 
\begin{itemize}
 
\item $\omega(\xi)\geq \left(\omega(x_0)\wedge \omega(x_0-)\right)+c$,
\item $\omega(\zeta-)\geq \left(\omega(x_0)\wedge \omega(x_0-)\right)+c\quad$ and
\item for all $x\in(\xi,\zeta)$, $x\neq x_0$, $\omega(x)\wedge \omega(x-)> \omega(x_0)\wedge \omega(x_0-)$.
\end{itemize}
Analogously, $\omega$ has a $c$-\textit{maximum} in $x_0$ if $-\omega$ has a $c$-minimum in $x_0$. Denote by $M_c(\omega)$ the set of $c$-\textit{extrema} of $\omega$. If $\omega$ takes pairwise distinct values in its local extrema and is continuous at these points, then we can check that $M_c(\omega)$ has no accumulation points and that the $c$-maxima and the $c$-minima alternate \label{valstand}. These assumptions are verified for $\alpha$-stable L�vy processes with $\alpha\in [1,2]$.  Thus there exists a triplet $\Delta_c=\Delta_c(\omega)=(\mathfrak p_c,\mathfrak m_c,\mathfrak q_c)$ of successive elements of $M_c$ such that 
\begin{itemize}\label{minvallee}
 \item $\mathfrak m_c$ and $0$ belongs to $[\mathfrak p_c,\mathfrak q_c]$,
\item $\mathfrak p_c$ and $\mathfrak q_c$ are $c$-maxima,
\item $\mathfrak m_c$ is a $c$-minimum.
\end{itemize}
This triplet is called the \textit{standard valley} with height $c$ of $\omega$. The stability of the environment $V$ allows us to reduce the study of a valley with height $c$ to a valley with height $1$. \label{stdval}

In the following, we denote
\begin{equation}\label{w+w-}
\left(\omega^+(x),x\geq0\right)=\left(\omega(x),x\geq0\right) \hbox{ , }\left(\omega^-(x),x\geq0\right) = \left(\omega((-x)-),x\geq0\right).
\end{equation}

Our main result gives the limit law of the local time process recentered on the minimum of the standard valley with height $\log t$.

\begin{theo}\label{cvloi}
Let $V$ be an $\alpha$-stable L�vy process with $\alpha\in [1,2]$ which is not a pure drift.\\
As $t\rightarrow \infty$, the process $\displaystyle \left(\frac{L_X(t,\mathfrak m_{\log t}+x)}{t},x\in\R\right)$
converges weakly in the Skorohod topology to the process $\displaystyle\left(\frac{e^{-\widetilde{V}(x)}}{\int_{-\infty}^{\infty}e^{-\widetilde{V}(y)}\ud y},x\in\R\right)$. The law of $\tilde{V}$ is $\tilde{\P}$ and under $\tilde{\P}(d\omega)$, $\omega^+$ and $\omega^-$ are independent, and distributed respectively as $\P^\uparrow$ and $\hat{\P}^\uparrow$, where $\P^\uparrow$ is the law of the L�vy process $V^+$ conditioned to stay positive, and $\hat{\P}^\uparrow$ is the law of the dual process $V^-$ conditioned to stay positive.
\end{theo}
Moreover, we obtain the localization of the favorite point $m^*(t)$ of the diffusion $X$.

\begin{theo}\label{cvptfav}
With the same assumptions as in previous theorem, for any $\delta \geq 0$,
 \begin{equation*}
\lim_{t\rightarrow+\infty} \mathcal{P}\left(\left|m^*(t)-\mathfrak{m}_{\log t}\right|\leq \delta \right)=1
 \end{equation*}
where $m^*(t)=\inf \big\{ x\in \R,\ L_X(t,x)=\sup \{ L_X(t,y), y\in\R\} \big\}$.
\end{theo}
The previous theorems imply the following corollary.

\begin{cor}\label{thlimsup}
With the same assumptions as before, we get
\begin{displaymath}
	\frac{L^*_X(t)}{t}   \xrightarrow[t\rightarrow \infty]{\mathcal L}     \frac{1 }{\int_{-\infty}^{\infty}e^{-\widetilde{V}(y)}\ud y} \label{eqthlimlaw}
\end{displaymath}
where $\widetilde{V}$ is the same process as in the previous theorem and $\xrightarrow{\mathcal L}$ denotes the convergence in law.
\end{cor}

\begin{proof} Recall that a L�vy process is continuous in probability. Thus, for any $x>0$, 
$$\mathcal{P}\big(V(x)=V(x-)\textrm{ and }V(-x)=V((-x)-)\big)=1.$$ 
Fix $\delta>0$. Theorem \ref{cvptfav} shows that it is sufficient to study the supremum of the process $x\mapsto L_X(t,\mathfrak{m}_{\log t}+x )$ on the compact set $[-\delta, \delta]$. %because $m^*(t)\in[\mathfrak{m}_{\log t}-\delta,\mathfrak{m}_{\log t}+\delta]$ $\mathcal P$-a.s. when $t\rightarrow +\infty$
Proposition VI.2.4 of \cite{js:ltsp} tells us that the function $F\mapsto \sup_{[-\delta,\delta]}F$ defined on $\mathcal V$ is continuous for the Skorohod topology for $F$ such that $F(\pm \delta)=F((\pm \delta)-)$. From Theorem \ref{cvloi} the result follows.
\end{proof}

The proof of Theorems \ref{cvloi} and \ref{cvptfav} is based on the study of the law of the environment. Indeed, for large times the diffusion is located in a valley whose right slope looks like the L�vy process $V^+$ conditioned to stay positive and the left slope like $V^-$ conditioned to stay positive (see Proposition \ref{vallee} for a rigorous statement). And, for large $t$ its local time process, normalized and centered at $\mathfrak m_{\log t}$, behaves like $e^{- V_{\mathfrak m_{\log t}}} / \int e^{- V_{\mathfrak m_{\log t}}}$ where $V_{\mathfrak m_{\log t}}$ is the environment recentered at $\mathfrak m_{\log t}$ (see Theorem \ref{limL} for more details). 



The law of the valley has already been obtained by Tanaka \cite{t:ltoddpbe} in the special case of a Brownian environment; in this case, the right and left slopes are both 3-dimensional Bessel processes. In \cite{t:lpcspdre}, Tanaka proved similar results in the L�vy environment case.


This paper is organized as follows. In Section \ref{sec1}, we define the environment $V$ and we give the asymptotic law of the standard valley $V$ on $\R_+$ (Proposition \ref{convergence}) and in Section \ref{sec2}, we prove the convergence of the local time: Theorem \ref{cvloi} is a consequence of Theorem \ref{limL} and Proposition \ref{conv exp} and we also show the localization of the favorite point: Theorem \ref{cvptfav} is the last statement of Theorem \ref{limL}.




\section{standard valley of a L�vy process.}\label{sec1}






\subsection{L�vy process on $\R$.}\label{general}
 A $\cv$-valued random process $V$ is a L�vy process on $\R$ if its increments are independent and stationary (see \cite{c:mvbmrlp}) i.e.
\begin{enumerate}
\item $V(0)=0$,
\item if $x_0 < x_1 < \dots < x_n$ are real numbers, then the random variables $(V(x_{i+1})-V(x_i), 0\leq i \leq n-1)$ are independent,
\item for all $x$ and $y$, the law of $V(x+y)-V(x)$ only depends on $y$.
\end{enumerate}
Denote by $\P$ the law of the L�vy process $V$.


\begin{rem}
An equivalent definition of a L�vy process indexed by $\R$ is given by Cheliotis in \cite{c:oddare}: let $(V^+(t),t\geq0)$ be an $\R$-valued L�vy process starting from 0 and $(V^-(t), t\geq 0)$ a process independent of $V^+$ and with the same law as $-V^+$, we define $(V(t),t\geq 0 )$ by 
\begin{equation}\label{def levy}
\forall t\in\R,\ V(t)=\ind_{t\geq 0}V^+(t) + \ind_{t\leq 0}V^-((-t)-) .
\end{equation}
This process has c�d-l�g paths and satisfies the three assumptions of the previous definition. Conversely, if the three assumptions hold for a process, then it can be decomposed in two L�vy processes as in Formula \reff{def levy}.
\end{rem}








Define $\underline{V}^+$ the infimum process of $V^+$ and $\overline{V}^+$ the supremum process: for any $t\geq0$, $\underline{V}^+(t)=\inf_{0\leq s \leq t}V^+(s)$ and $\overline{V}^+(t)=\sup_{0\leq s \leq t}V^+(s)$. Let $\mathcal{N}$ be the excursion measure of the process $V^+-\underline{V}^+$ away from 0. We denote by $L$ a local time of $V^+-\underline{V}^+$ in 0 and $L^{-1}(t)=\inf\{ s\geq0 ;  L(s)>t \}$ its right continuous inverse.

Denote by $\cv^+$ the set of c�d-l�g functions $\omega:\R_+ \rightarrow \R$ endowed with the Skorohod topology. For $c>0$ fixed, we define the first passage time above a level $c$ of $\omega$ and the last previous time in which $\omega$ reaches $0$: for all $\omega\in\cv^+$,
$$\tau_c(\omega) = \inf\{ t\geq 0 ; \omega(t) \geq c \} \hbox{ and } n_c(\omega) = \sup\{ t\leq \tau_c(\omega) ; \omega(t) = 0 \hbox{ or }\omega(t-) = 0\}.$$
We then denote  
$$\underline{\tau}^+_c = \tau_c\left( V^+-\underline{V}^+ \right) \hbox{ , } m^+_c = n_c\left( V^+-\underline{V}^+ \right) \hbox{ and }$$
\begin{align*}
 J^+_c&=\left(V^+(m^+_c)+c\right)\vee\overline{V}^+(m^+_c).
\end{align*}
Similarly, we define analogue variables $\underline{\tau}^-_c$, $m^-_c$ and $J^-_c$ using the process $V^-$. 
\begin{center}
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\caption{Standard valley with height $c$}

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\end{center}
\begin{rem}\label{m=m-oum+}
We show that the minimum of the standard valley with height $c$ defined in the introduction satisfies $\mathfrak m_c=m^+_c$ if $J^+_c<J^-_c$ and $\mathfrak m_c=-m^-_c$ otherwise (see \cite{kesten} for a proof in the Brownian case which is also true in our case). Thus to study the process $V$ in a neighborhood of $\mathfrak m_c$, it is sufficient to study $V^+$ in a neighborhood of $m^+_c$ and $V^-$ in a neighborhood of $m^-_c$.
\end{rem}





Therefore, in this section we study the L�vy process $V^+$ defined on $\R_+$. However, we still denote $\P$ its law and until the end of this section, all the probability measures are measures on $\cv^+$. In order to simplify notations, we write $V$, $m_c$ and $\underline{\tau}_c$ for $V^+$, $m_c^+$ and $\underline{\tau}^+_c$.


\subsection{Valley of the L�vy process on $\R_+$.}\label{loivallee}




We say that $(V(t),0\leq t\leq \underline{\tau}_c)$ is the standard valley with height $c$ of the L�vy process on $\R_+$.

Furthermore, we say that $0$ is regular for $(0,+\infty)$ (resp. for $(-\infty,0)$) if $\P( \inf\{ t>0 ; V(t)>0\}=0)=1$ (resp. $\P( \inf\{ t>0 ; V(t)<0\}=0)=1$). We will only work with L�vy process for which $0$ is regular for $(-\infty,0)$ and $(0,+\infty)$. Thereby we recall some properties of such processes stated in \cite{c:oddare}, which give us the existence of the triplet $(\mathfrak p_c,\mathfrak m_c,\mathfrak q_c)$ defined in the introduction.

For $\omega\in\cv^+$, the point $x_0$ is called a left local maximum (resp. minimum) if there exists $\varepsilon>0$ such that $\omega(x)\leq \omega(x_0-)$ (resp. $\omega(x)\geq \omega(x_0-)$) for all $x\in(x_0-\varepsilon,x_0)$. Similarly, we define a right local maximum and minimum.


 
\begin{lem}\label{continuite}(\cite{c:oddare}, Lemma 3)\\
Let $V$ be a L�vy process such that 0 is regular for $(-\infty,0)$ and $(0,+\infty)$, then $\P$-a.s.:
\begin{enumerate}
\item $V$ is continuous at its left local extrema and at its right local extrema,
\item In no two local extrema $V$ has the same value.
\end{enumerate}
\end{lem}

The point $(2)$ of the previous Lemma gives us the uniqueness of the minimum $m_c$ such that $V(m_c)=\inf_{[0,\underline{\tau}_c]}V$. Moreover, $m_c$ is a local minimum, thus according to point $(1)$ the process $V$ is continuous in $m_c$, that means $ V(m_c)= V(m_c-)$.


\begin{rem}\label{continuite ext locaux}
If $\omega\in\cv^+$ is c�d-l�g on a bounded interval $I$, then there exists $x,y\in\bar{I}$ such that $\sup_{I}\omega = \omega(x)\vee \omega(x-)$ and $\inf_{I}\omega = \omega(y)\wedge \omega(y-)$. So if $\omega$ is continuous at its local extrema, we get:
$$\exists x\in \bar{I},\hbox{ s.t. }\sup_{I}\omega = \omega(x) \hbox{ and } \exists y\in \bar{I},\hbox{ s.t. }\inf_{I}\omega = \omega(y).$$
\end{rem}











\subsubsection{The pre-infimum and post-infimum processes.}

We now define the pre-infimum and post-infimum processes $(\Vg(t), 0\leq t \leq m_c)$ and $(\Vd(t), 0\leq t \leq \underline{\tau}_c-m_c)$ by
$$\Vg(t) = V((m_c -t)-) - V(m_c)\ \textrm{ and }\ \Vd(t) = V(m_c +t) - V(m_c).$$
When there is no possible mistake, we write $\underset{\leftarrow}{V}$ instead of $\Vg$ and $\underset{\rightarrow}{V}$ instead of $\Vd$. We also denote respectively $\underset{\leftarrow}{\P}^c$ and $\underset{\rightarrow}{\P}^c$ the laws of the processes $\underset{\leftarrow}{V}$ stopped at time $m_c$ and $\underset{\rightarrow}{V}$ stopped at time $\tau_c(\underset{\rightarrow}{V})$.

We now study the law of these two processes. Therefore, we need the notion of L�vy process conditioned to stay positive.










\subsubsection{L�vy process conditioned to stay positive.}\label{deflevypositif}

We present here Bertoin's construction by concatenation of the excursions of the L�vy process above $0$ (see Section 3, \cite{b:stiehlrwlp}).


The process $V$ is a semi-martingale, its local continuous martingale part is null or proportional to a standard Brownian motion. Denote by $\ell$ the local time in a semi-martingale sense of $V$ at 0, see for example \cite{p:sisde} and consider the time passed by $V$ in $(0,\infty)$ and its right continuous inverse,
$$A_t^+ = \int_0^t ds \ind_{V(s)>0} \hbox{ and }\alpha_t^+ = \inf\{ s\geq0 ; A_s^+>t \}.$$
We define the process $V^{\uparrow}$ by
$$
 V^{\uparrow}(t) =\left\{\begin{array}{lr}
  V(\alpha_t^+) + \frac{1}{2} \ell_{\alpha_t^+} + \sum_{0<s\leq \alpha_t^+} (0\vee V(s-)) \ind_{V(s) \leq 0} - (0\wedge V(s-))\ind_{V(s) >0}& \hbox{if}\ t<A_\infty^+,\\
 +\infty& \textrm{otherwise.}
 \end{array}\right.
$$
We denote by $\P^{\uparrow}$ the law of the process $V^{\uparrow}$ and $\hat{\P}^{\uparrow}$ the law of the dual process $\hat{V}=-V$ conditioned to stay positive.
\begin{rem}\label{def:vuparrow}
We will mainly use this expression to prove that when the process $V$  is stable, $V^\uparrow$ is stable too.
\end{rem}
We state some properties of the L�vy process conditioned to stay positive (see \cite{b:stiehlrwlp}):
\begin{itemize}
\item $\P$-a.s. for all $t>0$, $V^\uparrow(t)>0$,
\item in the case where $V$ has no Brownian part, $\ell=0$,
\item the process $V^\uparrow$ tends to $+\infty$ as $t$ goes to $+\infty$.
\end{itemize}


 For every  process $Y$ which is lower bounded, we define the future infimum: for all $t\geq0$, $\uu{Y}(t)=\inf_{s\geq t}Y(s)$. For $c>0$, we introduce the time
$$\hat{m}^\uparrow_c = n_c(\hat{V}^{\uparrow} - \uu{\hat{V}}^{\uparrow}) .$$

We write $\P^{\uparrow,c}$ (resp. $\hat{\P}^{\uparrow,c}$) for the law of the process $V^{\uparrow}$ (resp. $\hat{V}^{\uparrow}$) stopped at time $\tau_c(V^\uparrow)$ (resp. $\hat{m}^\uparrow_c$). 



















\subsubsection{Valley with height $c$.}
Unlike the Brownian case where the law of the post-infimum process is equal to the law of a 3 dimensional Bessel process, in the L�vy case, this law is only absolutely continuous with respect to the law of a L�vy process conditioned to stay positive. In order to have the law of the standard valley, we use Proposition 4.7 of Duquesne \cite{d:pdrlp} and therefore we have to introduce some notations. 

For $t\geq0$, define $U(t)=-V(L^{-1}(t))$ if $L(\infty)>t$ and $U(t)=+\infty$ otherwise (recall that $L$ is the local time of $V-\underline{V}$ in $0$). The process $(L^{-1},U)$ is called the ladder process, it is a subordinator. We then define the potential measure $\cu$ associated with $U$: for all measurable positive function $F$,
$$\int_{\R}\cu(dx)F(x) = \E \left[ \int_0^{L(\infty)} dv F(U(v)) \right].$$

For the excursion measure $\mathcal{N}$ and for all measurable non-negative function $f$ on $\cv$, we write
$$\mathcal{N}(f)=\int f(v)\mathcal{N}(dv).$$




\begin{prop}\label{vallee}
We consider a L�vy process $V$, which does not go to $+\infty$ neither to $-\infty$ and such that 0 is regular for $(0,+\infty)$ and for $(-\infty,0)$.

Then the pre-infimum and post-infimum processes $(\underset{\leftarrow}{V}(t), 0\leq t \leq m_c)$ and $( \underset{\rightarrow}{V}(t) , 0\leq t \leq \tau_c(\underset{\rightarrow}{V})) $ are independent. Moreover

\begin{enumerate}
\item The law  $\underset{\leftarrow}{\P}^{c}$ of the pre-infimum process $(\underset{\leftarrow}{V}(t), 0\leq t \leq m_c)$ is equal to the law $\hat{\P}^{\uparrow,c}$ of the process $(\hat{V}^{\uparrow} (t), 0\leq t \leq \hat{m}^\uparrow_c)$.

\item The law $\underset{\rightarrow}{\P}^c$ of the post-infimum process $(\underset{\rightarrow}{V}(t),0\leq t \leq \tau_c(\underset{\rightarrow}{V}) )$ is absolutely continuous with respect to the law $\P^{\uparrow,c}$ of the process $(V^{\uparrow} (t), 0\leq t \leq \tau_c(V^\uparrow))$.

More precisely, $\displaystyle \underset{\rightarrow}{\P}^{c}(d\omega) = f_c\left(\omega(\tau_c(\omega))\right) \P^{\uparrow,c}(d\omega)$

where for all $x\in\R_+$, $\frac{1}{f_c(x)}=\cu \left( [0, x) \right) \mathcal{N}(\tau_c(v) < \infty)$.
\end{enumerate}
\end{prop}






\begin{rem}\hspace{0.1cm}\label{rem 1ere prop}
\begin{enumerate}
\item When $V$ is a L�vy process with no positive jumps, $\omega(\tau_c(\omega))=c$, $\P^{\uparrow,c}-a.s$. Thus, $\omega\mapsto f_c(\omega(\tau_c(\omega)))$ is constant and equal to $1$ and the post-infimum process is equal in law to the process $V^\uparrow$ conditioned to stay positive.

\item $\P$-a.s., $\underset{\leftarrow}{V}$ is continuous in its local extrema (see Lemma \ref{continuite}), then if $c$ tends to $+\infty$, using point (1) of  Proposition \ref{vallee}, $\hat{V}^\uparrow$ and $V^\uparrow$ are continuous at their local extrema too.

\end{enumerate}
\end{rem}







\begin{proof}[Proof of the Proposition \ref{vallee}]



$(1)$  We begin with the proof of the first part of Proposition \ref{vallee} and the independence of the two processes for $c=1$. To simplify the notations, we will not write the index $1$. 

We want to express the pre-infimum process $\underset{\leftarrow}{V}$ in terms of the excursions of $V$ above its minimum and of the  associated jump of $\underline{V}$. We consider the couple of processes $(V-\underline{V} , -\underline{V})$ and we give an expression using excursions. We use the proof of Lemma 4 of \cite{b:sdtplspi}, remark that the result is still true for every L�vy process that does not converge to $-\infty$, see  Theorem 27 of \cite{d:ftlp}.

In order to use excursion theory, we denote by $\bar{\cv}$ the set of the c�d-l�g paths $v:[0,\infty)\rightarrow \R_+$ such that
\begin{itemize}
\item[(i)] $v(t) >0$ for $0<t<\sigma(v) = \inf\{ s>0, v(s)=0 \}$,
\item[(ii)] $v(0)=v(t)=0$ for $t\geq\sigma(v)$.
\end{itemize}

We consider the excursion process of $V-\underline{V}$ above 0 defined by:
\begin{align*}
p_1(t) &= 
\left\{
\begin{array}{ll}
r\mapsto(V-\underline{V})(L^{-1}(t-)+r)\ind_{[0,L^{-1}(t)-L^{-1}(t-)]}(r) & \hbox{ if } L^{-1}(t-)<L^{-1}(t)\\
0 & \hbox{ if } L^{-1}(t-)=L^{-1}(t)
\end{array}
\right. \\
\hbox{and the} & \hbox{ jumps process of $\underline{V}$ defined by:}\\
p_2(t) &=  \underline{V}(L^{-1}(t-)) - \underline{V}(L^{-1}(t)).
\end{align*}


Thus the process $p(\cdot)=(p_1(\cdot),p_2(\cdot))$ is a $\bar{\cv}\times \R_+$-valued Poisson point process (see for example \cite{gp:filpsm}). Remark that $p_1\left( L(\underline{\tau})\right)$ is the excursion starting from $m$, that is to say that $L(\underline{\tau})$ is the first time $t$ where the excursion $p_1(t)$ has a height bigger than $1$. Thus, the processes $(p(t) ; 0\leq t < L(\underline{\tau}) )$ and $(p(t) ; t\geq L(\underline{\tau}) )$ are independent. The pre-infimum process depends only on the first and the post-infimum process depends only on the second, then these two processes are independent too.

In order to obtain the process $\underset{\leftarrow}{V}$, we first invert the excursions of $V-\underline{V}$ then we invert the time in each of these excursions.

Denote by $\tilde{p}_i = \left(p_i(L(\underline{\tau})-t)\ind_{t<  L(\underline{\tau})} + p_i(t)\ind_{t \geq L(\underline{\tau})} \, ; t\geq0\right)$ the process $p_i$ where the order of the excursions before $L(\underline{\tau})$ is changed for $i=1$ and $i=2$. The law of the Poisson point process $p$ is stable by inversion of the order of its excursions before $L(\underline{\tau})$, we deduce the equality in law of the two processes $(\tilde{p}_1(t),\tilde{p}_2(t))$ and $(p_1(t),p_2(t))$.

Consider now the time inversion in the excursions defined for $v\in \bar{\cv}$ by 
$$[v](s) = 
\begin{cases}
v((\sigma(v)-s)-) &\hbox{ for }  0\leq s \leq \sigma(v)\\
0 & \hbox{ for }  s> \sigma(v).
\end{cases}
$$

Using this transformation in the previous law, the processes $([\tilde{p}_1(t)],\tilde{p}_2(t)\, ; t\geq0)$ and $([p_1(t)],p_2(t)\, ; t\geq0)$ have same law. With the first one we get the process $\underset{\leftarrow}{V}$. Indeed, $ \left([\tilde{p}_1(t)]\, ; t\geq0 \right)$ is the excursion process of $\underset{\leftarrow}{V}$ above its future infimum and $ \left(\tilde{p}_2(t)\, ; t\geq0 \right)$ represents the jumps of the future infimum of $\underset{\leftarrow}{V}$. We now use the couple $([p_1(t)],p_2(t)\, ; t\geq0)$ to get a L�vy process conditioned to stay positive.

\label{procr}We recall the notations of \cite{b:sdtplspi}, for $t\geq 0$, we use $g(t) = \sup\{ s<t ; V(s)=\underline{V}(s) \}$ and $d(t) = \inf\{ s>t ; V(s)=\underline{V}(s) \}$, the left and right extremities of the excursion interval of $V-\underline{V}$ away from 0 that contains $t$ and
$$\mathcal R_{V-\underline{V}}(t) =
\begin{cases}
(V-\underline{V})((d(t)+g(t)-t)-) & \hbox{ if  } g(t)<d(t) \\
0 & \hbox{otherwise}.
\end{cases}
$$

See Figure \ref{fig:V} and \ref{fig:RV}.

The process $L$ is also a local time in 0 for the process $\mathcal R_{V-\underline{V}}$. So the excursion process of $\mathcal R_{V-\underline{V}}$  above 0, parametrized by $L$, is the Poisson point process $([p_1(t)],t\geq0)$ and, similarly, the excursion process of $\left( (V-\underline{V})((m-t)-) \ind_{0\leq t < m} + \mathcal R_{V-\underline{V}}(t) \ind_{t\geq m} , t\geq0\right)$ above 0 is  the Poisson point process $([\tilde{p}_1(t)],t\geq0)$. We have an analogue equality for the jumps associated with every excursion. Then we get the equality in law of the processes

\begin{flushleft}
$\displaystyle \left( (V-\underline{V})((m-t)-) \ind_{0\leq t < m} + \mathcal R_{V-\underline{V}}(t) \ind_{t\geq m} ,\right. $
\end{flushleft}

\begin{flushright}
$\displaystyle  \left.(\underline{V}((m -t)-) - \underline{V}( m ))\ind_{0\leq t < m} - \underline{V}(d(t)) \ind_{t\geq m} \; ; \; t\geq 0\right)$
\end{flushright}

\begin{center}
and
\end{center}

$$\left( \mathcal R_{V-\underline{V}}(t)  , -\underline{V}(d(t) ; t \geq 0) \right) .$$

Adding the two parts for $t<m$ we get: 
\begin{align*}
\left( \underset{\leftarrow}{V}(t) ; 0\leq t < m\right) &\stackrel{\cl}{=} \left( \mathcal R_{V-\underline{V}}(t)  -\underline{V}(d(t)) ; 0\leq t < m\right),
\end{align*}

See Figure \ref{fig:RV} and \ref{fig:preinf}.








%------------------premier dessin----------------------%

\begin{center}
\begin{figure}[ht]
\caption{The L�vy process $(V(t);t\geq0)$}\label{fig:V}



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\end{figure}
\end{center}




%----------------fin dessin 1-------------------




%----------------debut dessin 2----------------------





\begin{center}
\begin{figure}[ht]
\caption{The process $\left( \mathcal R_{V-\underline{V}}(t)  -\underline{V}(d(t)); t\geq 0\right)$}\label{fig:RV}




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\end{figure}
\end{center}









%--------------fin dessin 2 -------------------






%-----------------dessin 3 ---------------------

\begin{center}
\begin{figure}[ht]
\caption{The pre-infimum process    } \label{fig:preinf}
%  $(V((m-t)-)-V(m) ; t\geq0)$   ou    $(\underset{\leftarrow}{V}(t) ; t\geq0)$
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\end{figure}
\end{center}


%-----------------------fin dessin 3--------------------------





















We now use Theorem 28 of \cite{d:ftlp} : under $\P$, the law of the process $\left( \mathcal R_{V-\underline{V}}(t)  -\underline{V}(d(t)) ,t\geq0\right)$ is $\hat{\P}^{\uparrow}$.

Now we express the final time $m$ in terms of the process $\left( \mathcal R_{V-\underline{V}}(t)  -\underline{V}(d(t)), t\geq 0\right)$.

Recall that $\underline{\tau} = \inf\{ t\geq 0 ; V(t)-\underline{V}(t) \geq 1 \}$ and that $m = \sup\{ t\leq \underline{\tau} ; V(t) - \underline{V}(t) = 0 \}$. We define the variable $\underline{\tau}' = \tau(  \mathcal R_{V-\underline{V}}) = \inf\{t\geq 0 ; \mathcal R_{V-\underline{V}}\geq 1    \}$. Notice that $\underline{\tau}$ and $\underline{\tau}'$ belong to the same excursion interval of $V-\underline{V}$. The variable $m$ remains stable by time inversion in every excursions of $V-\underline{V}$ and by the inversion of the excursions before $m$, then $m=\sup\{t\leq \underline{\tau}' ; \mathcal R_{V-\underline{V}}(t)=0\}$. We write 
$$Y(t) := \mathcal R_{V-\underline{V}}(t) - \underline{V}(d(t)).$$
We prove that
$$\mathcal R_{V-\underline{V}}(t) = Y(t) - \uu{Y}(t),$$
it is enough to prove that, for a fixed $t\geq0$, the post-infimum process $\uu{Y}(t) = \inf_{s\geq t}Y(s)$ is equal to $- \underline{V}(d(t))$.

On the interval $[t,d(t))$, the process $(\underline{V}(d(s)),t\leq s < d(t))$ is constant and $$\inf_{t\leq s < d(t)} \mathcal R_{V-\underline{V}}(s) = \mathcal R_{V-\underline{V}}(d(t)-)=0$$ then $\inf_{t\leq s < d(t)}Y(s) = - \underline{V}(d(t))$. The process $\mathcal R_{V-\underline{V}}$ is non-negative and vanishes at times  $\{d(t)-,t\geq 0\}$ (see Corollary 1, \cite{r:nirlp}), and the process $- \underline{V}(d(.))$ increases by jumps at times $\{d(t),t\geq 0\}$, then $\uu{Y}(t)=\inf_{t\leq s < d(t)}Y(s)$ and so $\uu{Y}(t) =- \underline{V}(d(t))$. We deduce that 

$$m=\sup\{t\leq \underline{\tau}' ; Y(t) - \uu{Y}(t)=0\}.$$

Finally, under $\P$, the law of the process $\left( \mathcal R_{V-\underline{V}}(t)  -\underline{V}(d(t)) \right)_{0\leq t\leq m}$ is the law of the process  $\hat{V}^{\uparrow}$ stopped at time $\hat{m}^{\uparrow}$ and the point (1) is proved.


\vspace{0.5cm}






$(2)$ Now we consider the post-infimum process. Let $F$ be a bounded measurable function on $\cv^+$. We can write 
$$\underset{\rightarrow}{\E}^{c} \left[ F\left( \omega  \right)\right] = \mathcal{N} \left( F\left( v(\cdot \wedge \tau_c)  \right) | \tau_c(v) < \infty \right).$$

We use Proposition 4.7 of \cite{d:pdrlp}: for all bounded measurable function $G$ on $\cv^+$, 

\begin{align*}
\E^{\uparrow,c}\left[ G(\omega) \right] &= \mathcal{N}\left( G\left( v(\cdot \wedge \tau_c) \right)  \cu \left( [0, v(\tau_c) ) \right) | \tau_c <\infty \right) \mathcal{N}\left( \tau_c(v) <\infty \right)\\
&= \mathcal{N}\left( \frac{G( v(\cdot\wedge \tau_c) )}{ f_c(v(\tau_c))} \Big | \tau_c(v) <\infty \right)
\end{align*}
where we recall that for all $x\in\R_+$, $\frac{1}{f_c(x)} = \cu \left( [0, x) \right) \mathcal{N}(\tau_c(v) < \infty)$. Using the function $G( v(\cdot\wedge \tau_c) ) = F\left( v(\cdot \wedge \tau_c)  \right) f_c(v(\tau_c))$, we obtain the second point of the proposition.



\end{proof}








\subsection{Stable L�vy process}



\begin{defi}(\cite{b:pl}, Section VIII)\\
The L�vy process $V$ is stable with index $\alpha \in (0,2]$ if, for all $c>0$
$$ \left(c^{-1} V(c^\alpha t),t\geq 0\right) \stackrel{\cl}{=} \left(V(t),t\geq 0\right) .$$
\end{defi}

For $\alpha\in(0,1)\cup(1,2)$, the Fourier transform of an $\alpha$-stable L�vy process $V$ is : for all $t\geq0$ and for all $\lambda\in \R$, $\E \left[ e^{i\lambda V(t)}\right] = e^{-t\psi(\lambda)}$ with
$$\psi(\lambda) = k |\lambda|^\alpha \left(1-i \beta \hspace{0,2cm} sgn(\lambda)\tan(\frac{\pi \alpha}{2}) \right)$$
where $k>0$ and $\beta\in[-1,1]$.\\
The L�vy measure of the process $V$ is absolutely continuous with respect to the Lebesgue measure, it can be written as follows:
$$\pi(dx) = c^+ x^{-\alpha-1} \ind_{x > 0} dx + c^-  x ^{-\alpha-1} \ind_{x<0} dx$$
where $c^+$ and $c^-$ are positive numbers non equal to 0 together and such that $\beta = \frac{c^+ - c^-}{c^+ + c^-}$.

In the quadratic case ($\alpha=2$), $V$ is a Brownian motion and its characteristic exponent can be written $\psi(\lambda) = k \lambda^2$ where $k$ is a positive number. The L�vy measure is then trivial.

In the case $\alpha=1$, $V$ is a Cauchy process and $\psi(\lambda)=k|\lambda| + id\lambda$ with  $k>0$ and $d\in \R$. The L�vy measure is then proportional to $x^{-2} dx$.


\begin{rem}\label{stabmc} If the process $V$ is $\alpha$-stable, then the pre-infimum processes $\left(c^{-1}\Vg(c^\alpha t),t\geq0\right)$ and $\Vg[1]$ (stopped at correct times) have the same law. This also holds for the post-infimum processes $\left(c^{-1}\Vd(c^\alpha t),t\geq0\right)$ and $\Vd[1]$. 
\end{rem}

\begin{lem}Let $V$ be an $\alpha$-stable L�vy process which is not a pure drift. Then $V$ is recurrent if and only if $\alpha \geq1$.
\end{lem}

\begin{proof}We use a  condition of recurrence given in \cite{b:pl}, Theorem I.17: $V$ is recurrent if and only if $\int_{-r}^r \Re \left[ \frac{1}{\psi(\lambda)}\right]d\lambda = +\infty$ where $\Re[z]$ is the real part of the complex number $z$.

For $\alpha=1$, $\int_{-r}^r \frac{k|\lambda|}{(k^2+d^2)\lambda^2}d\lambda = +\infty$ because $k\neq0$.

For $\alpha\neq 1$, $\int_{-r}^r \frac{1}{k|\lambda|^\alpha(1+\beta^2 \tan^2(\frac{\pi \alpha}{2}))}d\lambda<+\infty$ if and only if $\alpha <1$. 
\end{proof}

Then we use a stable L�vy process with index $\alpha\in [1,2]$ and which is not a pure drift. We need others properties for $V$, mainly to use the results of Section \ref{loivallee}.

\begin{lem}
Let $V$ be a stable L�vy process with index $\alpha\in [1,2]$ and which is not a pure drift. Then,
\begin{itemize}
\item 0 is regular for $(0,+\infty)$ and for $(-\infty,0)$,
\item $\P$-a.s., $\underset{t\rightarrow +\infty}{\lim\sup}V(t) =+\infty$ and  $\underset{t\rightarrow +\infty}{\lim\inf}V(t) = -\infty$.
\end{itemize}
\end{lem}

\begin{proof}\hspace{1cm}

$\bullet$ Rogozin criterion (see Prop. VI.11 of \cite{b:pl})  gives a necessary and sufficient condition to have the regularity in 0:

0 is regular for  $(0,+\infty)$ (resp. for $(-\infty,0)$) if and only if $\displaystyle \int_0^1 t^{-1}\P(V(t)\geq 0)dt = +\infty$ (resp. $\displaystyle \int_0^1 t^{-1}\P\left(V(t) \leq 0\right)dt = +\infty)$.

Remark, using the stability property of $V$, that $\P(V(t)\geq 0)$ does not depend on $t$. Moreover this probability is positive if $V$ is not a pure drift, see \cite{b:pl}, VIII.1. 

With the previous criterion, $0$ is regular for $(0,+\infty)$ and for $(-\infty,0)$.


$\bullet$ We have $\int_1^\infty t^{-1}\P(V(t)\geq0)dt = \int_1^\infty t^{-1}\P(V(t)\leq0)dt = +\infty$, this implies that $\P$-a.s., $\underset{t\rightarrow +\infty}{\lim\sup}V(t) =+\infty$ and  $\underset{t\rightarrow +\infty}{\lim\inf}V(t) = -\infty$, see \cite{b:pl}, Theo. VI.12.

\end{proof}






\begin{lem}\label{stabiliteuparrow}
We consider a stable L�vy process $V$ with index $\alpha\in(0,2]$. Then the process $V^{\uparrow}$ is an $\alpha$-stable process.
\end{lem}

\begin{proof}
In the case where $\alpha=2$, the stable L�vy process $V$ is a standard Brownian motion, $V^{\uparrow}$ is a $3$-dimensional Bessel process (see \cite{mck:ensd}) and then is stable with index $2$.

If $\alpha\in(0,2)$, the stable L�vy process has no Brownian part. Recall that $\alpha_t^+$ is the right continuous inverse of $A_t^+=\int_0^t \ind_{V(s)>0} ds $. These two processes are stable with index $1$. We use the definition given in Section \ref{deflevypositif}:
$$V^{\uparrow}(t) = V(\alpha_t^+) + \sum_{0<s\leq \alpha_t^+} \left(0\vee V(s-) \right)\ind_{V(s) \leq 0} - \left(0\wedge V(s-)\right)\ind_{V(s) >0}, $$
We deduce that the process $V^{\uparrow}$ is stable with index $\alpha$.

\end{proof}


\begin{prop}\label{convergence}
We consider a stable L�vy process with index $\alpha\in [1,2]$ and which is not a pure drift. When $c$ converges to $+\infty$:
\begin{enumerate}
\item the law $\underset{\leftarrow}{\P}^{c}$ converges weakly (in the Skorohod sense) to the law $\hat{\P}^{\uparrow}$.

\item the law $\underset{\rightarrow}{\P}^{c}$ converges weakly (in the Skorohod sense) to the law $\P^{\uparrow}$.


\end{enumerate}
\end{prop}

To prove the second point, we need the following lemma.


\begin{lem}\label{levy renormalise}
We fix $\varepsilon>0$ and we define 
$$
\sigma_\varepsilon=\sigma_\varepsilon(V^\uparrow)=\inf\left\{t>0/V^\uparrow(t)-\uu{V}^\uparrow(t)=0 \textrm{ and } \exists s<t, V^\uparrow(s)-\uu{V}^\uparrow(s)>\varepsilon\right\}.
$$
Then the process $\left(V^{\uparrow}_{\sigma_\varepsilon}(t),t\geq0\right)=\left(V^\uparrow(\sigma_\varepsilon+t)-V^\uparrow(\sigma_\varepsilon),t\geq0\right)$ is independent of the process $\left(V^\uparrow(t),0\leq t\leq \sigma_\varepsilon\right)$ and its law is given by $\P^\uparrow$.
\end{lem}
\begin{proof}

Thanks to Theorem 28 of \cite{d:ftlp}, the law of the process $\mathcal{R}_{\overline{V}-V}+V(d(\cdot))$ is $\P^\uparrow$ (where $\mathcal{R}$ and $d$ are defined like page \pageref{procr} but using the process $\overline{V}-V$). It is enough to prove the result in the case where $V^\uparrow=\mathcal{R}_{\overline{V}-V}+V(d(\cdot))$. But in this case, $\sigma_\varepsilon$ can be written as
$$
\sigma_\varepsilon=\inf\left\{t>0/\overline{V}(t)-V(t)=0 \textrm{ and } \exists s<t, \overline{V}(s)-V(s)>\varepsilon\right\}
$$
then $\sigma_\varepsilon$ is a stopping time for $V$. 

Thus the process $\left(V_{\sigma_\varepsilon}(t),t\geq0\right)=\left(V(\sigma_\varepsilon+t)-V(\sigma_\varepsilon),t\geq0\right)$ is a L�vy process with law $\P$ independent of $\left(V(t),0\leq t\leq \sigma_\varepsilon\right)$. We can check that  $V^{\uparrow}_{\sigma_\varepsilon}(t)=\mathcal{R}_{\overline{V}_{\sigma_\varepsilon}-V_{\sigma_\varepsilon}}(t)+V_{\sigma_\varepsilon}(d(t))$, this proves the Lemma.
\end{proof}











\begin{proof}[Proof of Proposition \ref{convergence}]

$(1)$ Thanks to Proposition \ref{vallee}, the laws of $\underset{\leftarrow}{\P}^{c}$ and $\hat{\P}^{\uparrow,c}$ are equal. When $c\rightarrow \infty$, $\hat{\P}^{\uparrow}$-a.s., $\hat{m}^\uparrow_c$ tends to infinity, then $\underset{\leftarrow}{\P}^{c}=\hat{\P}^{\uparrow,c}$ converges to $\hat{\P}^{\uparrow}$.

\vspace{.4cm}

\noindent(2) Now we prove that for a continuous bounded function $F$ in the Skorohod topology such that $F(\omega)=F(\omega\ind_{[-K,K]})$, we have
\[
\lim_{c\rightarrow \infty} \Ed\left[ F(\omega) \right]=\E^\uparrow\left[ F(\omega) \right].
\]

As $\P$-a.s., $\lim_{c\rightarrow\infty}\tau_c(\Vd)=\infty$, so $\lim_{c\rightarrow\infty} \underset{\rightarrow}{\P}^c(K\leq \tau_c(\omega))=1$ and thus
$$\lim_{c\rightarrow \infty} \Ed\left[ F(\omega) \right]=\lim_{c\rightarrow\infty}\Ed\left[ F(\omega), K\leq \tau_c(\omega) \right].$$
We first use the stability of $V$. Thanks to Remark \ref{stabmc},
\begin{align*}
\Ed\left[ F(\omega), K\leq \tau_c(\omega) \right]&= \Ed[1]\left[ F\left(  c\,\omega(c^{-\alpha}\cdot) \right) ,K\leq c^{\alpha} \tau_1(\omega)  \right],\\
&= \E^\uparrow\left[ F\left( c\,\omega(c^{-\alpha}\cdot)\right)f_1\left(\omega(\tau_1)\right) ,K\leq c^{\alpha}\tau_1(\omega)  \right]
\end{align*}
where we use the result (2) of Proposition \ref{vallee}, for the second equality. 

We now prove the asymptotic independence of $F\left( c\,\omega(c^{-\alpha}\cdot)\right)$ and of $f_1\left(\omega(\tau_1)\right)$ under $\P^\uparrow$. Then we express these two terms with respect to the processes $(\omega(t),0\leq t\leq \sigma_\varepsilon)$ and $(\omega(t),\sigma_\varepsilon\leq t)$ where, for $\varepsilon>0$, we recall that:
$$
\sigma_\varepsilon=\sigma_\varepsilon(\omega)=\inf\left\{t>0/\omega(t)-\uu{\omega}(t)  =0 \textrm{ and } \exists s<t, \omega(s)-\uu{\omega}(s)>\varepsilon\right\}.
$$ 




\begin{center}
\begin{figure}[ht]
\caption{The process $V^\uparrow$}

\label{vfleche}

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\end{figure}
\end{center}



We fix $\varepsilon>0$, we remark that $\lim_{c\rightarrow +\infty} \P^\uparrow\left( K \leq c^{\alpha} \tau_1 \right)=\lim_{c\rightarrow +\infty} \P^\uparrow\left( K\leq c^{\alpha}\sigma_\varepsilon  \right)=1$. Then we can replace $K \leq c^{\alpha} \tau_1$ by $K\leq c^{\alpha}\sigma_\varepsilon$ in the expectation. 

Moreover, $\P^\uparrow$-a.s., $\lim_{\varepsilon\rightarrow 0}\sigma_\varepsilon=0$. Indeed, if there exists $\delta>0$ such that for all $\varepsilon>0$, $\sigma_\varepsilon>\delta$, then the excursion of $\omega-\uu{\omega}$ containing time $\delta$ is the first one. And this is $\P^\uparrow$-a.s. impossible because 0 is regular for the process $V$. Thus $\lim_{\varepsilon \rightarrow 0} \P^\uparrow\left( \sigma_\varepsilon < \tau_1\right)=1$. By conditioning with respect to $\left( \omega(t), 0\leq t \leq \sigma_\varepsilon\right)$ and using that $\{ \sigma_\varepsilon<\tau_1^\uparrow \}=\{\forall t\leq \sigma_\varepsilon, V_t^\uparrow <1\}$ is measurable with respect to  $\left( \omega(t), 0\leq t \leq \sigma_\varepsilon\right)$, we get: 

\begin{flushleft}
\begin{equation}\label{hypotheses}
 \E^\uparrow\left[ F\left(  c\,\omega(c^{-\alpha}\cdot)\right)f_1\left(\omega(\tau_1)\right) ,  c^{-\alpha}K\leq \sigma_\varepsilon < \tau_1 \right] = \hspace{5cm}
 \end{equation}
\end{flushleft}
 
\begin{flushright}
 \begin{equation*}
  \hspace{3cm}\E^\uparrow\left[ F\left(  c\,\omega(c^{-\alpha}\cdot)\right) \, \ind_{c^{-\alpha}K\leq\sigma_\varepsilon< \tau_1} \, \E^\uparrow\left[ f_1\left(\omega(\tau_1)\right) \Big | \left(\omega(t),0\leq t \leq \sigma_\varepsilon\right)\right] \right].
  \end{equation*}
\end{flushright}

We compute the limit of the conditional expectation when $\varepsilon$ converges to 0. When $\sigma_\varepsilon < \tau_1$, we can write $\omega(\tau_1) = \omega(\sigma_\varepsilon) + \omega_{\sigma_\varepsilon} ({\tau_{1-\omega(\sigma_\varepsilon)}})$ where $\omega_{\sigma_\varepsilon}$ is the path starting from $\sigma_\varepsilon$ : $\left(\omega_{\sigma_\varepsilon}(t),t\geq0\right)=\left(\omega(\sigma_\varepsilon+t)-\omega(\sigma_\varepsilon),t\geq0\right)$, see Figure \ref{vfleche}. Thanks to Lemma \ref{levy renormalise},

\begin{align*}
\E^\uparrow\left[ f_1\left(\omega(\tau_1)\right) \Big | \left( \omega(t),0\leq t \leq \sigma_\varepsilon \right) \right] = \chi(\omega(\sigma_\varepsilon))
\hbox{ where } \chi(x) = \E^\uparrow\left[f_1\left(  x + \omega(\tau_{1-x})  \right) \right].
\end{align*}
Remark that $\P^\uparrow$-a.s, $\lim_{\varepsilon\rightarrow 0}\omega(\sigma_\varepsilon)=0$. Prove that $\P^\uparrow$-a.s., $\lim_{y\rightarrow 1-}\omega(\tau_y)=\omega(\tau_1)$.  $\P^\uparrow$-a.s., $\omega$ is continuous at its local extrema (see (2) of Remark \ref{rem 1ere prop}). Thus, thanks to Remark \ref{continuite ext locaux}, $\P^\uparrow$-a.s., $\sup_{[0,\tau_1)}\omega$ is attained in a point $x_0\in[0,\tau_1]$ and  $\omega(x_0)\leq 1$.

If $x_0=\tau_1$, then $\omega(\tau_1)=1$ and $\lim_{y\rightarrow 1-}1-\omega(\tau_y) \leq \lim_{y\rightarrow 1-}1-y =0$. And if $x_0<\tau_1$, then $\omega(x_0)<1$ and then, for $1-y$ small enough, $\tau_1=\tau_y$, thereby $\lim_{y\rightarrow 1-}\omega(\tau_y)=\omega(\tau_1)$.





Moreover, in the case where $V$ is a stable process with index $\alpha\in[1,2)$, an explicit expression of $\cu$ is given in Example 7 of \cite{dk:oulp}:
$$\cu\left( [  0, x )\right) = \frac{x^{\alpha\rho}}{\Gamma(\alpha \rho+1)} \hbox{ where }\rho=\P(V(t)\geq0)\hbox{ does not depend on $t$}.$$
Thus using the expression of $f_c$ given in Proposition \ref{vallee}, we get: 
\begin{equation}\label{expressionf1}
f_1(x)=\frac{\Gamma(\alpha\rho+1)}{x^{\alpha\rho}\mathcal{N}(\tau_1(\omega) < \infty)},
\end{equation}
then $f_1$ is continuous on  $\R_+^*$ and
\begin{eqnarray}\label{conv ki}
 \lim_{\varepsilon\rightarrow 0}\E^\uparrow\left[ f_1\left(\omega(\tau_1)\right) \Big | \left( \omega(t),0\leq t \leq \sigma_\varepsilon\right) \right]=\nonumber\lim_{\varepsilon\rightarrow 0} \chi(\omega(\sigma_\varepsilon)) &=&
 \E^\uparrow\left[ f_1\left(\omega(\tau_{1}) \right) \right],  \\
 &=& 1 \textrm{, $\P^\uparrow$-a.s.},\label{limite psi}
\end{eqnarray}
because $f_1\left(\omega(\tau_{1}) \right)$ is the density of $\underset{\rightarrow}{\P}^1$ with respect to $\P^{\uparrow,1}$.

Using the stability of the process $V^\uparrow$ given in Lemma \ref{stabiliteuparrow}, we get that for all $c\geq0$,
$$ \E^\uparrow\left[ F\left(  c\,\omega(c^{-\alpha}\cdot)\right)\right]= \E^\uparrow\left[ F\left(\omega\right)\right]=\E\left[ F\left(V^\uparrow \right)\right].$$
So to prove the proposition, it is enough to show that 

$$ \lim_{\varepsilon\rightarrow 0} \lim_{c\rightarrow \infty}\left|   \E^\uparrow\left[ F\left(  c\,\omega(c^{-\alpha}\cdot) \right) \, \ind_{c^{-\alpha}K\leq\sigma_\varepsilon}  \chi\left(\omega(\sigma_\varepsilon) \right)\ind_{\sigma_\varepsilon < \tau_1}\right]    -    \E^\uparrow\left[ F\left(  c\,\omega(c^{-\alpha}\cdot)\right)\right]   \right|=0.$$

We introduce $  \E^\uparrow\left[ F\left(  c\,\omega(c^{-\alpha}\cdot) \right)\ind_{c^{-\alpha}K\leq\sigma_\varepsilon} \right]  $, and we get the following upper bound :

\begin{flushleft}
$\displaystyle \left|   \E^\uparrow\left[ F\left(  c\,\omega(c^{-\alpha}\cdot) \right) \, \ind_{c^{-\alpha}K\leq\sigma_\varepsilon}  \chi\left(\omega(\sigma_\varepsilon) \right)\ind_{\sigma_\varepsilon < \tau_1}\right]    -    \E^\uparrow\left[ F\left(  c\,\omega(c^{-\alpha}\cdot)\right)\right]   \right| \leq$
\end{flushleft}

\begin{flushright}
$\displaystyle           \E^\uparrow\left[\left|  F\left(  c\,\omega(c^{-\alpha}\cdot) \right) \, \ind_{c^{-\alpha}K\leq\sigma_\varepsilon}  \left(\chi\left(\omega(\sigma_\varepsilon) \right)\ind_{\sigma_\varepsilon < \tau_1}-1\right)\right|\right]      +      \E^\uparrow\left[\left| F\left(  c\,\omega(c^{-\alpha}\cdot) \right)\left(\ind_{c^{-\alpha}K\leq\sigma_\varepsilon}-1\right)\right|  \right]                  .   $
\end{flushright}
For $\varepsilon$ fixed, we have $\lim_{c\rightarrow +\infty} \P^\uparrow\left(c^{-\alpha} K\leq \sigma_\varepsilon  \right)=1$. Then for the second part, we get :

$$\displaystyle  \lim_{c\rightarrow \infty} \E^\uparrow\left[  \left| F\left(  c\,\omega(c^{-\alpha}\cdot) \right)\left( \ind_{c^{-\alpha}K\leq\sigma_\varepsilon}  -    1\right) \right|  \right]            \leq  \left\| F \right\|_\infty  \lim_{c\rightarrow \infty}   \E^\uparrow\left[ \left|  \ind_{c^{-\alpha}K\leq\sigma_\varepsilon}  -    1 \right| \right]=0,$$
and for the first part, we have : 

$$\displaystyle \underset{c\rightarrow \infty}{\overline{\lim} }  \E^\uparrow\left[\left| F\left(  c\,\omega(c^{-\alpha}\cdot) \right) \, \ind_{c^{-\alpha}K\leq\sigma_\varepsilon}  \left(1-\chi\left(\omega(\sigma_\varepsilon) \right)\ind_{\sigma_\varepsilon < \tau_1}\right)\right|\right]  \leq   \displaystyle   \left\| F \right\|_\infty   \E^\uparrow\left[\left|  1- \chi\left(\omega(\sigma_\varepsilon) \right)\ind_{\sigma_\varepsilon < \tau_1} \right|  \right] . $$



Let $\varepsilon$ tends to 0, we have $\P^\uparrow$-a.s, $\lim_{\varepsilon\rightarrow 0}\sigma_\varepsilon=0$ and we use \reff{conv ki} to get the convergence : 

$$ \lim_{\varepsilon\rightarrow 0} \lim_{c\rightarrow \infty}   \E^\uparrow\left[\left| F\left(  c\,\omega(c^{-\alpha}\cdot) \right) \, \ind_{c^{-\alpha}K\leq\sigma_\varepsilon}  \left(1-\chi\left(\omega(\sigma_\varepsilon) \right)\ind_{\sigma_\varepsilon < \tau_1}\right)\right|\right]   =0.$$


This completes the proof.



\end{proof}

































\section{Diffusion in a random environment}\label{sec2}
\subsection{Asymptotic behavior of a diffusion in a stable environment}
We suppose that the diffusion $X$ in the random environment $V$ is defined on a probability space $\left(\Omega,\mathcal{F},\mathcal{P}\right)$ and we write, as previously, $\Pp$ the law of $V$. But we take a general case for the environment: \emph{ in all this section, we only suppose that $V$ is an $\alpha$-stable c�d-l�g process on $\R$ ($\alpha>0$) and not necessary a L�vy process}. We will prove that the asymptotic behavior of $L_X$, the local time of the diffusion, depends only on the environment.

We first give some properties of $X$. We can prove (see for example \cite{s:swvsc}) that there exists a Brownian motion $B$ independent of $V$ such that $\mathcal{P}$-a.s.,
\begin{equation}\label{eqx}
 \forall t>0,\quad X(V,t)=S_V^{-1}\left(B\left(T_V^{-1}(t)\right)\right)
\end{equation}
where
\begin{equation*}
	\forall x \in \mathbb{R},\quad S_V(x):=\int_0^xe^{V(y)}\ud y \label{eqS}
\end{equation*}
and
\begin{equation*}
\forall t\geq 0,\quad T_{V}(t):=\int_0^te^{-2V(S_V^{-1}(B(s)))}\ud s.\label{eqT}
\end{equation*}
In order to simplify the notations, we write $S$ and $T$ respectively for $S_V$ and $T_{V}$, when there is no confusion.
From Formula $(\ref{eqx})$ and the occupation time Formula \reff{toccu}, we deduce an expression of the local time process of $X$ using the local time process $L_B$ of the Brownian motion $B$. For all $x\in\R$ and all $t\geq0$,
\begin{eqnarray}
	L_X(t,x)=e^{-V(x)}L_B(T^{-1}(t),S(x)).\label{eqLx} \label{tsc}
\end{eqnarray}

To study the asymptotic behavior of the diffusion, we first work in a quenched environment.

As Brox in the Brownian environment case  (see \cite{b:oddpwm}), we introduce the process $X_c(V,t):=X(c V,t)$. For all $x\in\R$, we write $V^c(x):=c^{-1}V(c^\alpha x)$. There is a direct link between the law of $X_{c}$ and the law of $X$:
\begin{lem} \label{egx}
For every fixed $c$ , conditionally on $V$, the processes $\displaystyle\left( c^{-\alpha}X(V,c^{2\alpha}t); t\geq 0\right)$ and $\displaystyle\left( X_c(V^c,t); t\geq 0 \right)$ have the same law.
\end{lem}

\begin{proof}
We can easily check the result with Formula $(\ref{eqx})$.
\end{proof}

With the definition of the local time (Formula \reff{toccu}), we get an analogous relation for the local time processes:
\begin{cor}\label{eglx}
For every fixed $c$, conditionally on $V$,\\
the processes $\displaystyle\left( L_{X_c(V^c,\cdot)}(t,x) ;t\geq0, x\in \R\right)$ and $\displaystyle\left( c^{-\alpha} L_{X(V,\cdot)}(c^{2\alpha}t,c^\alpha x); t\geq0, x\in \R\right)$ have the same law.
\end{cor}

We will first give a convergence result for the process $L_{X_c}$ and then we will deduce a convergence result for the process $L_{X}$. The asymptotic behavior of $L_{X_c}$ with a quenched environment can be only described with respect to the environment if this one is a "good" environment in the sense of the following definition: 
\begin{defi}
We call a good environment every path $\omega\in\mathcal{V}$ that checks
\begin{enumerate}
\item$\displaystyle\liminf_{x\rightarrow\pm\infty}\omega(x)=-\infty \textrm{ and }\limsup_{x\rightarrow\pm\infty}\omega(x)=+\infty$,
\item $\omega$ is continuous at its  local extrema and the values of $\omega$ in these points are pairwise distinct.
\end{enumerate}
Denote by $\tV$ the set of good environments.
\end{defi}

Remark that if the paths of $V$ are in  $\tV$, then the minimum $\mathfrak m_c(V)$ of the standard valley with height $c$ (see page \pageref{valstand}) is well defined.

Then we can define, for every $c>0$ and every $r>0$,
\begin{align*}
	&a_{cr}=a_{cr}(V_{\mathfrak m_c}):=\sup\left\{x\leq0/V_{\mathfrak m_c}(x)>cr\right\}\textrm{ and} \\
& b_{cr}=b_{cr}(V_{\mathfrak m_c}):=\inf\left\{x\geq0/V_{\mathfrak m_c}(x)>cr\right\}
\end{align*}
where for $x\in\R$, $(V_x(y), y\in\R)$ is the recentered environment in $x$: for all $y\in\R$,
\begin{displaymath}
	V_x(y):=V(x+y)-V(x). \label{shdipo}
\end{displaymath}
Now we can give the result in a quenched environment. We denote by $\mathcal{P}_V$ the probability measure $\mathcal{P}(\cdot|V)$.

\begin{prop}\label{cle}$ $\\
We fix a real number $r\in(0,1)$ and we choose an $\R$-valued function $h$ such that $\underset{c\rightarrow\infty}{\lim}h(c)=1$.
If $V$ is a good environment, then for all $\delta>0$,
\begin{displaymath}
\lima \mathcal{P}_V\left(\sup_{a_r\leq x\leq b_r}\left|\frac{L_{X_c}(e^{c h(c)},\mathfrak m_1+x)}{e^{c h(c)}}\frac{\int_{a_{r}}^{b_{r}}e^{-c V_{\mathfrak m_1}(y)}\ud y}{e^{-c V_{\mathfrak m_1}(x)}}-1\right|\leq \delta \right)=1
\end{displaymath}
and
\begin{displaymath}
\lima \mathcal{P}_V\left(\sup_{x\in\R\setminus[a_r,b_r]}L_{X_c}(e^{c h(c)},\mathfrak m_1+x)\leq \frac{\delta e^{c h(c)}}{\int_{a_{r}}^{b_{r}}e^{-c V_{\mathfrak m_1}(y)}\ud y} \right)=1.
\end{displaymath}
\end{prop}

\begin{proof}
The ideas of the proof are essentially the same as in Sections 2.1 and 2.2 of \cite{ad:llltbd}. For the first one, we proceed as in Proposition 2.1 of \cite{ad:llltbd} with $[a_r,b_r]$ instead of a compact $[-K,K]$. 
The proof of the second formula is the same as the one of Proposition 2.5 of \cite{ad:llltbd} but instead of computing with the integral of the local time (see Formula (34) of \cite{ad:llltbd}), we use the supremum on $\R\setminus[a_r,b_r]$ of the same local time. In both cases, same arguments as the ones of Proposition 3.1 of \cite{ad:llltbd} used with the equivalent of Proposition 2.5 of \cite{ad:llltbd} allow to write the formula with a deterministic time instead of a random time.
\end{proof}

Denote by $m^*_c(t)$ the favorite point of $X_c$,
$$m^*_c(t)=\inf\left\{x\in\R,\ L_{X_c}(t,x)=\sup_{y\in\R}L_{X_c}(t,y)\right\}.$$
The previous proposition implies that this point is localized in the neighborhood of $\mathfrak{m}_1$:
\begin{prop}\label{cvgmc}
If $V$ is a good environment $\P$-a.s. then for any $\delta>0$ such that
\begin{equation}\label{condlim}
\lim_{\varepsilon\rightarrow0}\limsup_{c\rightarrow\infty}\mathbb{P}\left(\inf_{[a_{r},-\delta/c^\alpha]\cup[\delta/c^\alpha,b_{r}]}cV_{\mathfrak m_1}\leq\varepsilon\right)=0
\end{equation}
we have
 \begin{displaymath}
 \lima \mathcal{P}\left(\left|m^*_c(e^{ch(c)})-\mathfrak{m}_1\right|\leq \delta/c^\alpha \right)=1.
 \end{displaymath}
\end{prop}
\begin{proof}
 Let $\varepsilon\in(0,1/2)$, for any $c>1$, we define two events:
 $$
 A_c:=\left\{\sup_{a_r\leq x\leq b_r}\left|\frac{L_{X_c}(e^{c h(c)},\mathfrak m_1+x)}{e^{c h(c)}}\frac{\int_{a_{r}}^{b_{r}}e^{-c V_{\mathfrak m_1}(y)}\ud y}{e^{-c V_{\mathfrak m_1}(x)}}-1\right|\leq \varepsilon \right\}
 $$
 and
$$
 B_c:=\left\{\sup_{x\in\R\setminus[a_r,b_r]}L_{X_c}(e^{c h(c)},\mathfrak m_1+x)\leq \frac{\varepsilon e^{c h(c)}}{\int_{a_{r}}^{b_{r}}e^{-c V_{\mathfrak m_1}(y)}\ud y} \right\}.
 $$
 On $A_c\cap B_c$, we have the following inequality
 $$L_{X_c}(e^{ch(c)},m^*_c(e^{ch(c)}))\geq L_{X_c}(e^{ch(c)},\mathfrak{m}_1)\geq \frac{(1-\varepsilon)e^{c h(c)}}{\int_{a_{r}}^{b_{r}}e^{-c V_{\mathfrak m_1}(y)}\ud y}.$$
Thereby $m^*_c(e^{ch(c)})\in(a_r,b_r)$ and so
 $$L_{X_c}(e^{ch(c)},m^*_c(e^{ch(c)}))\leq \frac{(1+\varepsilon)e^{c h(c)}}{\int_{a_{r}}^{b_{r}}e^{-c V_{\mathfrak m_1}(y)}\ud y}e^{-c V_{\mathfrak m_1}(m^*_c(e^{ch(c)}))}.$$
 Necessarily, 
 $$c V_{\mathfrak m_1}(m^*_c(e^{ch(c)})\leq\log\frac{1+\varepsilon}{1-\varepsilon}.$$
 Denote 
 $$
 \mathcal{E}^\varepsilon_c=\left\{V\in\tV\ ,\ \inf_{[a_r,-\delta/c^\alpha]\cup[\delta/c^\alpha,b_r]}cV_{\mathfrak m_1}>\log\frac{1+\varepsilon}{1-\varepsilon}\right\}.
 $$
 If $V\in\mathcal{E}^\varepsilon_c$, then on $A_c\cap B_c$, 
 $$\left|m^*_c(e^{ch(c)})-\mathfrak{m}_1\right|\leq \delta/c^\alpha.$$
 Thus,
 \begin{align*}
  \mathcal{P}\left(\left|m^*_c(e^{ch(c)})-\mathfrak{m}_1\right|>\delta/c^\alpha \right)&\leq \mathcal{P}(\overline{A_c})+\mathcal{P}(\overline{B_c})+\mathcal{P}(\overline{\mathcal{E}^\varepsilon_c})
 \end{align*}
 and therefore
 \begin{align*}
  \limsup_{c\rightarrow\infty}\mathcal{P}\left(\left|m^*_c(e^{ch(c)})-\mathfrak{m}_1\right|>\delta/c^\alpha \right)&\leq\lim_{\varepsilon\rightarrow0}\limsup_{c\rightarrow\infty}\mathcal{P}(\overline{\mathcal{E}^\varepsilon_c})=0
 \end{align*}
according to Hypothesis \eqref{condlim}.
\end{proof}


Now we can come back to the process $L_{X}$ and prove the following theorem, main result of this section:

\begin{theo}\label{limL}
Fix a real number $r\in(0,1)$. If the environment $V$ is a good environment $\P$-a.s. and there exists $\alpha>0$ such that $V$ is $\alpha$-stable then for all $\delta>0$,
\begin{displaymath}
\lima \mathcal{P}\left(\sup_{a_{cr}\leq x\leq b_{cr}}\left|\frac{L_{X}(e^{c},\mathfrak m_c+x)}{e^{c}}\frac{\int_{a_{cr}}^{b_{cr}}e^{- V_{\mathfrak m_c}(y)}\ud y}{e^{-V_{\mathfrak m_c}(x)}}-1\right|\leq \delta \right)=1.
\end{displaymath}
and
\begin{displaymath}
\lima \mathcal{P}\left(\sup_{x\in\R\setminus[a_{cr},b_{cr}]}L_{X}(e^{c},\mathfrak m_c+x)\leq \frac{\delta e^{c}}{\int_{a_{cr}}^{b_{cr}}e^{-V_{\mathfrak m_c}(y)}\ud y} \right)=1.
\end{displaymath}
Recall the definition of $m^*$ of Theorem \ref{cvptfav}:
$$m^*(t)=\inf\left\{x\in\R,\ L_{X}(t,x)=\sup_{y\in\R}L_{X}(t,y)\right\}.$$
If for any $\delta>0$, we have
\begin{equation}\label{condlima}
\lim_{\varepsilon\rightarrow0}\limsup_{c\rightarrow\infty}\mathbb{P}\left(\inf_{[a_{cr},-\delta]\cup[\delta,b_{cr}]}V_{\mathfrak m_c}\leq\varepsilon\right)=0
\end{equation}
then 
 \begin{displaymath}
 \lima \mathcal{P}\left(\left|m^*(e^{c})-\mathfrak{m}_c\right|\leq \delta \right)=1.
 \end{displaymath}
\end{theo}

\begin{rem}
We deal in this article with stable L�vy processes and we will see later that they verify  Hypothesis \eqref{condlima}. Moreover, the previous theorem also applies when the environment $V$ is a fractional Brownian motion.
\end{rem}



\begin{proof}
We fix $c>0$ and we recall that $V^c(\cdot):=c^{-1}V(c^\alpha\cdot)$. By the stability of $V$, the processes $V$ and $V^c$ have same law. We first remark that, as the paths of $V$ are in $\tV$, $\mathcal{P}$-a.s., the standard valley is well defined and therefore 
 
\begin{align*}
\mathfrak m_1(V^c)&=c^{-\alpha}\mathfrak m_c(V),\\
 a_{ r}(V^c_{\mathfrak m_1(V^c)})&=c^{-\alpha}a_{c r}(V_{\mathfrak m_c(V)}),\\
 b_{r}(V^c_{\mathfrak m_1(V^c)})&=c ^{-\alpha} b_{c r}(V_{\mathfrak m_c(V)}),
\end{align*}
and for all $x\in\R$,
\begin{equation}\label{scaleV}
V^c_{\mathfrak m_1(V^c)}(x)=\frac{1}{c}V_{\mathfrak m_c(V)}(c^\alpha x).
\end{equation}

Thus, using the substitution $z=c^\alpha y$, we get that

\begin{equation*}
\int_{a_r}^{b_r}e^{-c V_{\mathfrak m_1}(y)}dy=c^{-\alpha}\int_{a_{cr}}^{b_{cr}}e^{V_{\mathfrak m_c}(z)}dz.
\end{equation*}

Now we replace $t$ by $c^{-2\alpha}e^{c}$ and $x$ by $\mathfrak m_1(V^c)+c^{-\alpha}x$ in Corollary \ref{eglx}, then we get that, conditionally on $V$,

$\displaystyle\left(L_{X_c( V^c,\cdot)}(c^{-2\alpha}e^{c},\mathfrak m_1(V^c)+\ax),x\in\R\right)$ and $\displaystyle\left(\frac{1}{c^\alpha}L_{X(V,\cdot)}(e^c,\mathfrak m_c(V)+x), x\in\R\right)$ have the same law.
Moreover the processes $V$ and $V^c$ have the same law, so we obtain that
 
$\displaystyle\left(L_{X_c( V,\cdot)}(c^{-2\alpha}e^{c},\mathfrak m_1(V)+\ax), x\in\R\right)$ and $\displaystyle\left(\frac{1}{c^\alpha}L_{X(V,\cdot)}(e^c,\mathfrak m_c(V)+x), x\in\R\right)$ have the same law without conditioning.
Thus for all $c>0$, $K>0$, $r\in(0,1)$ and $\delta>0$,
\begin{align*}
&\mathcal{P}\left(\sup_{a_{cr}\leq x\leq b_{cr}}\left|\frac{L_{X(V,\cdot)}(e^{c},\mathfrak m_c(V)+x)}{e^{c}}\frac{\int_{a_{c r}}^{b_{c r}}e^{-V_{\mathfrak m_c}(y)}\ud y}{e^{-V_{\mathfrak m_c}(x)}}-1\right|<\delta\right)\quad=\\
&\mathcal{P}\left(\sup_{a_r\leq x\leq b_r}\left|\frac{L_{X_c( V,\cdot)}(c^{-2\alpha}e^{c},\mathfrak m_1(V)+x)}{c^{-2\alpha}e^{c}}\frac{\int_{a_{r}}^{b_{r}}e^{-c V_{\mathfrak m_1}(y)}\ud y}{e^{-c V_{\mathfrak m_1}(x)}}-1\right|<\delta\right).
\end{align*}
To finish, it is enough to remark that $c^{-2\alpha}e^{c}=e^{c(1-\frac{2\alpha}{c}\log c)}$ and $\limab(1-\frac{2\alpha}{c}\log c)=1$. Then, using Proposition \ref{cle}, we obtain the first convergence of Theorem \ref{limL}. The other ones are proved in the same way.
\end{proof}












 


\subsection{Limit law of the renormalized local time}
\emph{Now we suppose again that our environment is an $\alpha$-stable L�vy process with $\alpha\in[1,2]$ and is not a pure drift}.


\begin{lem}
Hypothesis \reff{condlima} is true for such a L�vy process:
\begin{equation*}
\lim_{\varepsilon\rightarrow0}\limsup_{c\rightarrow\infty}\mathbb{P}\left(\inf_{[a_{cr},-\delta]\cup[\delta,b_{cr}]}V_{\mathfrak m_c}\leq\varepsilon\right)=0.
\end{equation*}
Then, by Theorem \ref{limL}, we have the convergence in probability of $m^*(e^{c})-\mathfrak{m}_c$ to $0$.
\end{lem}
\begin{proof}
Recall that $\mathfrak{m}_c=m_c^+$ or $\mathfrak{m}_c=m_c^-$. As the proof is similar in the both cases, we only study the first one. If $\beta<\alpha$ then we use the convention $[\alpha,\beta]=\emptyset$ and $\inf_{[\alpha,\beta]}V_{\mathfrak m_c}=+\infty$.

\begin{align*}
&\mathbb{P}\left(\inf_{[a_{cr},-\mathfrak{m}_c]\cup[-\mathfrak{m}_c,-\delta]\cup[\delta,b_{cr}]}V_{\mathfrak m_c}\leq\varepsilon,\mathfrak{m}_c=m_c^+\right) \leq \\
&\mathbb{P}\left(\inf_{[a_{cr},-\mathfrak{m}_c]}V_{\mathfrak m_c}\leq\varepsilon,\mathfrak{m}_c=m_c^+\right)
+ \mathbb{P}\left(\inf_{[-m^+_c,-\delta]}V_{m^+_c}\leq\varepsilon\right)  
+ \mathbb{P}\left(\inf_{[\delta,b_{cr}]}V_{m^+_c}\leq\varepsilon \right).
\end{align*}

We use the stability of $V$, $\mathfrak{m}_c$, $a_{cr}$ and $b_{cr}$ (see \reff{scaleV} above) to get for any $\varepsilon>0$:

\begin{align*}
\mathbb{P}\left(\inf_{[a_{cr},-\mathfrak{m}_c]}V_{\mathfrak m_c}\leq\varepsilon,\mathfrak{m}_c=m_c^+\right) & = \mathbb{P}\left(\inf_{[a_{r},-\mathfrak{m}_1]}V_{\mathfrak m_1}\leq \varepsilon c^{-1},\mathfrak{m}_1=m_1^+\right)\\
& \xrightarrow[c\rightarrow \infty]{} \mathbb{P}\left(\inf_{[a_{r},-\mathfrak{m}_1]}V_{\mathfrak m_1}\leq\ 0,\mathfrak{m}_1=m_1^+\right).
\end{align*}
By uniqueness of the minima (Lemma \ref{continuite}), this last probability is equal to 0. 

$\P$-a.s. $\inf_{[\delta,+\infty)}\hat{V}^\uparrow>0$, then Point (1) of Proposition \ref{vallee} gives us that 
$$\lim_{\varepsilon\rightarrow0}\limsup_{c\rightarrow\infty}\mathbb{P}\left(\inf_{[-m^+_c,-\delta]}V_{m^+_c}\leq\varepsilon\right)=0.$$


Using now the stability of $V$ and then Point (2) of Proposition \ref{vallee}, we obtain:

\begin{align*}
& \mathbb{P}\left(\inf_{[\delta,b_{cr}]}V_{m^+_c}\leq\varepsilon\right)  = \mathcal{P}\left(\inf_{[c^{-\alpha}\delta,b_{r}]}V_{m^+_1}\leq \varepsilon c^{-1}\right) =\int \ind_{\left\{ \inf_{[c^{-\alpha}\delta,\tau_{r}(\omega)]}\omega\leq \varepsilon c^{-1} \right\}} \underset{\rightarrow}{\P}^1(d\omega)\\
&=\int \ind_{\left\{ \inf_{[c^{-\alpha}\delta,\tau_{r}(\omega)]}\omega\leq \varepsilon c^{-1} \right\}} f_1\left(\omega(\tau_1(\omega))\right) \P^{\uparrow,1}(d\omega) \leq \frac{\Gamma(\alpha\rho+1)}{\mathcal N(\tau_1<\infty)} \mathbb{P}\left(\inf_{[\delta,\tau_{cr}(V^\uparrow)]}V^\uparrow\leq\varepsilon\right)
\end{align*}
where the last inequality comes from the expression of $f_1$ given by \reff{expressionf1}. Finally,

$$\lim_{\varepsilon\rightarrow0}\limsup_{c\rightarrow\infty}\mathbb{P}\left(\inf_{[\delta,b_{cr}]}V_{m^+_c}\leq\varepsilon\right)\leq \frac{\Gamma(\alpha\rho+1)}{\mathcal N(\tau_1<\infty)} \mathbb{P}\left(\inf_{[\delta,\infty)}V^\uparrow=0\right)=0.$$
This finishes the proof.
\end{proof}




According to Theorem \ref{limL}, to obtain the weak convergence of the local time process, we only have to study the convergence of $e^{-V_{\mathfrak m_c}}/\int_{a_{rc}}^{b_{rc}} e^{-V_{\mathfrak m_c}}$, what we do now.

For $\omega\in\cv$, recall the definition of $\omega^+$ and $\omega^-$ given in \reff{w+w-} and notations of the first part: 
\begin{align*}
\underline{\tau}^+_c &= \inf\{ t\geq 0 ; V^+(t)-\underline{V}^+(t) \geq c \},\\
m^+_c &= \sup\{ t\leq \underline{\tau}^+_c ; V^+(t) - \underline{V}^+(t) = 0 \}\textrm{ and}\\
J_c^+&=\left(V^+(m^+_c)+c\right)\vee\overline{V}^+(m^+_c)
\end{align*}
and similar variables defined using $V^-$,  $\underline{\tau}^-_c$, $m^-_c$ and $J^-_c$. As said before, we can prove that the minimum of the standard valley  $\mathfrak m_c$ is equal to $m^+_c$ if $J^+_c<J^-_c$ and $-m^-_c$ otherwise.








\begin{prop}\label{conv exp}
Let $V$ be a stable L�vy process with index $\alpha\in[1,2]$ which is not a pure drift. We fix $r\in(0,1)$.

When $c\rightarrow\infty$, the process $\displaystyle \frac{e^{-V_{\mathfrak m_c}}}{\int_{a_{rc}}^{b_{rc}} e^{-V_{\mathfrak m_c}(y)}dy  }$ converges weakly (in the Skorohod sense) to $\displaystyle \frac{e^{-\tilde{V}}}{\int_{-\infty}^{+\infty} e^{-\tilde{V}(y)}dy  }$ where the law of $\tilde{V}$ is $\tilde{\P}$ and, under $\tilde{\P}(d\omega)$, $\omega^+$ and $\omega^-$ are independent, and distributed respectively as $\P^\uparrow$, the law of the L�vy process $V$ conditioned to stay positive, and $\hat{\P}^\uparrow$ the law of the dual process $V^-$ conditioned to stay positive.
\end{prop}

In order to prove Proposition \ref{conv exp}, we need the following Lemmas. In the first one, we prove that the integral $\int_{-\infty}^{+\infty} e^{-\tilde{V}(y)}dy $ is finite.







\begin{lem}
 We have  
$$\hbox{$\tilde{\P}$-a.s. }\int_{-\infty}^{+\infty} e^{- \tilde{V}(y)} dy <\infty.$$
\end{lem}



\begin{proof}
It is enough to prove the integrability on $\R_+$ for the process $(\hat{V}^\uparrow(x),x\geq0)$ whose law is $\hat{\P}^{\uparrow}$. Fix $t\geq0$ and recall that $U(t) = -V(L^{-1}(t))$. We prove the integrability of $e^{-U_t}$. The process $U$ is a subordinator, then it is characterized by its Laplace exponent $\psi_U$ (see Section III.1, \cite{b:pl}): for all $\lambda\geq0$,
$$\E\left[  e^{-\lambda U(t)} \right] = e^{-t\psi_U(\lambda)}.$$

For all $t>0$, $U(t)>0$ then $\psi_U(1)> 0$, we obtain that $\E\left[  e^{- U(t)} \right] = e^{-t\psi_U(1)}$ is integrable on $\R_+$. 
We recall the notation of the Proposition \ref{vallee}, we get
$$U(t) = - \underline{V}(d(t)) \leq \mathcal R_{V-\underline{V}}(t) - \underline{V}(d(t)).$$

Thanks to this Proposition, the process $(\mathcal R_{V-\underline{V}}(t) - \underline{V}(d(t)) ; t\geq 0)$ is equal in law to the process $\hat{V}^{\uparrow}$. 
Thus, $\E\left[  e^{- \hat{V}^{\uparrow}(t)} \right]$ is also integrable on $\R_+$, we deduce that $e^{- \hat{V}^{\uparrow}}$ is $\hat{\P}^{\uparrow}$-a.s. integrable.\\
\end{proof}






\begin{lem}\label{equivalence integrale}
Let $\omega$ be a non-negative c�d-l�g function with $\omega(0)=0$ which is continuous in its extrema and such that $0$ is the unique point of minimum of $\omega$. Then for all $a\leq \alpha< 0< \beta \leq b$, we get
$$\int_a^b e^{-c \omega(y)}dy \underset{c\rightarrow +\infty}{\sim} \int_\alpha^\beta e^{-c \omega(y)}dy.$$
\end{lem}

\begin{proof}
\begin{align*}
\int_a^b e^{-c \omega(y)}dy - \int_\alpha^\beta e^{-c \omega(y)}dy &=\int_a^\alpha e^{-c \omega(y)}dy + \int_\beta^b e^{-c \omega(y)}dy\\
&\leq |\alpha-a| e^{-c \inf_{y\in(a,\alpha)}\omega(y)} + |b-\beta| e^{-c \inf_{y\in(\beta,b)}\omega(y)}.
\end{align*}
As $\omega$ is positive on $\R\setminus\{0\}$ and continuous at its extrema, for all $x\neq0$, $\inf_{(a,\alpha)}\omega>0$ and $\inf_{(\beta, b)}\omega>0$ (see Remark \ref{continuite ext locaux}), then $\lim_{c\rightarrow +\infty} \int_a^b e^{-c \omega(y)}dy - \int_\alpha^\beta e^{-c \omega(y)}dy =0$.

By the Laplace method,
$$\lim_{c\rightarrow +\infty} \frac{1}{c} \log \int_{a}^{b} e^{-c\omega(y)}dy = \sup_{(a,b)}(-\omega).$$
As $a\leq 0\leq b$ then $\sup_{(a,b)}(-\omega)=0$, thus $\int_a^b e^{-c\omega(y)}dy = e^{o(c)}$.
$$\lim_{c\rightarrow +\infty}   \frac{\int_a^b e^{-c\omega(y)}dy-\int_\alpha^\beta e^{-c\omega(y)}dy}{\int_a^b e^{-c\omega(y)}dy} \leq \lim_{c\rightarrow +\infty}e^{-c\inf_{(a,\alpha)}\omega-c\inf_{(\beta,b)}\omega-o(c)}=0.$$
This implies the equivalence.
\end{proof}

























\begin{proof}[Proof of Proposition \ref{conv exp}]
We follow the same idea as in the proof of Proposition \ref{convergence}.

It is enough to prove that for a bounded continuous function $F$ in the Skorohod topology and such that $F(\omega)=F(\omega\ind_{[-K,K]})$,
$$\lim_{c\rightarrow +\infty} \E\left[ F\left( \frac{e^{-V_{\mathfrak m_c}}}{\int_{a_{rc}}^{b_{rc}} e^{-V_{\mathfrak m_c}(y)}dy  } \right) \right] = \E\left[ F\left( \frac{e^{-\tilde{V}}}{\int_{-\infty}^{+\infty} e^{-\tilde{V}(y)}dy  } \right)  \right].$$

Using Formula (\ref{scaleV}) and the fact that $\mathfrak m_1\in\{-m_1^-,m_1^+\}$, 
\begin{flushleft}
$\displaystyle \E\left[ F\left( \frac{e^{-V_{\mathfrak m_c}}}{\int_{a_{rc}}^{b_{rc}} e^{-V_{\mathfrak m_c}(y)}dy  } \right) \right] =  \E\left[ F\left( \frac{e^{- c V_{\mathfrak m_1}(c^{-\alpha} \cdot) }}{c^\alpha\int_{a_{r}}^{b_{r}} e^{-c V_{\mathfrak m_1}(y)}dy  } \right) , \mathfrak m_1 = m_1^+\right] +$
\end{flushleft}
\begin{flushright}
$\displaystyle\E\left[ F\left( \frac{e^{- c V_{\mathfrak m_1}(c^{-\alpha} \cdot) }}{c^\alpha\int_{a_{r}}^{b_{r}} e^{-c V_{\mathfrak m_1}(y)}dy  } \right) , \mathfrak m_1 = -m_1^-\right].$
\end{flushright}

We will compute the two terms of the sum by the same method, thus we only study the first one. Thanks to Lemma \ref{equivalence integrale}, it suffices to take the integral from $-\mathfrak m_1\vee a_r$ to $b_r$.
By definition, the law of the process $\left(\tilde{V}(t),t\geq0\right)$ is $\P^\uparrow$ and the law of the process $\left(\tilde{V}((-t)-),t\geq0\right)$ is $\hat{\P}^\uparrow$, moreover they are independent. Then we write them respectively $V^\uparrow$ and $\hat{V}^\uparrow$. Using the fact that $\mathfrak m_1=m_1^+$ if and only if $J^+_1<J^-_1$ and the result of Proposition \ref{vallee}, we get that

\begin{flushleft}
$\displaystyle \E\left[ F\left( \frac{e^{- c V_{\mathfrak m_1}(c^{-\alpha} \cdot) }}{c^\alpha\int_{-\mathfrak m_1\vee a_{r}}^{b_{r}} e^{-c V_{\mathfrak m_1}(y)}dy  } \right) , \mathfrak m_1=m_1^+ , \frac{K}{c^{\alpha}}\leq \mathfrak m_1\wedge \tau_1(\underset{\rightarrow}{V}) \right] = $
\end{flushleft}
\begin{flushright}
\begin{equation}\label{reprise}
\displaystyle \hspace{2cm}\E\left[ F\left( \frac{e^{- c \tilde{V}(c^{-\alpha} \cdot) }}{c^\alpha\int_{-\tilde{a}_{r}}^{\tilde{b}_{r}} e^{-c \tilde{V}(y)}dy  } \right) f_1\left(V^\uparrow\left(\tau_1  \right) \right) , \tilde{J}_1^+\leq\tilde{J}_1^-, \frac{K}{c^{\alpha}}\leq \hat{m}^\uparrow_1\wedge \tau_1(V^\uparrow) \right]
\end{equation}
\end{flushright}
where $\tilde{a}_{r}$, $\tilde{b}_{r}$, $\tilde{J}_1^+$ and $\tilde{J}_1^-$ are defined similarly as $a_r$, $b_r$, $J_1^+$ and $J^-_1$ but using the process $\tilde{V}$ :
\begin{align*}
    \tilde{a}_{r}&:=\hat{m}^\uparrow_1\vee \inf\left\{x\geq0/\hat{V}^\uparrow(x)>r\right\},\\
    \tilde{b}_{r}&:=\inf\left\{x\geq0/V^\uparrow(x)>r\right\},\\
\tilde{J}_1^+&:= \left(1-\hat{V}^\uparrow(\hat{m}^\uparrow_1)\right)\vee \left(\sup_{(0,\hat{m}^\uparrow_1)}{\hat{V}^\uparrow} - \hat{V}^\uparrow(\hat{m}^\uparrow_1)\right),\\
\textrm{and }\tilde{J}_1^-&:= \left(1-W^{\uparrow}(m^{\uparrow}_1)\right)\vee \left(\sup_{(0,m^{\uparrow}_1)}{W^{\uparrow}} - W^{\uparrow}(m^{\uparrow}_1)\right)
   \end{align*}
where $W^{\uparrow}$ is a process with law $\P^\uparrow$ independent of $\tilde{V}$. To simplify the notation, we write $F_{\tilde{a}_r,\tilde{b}_r}$ for $F\left( \frac{e^{- c \tilde{V}(c^{-\alpha} \cdot) }}{c^\alpha\int_{-\tilde{a}_{r}}^{\tilde{b}_{r}} e^{-c \tilde{V}(y)}dy  } \right)$.

As in the proof of Proposition \ref{convergence}, we fix $\varepsilon>0$ and we separate the several terms of the expectation with respect to $(\tilde{V}(t), t \leq -\sigma^1_\varepsilon)$, $(\tilde{V}(t),-\sigma^1_\varepsilon \leq t\leq \sigma^2_\varepsilon)$ and $(\tilde{V}(t),\sigma^2_\varepsilon\leq t)$, where:
$$
\sigma^1_\varepsilon=\sigma_\varepsilon(\hat{V}^\uparrow)=\inf\left\{t>0/{\hat{V}^\uparrow}(t)-\uu{\hat{V}^\uparrow}(t)=0 \textrm{ and } \exists s<t, \hat{V}^\uparrow(s)-\uu{\hat{V}^\uparrow}(s)>\varepsilon\right\},
$$
$$
\sigma^2_\varepsilon=\sigma_\varepsilon(V^\uparrow)=\inf\left\{t>0/{V^\uparrow}(t)-\uu{V^\uparrow}(t)=0 \textrm{ and } \exists s<t, V^\uparrow(s)-\uu{V^\uparrow}(s)>\varepsilon\right\},
$$

to obtain the asymptotic independence of $F_{\tilde{a}_r,\tilde{b}_r}$, of $V^\uparrow(\tau_1)$ and of $\tilde{J}^+_1$.
 


\begin{center}
\begin{figure}[ht]
\caption{The process $\tilde{V}$.}

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\end{figure}
\end{center}












Study in detail the hypothesis $\tilde{J}^+_1 \leq \tilde{J}^-_1 $. Denote 
$$M_+ = \sup_{(0,\hat{m}^\uparrow_1)}\hat{V}^\uparrow  -  \hat{V}^\uparrow(\hat{m}_1^\uparrow)\hbox{ and }M_\varepsilon = \sup_{(0,\sigma^1_\varepsilon)}\hat{V}^\uparrow-\hat{V}^\uparrow(\hat{m}_1^\uparrow).$$

For $\varepsilon$ small enough, $M_\varepsilon < M_+$, in this case: $\sigma^1_\varepsilon < \hat{m}^\uparrow_1$, $\sup_{(0,\hat{m}^\uparrow_1)}\hat{V}^\uparrow = \sup_{(\sigma^1_\varepsilon,\hat{m}^\uparrow_1)}\hat{V}^\uparrow$ and $\hat{V}^\uparrow(\hat{m}^\uparrow_1) = \hat{V}^\uparrow(\sigma^1_\varepsilon) + \hat{V}^\uparrow_{\sigma^1_\varepsilon}(\hat{m}_1^\uparrow - \sigma^1_\varepsilon)$. Moreover, for $\varepsilon$ small enough, $\sigma^2_\varepsilon < \tau_1(V^\uparrow)$, thus $$\tilde{J}^+_1 = \phi_\varepsilon\left(\hat{V}^\uparrow(\sigma^1_\varepsilon) \right) \hbox{ with }\phi_\varepsilon(x) = \left(1-   x - \hat{V}^\uparrow_{\sigma^1_\varepsilon}(\hat{m}_1^\uparrow - \sigma^1_\varepsilon) \right)\vee \left(  \sup_{(\sigma^1_\varepsilon,\hat{m}^\uparrow_1)}\hat{V}^\uparrow_{\sigma^1_\varepsilon} + \hat{V}^\uparrow_{\sigma^1_\varepsilon}(\hat{m}^\uparrow_1-\sigma^1_\varepsilon) \right).$$


We follow the idea of the proof of Proposition \ref{convergence} with Lemma \ref{levy renormalise}. We use Lemma \ref{equivalence integrale} for the integral in the function $F$ and we remark that 
$$\lim_{c\rightarrow +\infty} \P\left( \frac{K}{c^{\alpha}} \leq \hat{m}^\uparrow_1\wedge \tau_1(V^\uparrow) \right)=\lim_{c\rightarrow +\infty} \P\left( \frac{K}{c^{\alpha}}\leq \sigma^1_\varepsilon \wedge \sigma^2_\varepsilon \right)=1.$$ 
Recall \reff{reprise}, we obtain




\begin{flushleft}
$
\displaystyle \lim_{c\rightarrow +\infty}\E\left[ F_{\tilde{a}_r,\tilde{b}_r}\; f_1\left(V^\uparrow\left(\tau_1  \right) \right) , \tilde{J}_1^+\leq\tilde{J}_1^-, \frac{K}{c^{\alpha}}\leq \hat{m}^\uparrow_1\wedge \tau_1(V^\uparrow) \right] =
$
\end{flushleft}


\begin{flushright}
$
\displaystyle \lim_{\varepsilon\rightarrow 0}\lim_{c\rightarrow +\infty} \E \left[ F_{\sigma_\varepsilon^1,\sigma_\varepsilon^2} \;   \chi_2\left(V^\uparrow(\sigma_\varepsilon^2)\right)\,\chi_1\left(\hat{V}^\uparrow(\sigma_\varepsilon^1)\right) , \frac{K}{c^\alpha}\leq  \sigma_\varepsilon^1\wedge \sigma_\varepsilon^2, \sigma_\varepsilon^1\leq \hat{m}^\uparrow_1, \sigma_\varepsilon^2\leq \tau_1(V^\uparrow) \right]
$
\end{flushright}
where
\begin{align*}
\chi_1(x) &= \P\left( \left(1-   x - \hat{V}^\uparrow(\hat{m}_1^\uparrow) \right)\vee M_+    <\tilde{J}^-_1 \right) \hbox{ and}\\
\chi_2(x) &= \E\left[  f_1\left(x+V^\uparrow(\tau_{1-x})\right) \right].
\end{align*}
Moreover, $\lim_{\varepsilon\rightarrow 0} \chi_1\left(\hat{V}^\uparrow(\sigma_\varepsilon^1)\right) = \P(\tilde{J}^+_1 \leq \tilde{J}^-_1)=\P(\mathfrak m_1>0)$ and $\lim_{\varepsilon\rightarrow 0} \chi_2\left(V^\uparrow(\sigma_\varepsilon^2)\right) = 1$ thanks to \reff{limite psi}. Then we obtain:

$$\lim_{\varepsilon\rightarrow 0}\lim_{c\rightarrow +\infty} \left| \E\left[ F_{\tilde{a}_r,\tilde{b}_r} \left(\P(\mathfrak m_1>0)  -    \chi_1\left(\hat{V}^\uparrow(\sigma_\varepsilon^1)\right) \,  \chi_2\left(V^\uparrow(\sigma_\varepsilon^2)\right) \right)\right] \right|=0.$$


We use the stability of $V^\uparrow$ and of $\hat{V}^\uparrow$ (see Lemma \ref{stabiliteuparrow}). Moreover, for all $r\in(0,1)$, $\displaystyle \lim_{c\rightarrow +\infty} \tilde{a}_{cr}=-\infty$ and $\displaystyle \lim_{c\rightarrow +\infty} \tilde{b}_{cr}=+\infty$  a.s., then $\frac{e^{-\tilde{V}}}{\int_{a_{rc}}^{b_{rc}} e^{-\tilde{V}(y)}dy  }$ converges to $\frac{e^{-\tilde{V}}}{\int_{-\infty}^{+\infty} e^{-\tilde{V}(y)}dy  }$ in the Skorohod topology on $[-K,K]$. Thus 
$$ \lim_{c\rightarrow +\infty} \E\left[ F\left( \frac{e^{- c V_{\mathfrak m_1}(c^{-\alpha} \cdot) }}{c^\alpha\int_{a_{r}}^{b_{r}} e^{-c V_{\mathfrak m_1}(y)}dy  } \right) ,\mathfrak m_1 = m_1^+\right]  =\E\left[ F\left( \frac{e^{-\tilde{V}}}{\int_{-\infty}^{+\infty} e^{-\tilde{V}(y)}dy  } \right)  \right]\P(\mathfrak m_1>0).$$

We similarly study the case where $\mathfrak m_1=-m_1^-$, we obtain:

$$\lim_{c\rightarrow +\infty} \E\left[ F\left( \frac{e^{- c V_{\mathfrak m_1}(c^{-\alpha} \cdot) }}{c^\alpha\int_{a_{r}}^{b_{r}} e^{-c V_{\mathfrak m_1}(y)}dy  } \right) , \mathfrak m_1 = -m_1^-\right] =\E\left[ F\left( \frac{e^{-\tilde{V}}}{\int_{-\infty}^{+\infty} e^{-\tilde{V}(y)}dy  } \right)  \right]\P(\mathfrak m_1<0).$$

Then we obtain the result.








\end{proof}


Using Theorem \ref{limL} and Proposition \ref{conv exp}, we deduce Theorem \ref{cvloi} given in the introduction.




\vspace{1cm}
\textbf{Acknowledgments: }
 We are grateful to Romain Abraham and Pierre Andreoletti for helpful discussions. We also want to thank an anonymous referee for careful reading and many suggestions.














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\newblock { Adv. Appl. Prob.}, vol 12, pp 893-902, 1980.

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Y. HU, Z. SHI et M. YOR.
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J. Jacod, A.N. SHIRYAEV.
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\newblock {A Series of Comprehensive Studies in Mathematics 288. Springer Verlag}, 1987.

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K. KAWAZU and H. TANAKA.
\newblock {\em On the maximum of diffusion process in a drifted Brownian environment},
\newblock {S�minaire de Probabilit� XXVII}, Lectures Notes in Math., pp 78-85, 1991.

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K. KAWAZU and H. TANAKA.
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\newblock { Theory Probab. Appl.}, vol 38, pp 382-385, 1985.

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Z. SHI.
\newblock {\em Sinai's walk via stochastic calculus},
\newblock { Milieux Al\'eatoires, Soc. Math. France, Paris.}, pp 53-74, 2001.

\bibitem{s:ltcre}
Z. SHI.
\newblock {\em A Local time curiosity in random environment},
\newblock {Stoch. Proc. App. }, vol 76, pp 231-250, 1998.

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A. SINGH. 
\newblock {\em Rates of convergence of a transient diffusion in a spectrally negative potential},
\newblock { Ann. Probab.}, vol 36, pp 279-318, 2008.

\bibitem{t:atebmdbp}
M. TALET. 
\newblock {\em Annealed tail estimates for a Brownian motion in a drifted Brownian potential},
\newblock { Ann. Probab.}, vol 35, pp 32-67, 2007.

\bibitem{t:lpcspdre}
H. TANAKA. 
\newblock {\em L�vy processes conditioned to stay positive and diffusions in random environments},
\newblock { Advanced Studies in Pure Math.}, vol 39, pp 355-376, 2004.


\bibitem{Tanaka}
H. TANAKA. 
\newblock {\em Localization of a Diffusion Process in a One-Dimensional Brownian Environmement},
\newblock { Comm. Pure Appl. Math.}, vol 17, pp 755-766, 1994.

\bibitem{t:ltoddpbe}
H. TANAKA. 
\newblock {\em Limit theorem for one-dimensional diffusion process in Brownian Environmement},
\newblock { Lectures Notes in Math., Springer, Berlin et New-York }, 1322, pp 156-172, 1988.


\end{thebibliography}


\end{document}
0909.2929.extracted_bib
\begin{thebibliography}{10}

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J. BERTOIN.
\newblock {\em Sur la d\'ecomposition de la trajectoire d'un processus de L\'evy spectrallement positif en son infimum},
\newblock { Ann. Inst. H. Poincar\'e}, vol. 27, 537-547, 1991.

\bibitem{b:oddpwm}
Th. BROX.
\newblock {\em A one-dimensional diffusion process in a Wiener medium},
\newblock { Ann. Probab.}, vol 14, 4, pp 1206-1218, 1986.

\bibitem{c:mvbmrlp}
P. CARMONA.
\newblock {\em The mean velocity of a Brownian Motion in a random L\'evy potential},
\newblock { Ann. Probab.}, vol 25, 4, pp 1774-1788, 1997.

\bibitem{c:oddare}
D. CHELIOTIS.
\newblock {\em One dimensional diffusion in an asymmetric random environment},
\newblock { Ann. Inst. H. Poincar\'e}, vol 42, 6, pp 715-726, 2006.

\bibitem{d:assdpdbp}
A. DEVULDER. 
\newblock {\em Almost sure asymptotics for a diffusion process in a drifted Brownian potential},
\newblock { Preprint}, 2006.

\bibitem{d:mltdpdbp}
A. DEVULDER.
\newblock {\em The maximum of the local time of a diffusion process in a drifted Brownian potential},
\newblock { Preprint}, 2006.

\bibitem{d:ftlp}
R. A. DONEY.
\newblock {\em Fluctuation theory for L\'evy processes},
\newblock { Ecole d'\'et\'e de Probabilit\'es de St-Flour XXXV}, 2005.

\bibitem{dk:oulp}
R. A. DONEY and A. E. KYPRIANOU.
\newblock {\em Overshoots and Undershoots of L\'evy processes},
\newblock { Ann. Appl. Prob.}, vol 16, 1, pp 91-106, 2006.

\bibitem{d:pdrlp}
T. DUQUESNE.
\newblock {\em Path decompositions for real Levy processes}
\newblock { Ann. Inst. H. Poincar\'e}, vol 39, 2, pp 339-370, 2003. 

\bibitem{gp:filpsm}
P. GREENWOOD and J. PITMAN.
\newblock {\em Fluctuation identities for L\'evy processes and splitting at the maximum},
\newblock { Adv. Appl. Prob.}, vol 12, pp 893-902, 1980.

\bibitem{hs:ltsrwre}
Y. HU and Z. SHI.
\newblock {\em The local time of simple random walk in random environment},
\newblock { Journal of Theorical Prob.}, vol 11, pp 765-793, 1997.

\bibitem{hs:lssrwre}
Y. HU and Z. SHI.
\newblock {\em The limits of Sinai's simple random walk in random environment},
\newblock { Ann. Probab.}, vol 26, pp 1477-1521, 1998.

\bibitem{hs:pmvsre}
Y. HU and Z. SHI.
\newblock {\em The problem of the most visited site in random environment},
\newblock { Probab. Theory Relat. Fields}, vol 116, pp 273-302, 2000.

\bibitem{hsy:rcddbp}
Y. HU, Z. SHI et M. YOR.
\newblock {\em Rates of convergence of diffusions with drifted Brownian potentials},
\newblock { Trans. Am. Math. Soc.}, vol 351, pp 3915-3934, 1999.

\bibitem{js:ltsp}
J. Jacod, A.N. SHIRYAEV.
\newblock {\em Limit Theorems for Stochastic Processes},
\newblock {A Series of Comprehensive Studies in Mathematics 288. Springer Verlag}, 1987.

\bibitem{kt:omdpdbe}
K. KAWAZU and H. TANAKA.
\newblock {\em On the maximum of diffusion process in a drifted Brownian environment},
\newblock {S�minaire de Probabilit� XXVII}, Lectures Notes in Math., pp 78-85, 1991.

\bibitem{kt:adpbewd}
K. KAWAZU and H. TANAKA.
\newblock {\em A diffusion process in a Brownian environment with drift},
\newblock {J. Math. Soc. Japan}, vol 49, pp 189-211, 1997.

\bibitem{kesten}
H. KESTEN.
\newblock {\em The limit distribution of Sinai's random walk in random environment},
\newblock {Physica}, vol 138A, pp 99-309, 1986.

\bibitem{mck:ensd}
H. P. McKEAN.
\newblock {\em Excursions of non-singular diffusion},
\newblock { Z. Wahrsch.}, vol 1, pp 230-239, 1963.

\bibitem{NevPit}
J. NEVEU and J. PITMAN.
\newblock {\em Renewal property of the extrema and tree property of the excursion of a one-dimensional Brownian motion. S\'eminaire de Probabilit\'ees XXIII},
\newblock { Lecture Notes Math.}, vol 1372, pp 239-247, 1989.

\bibitem{p:sisde}
P. PROTTER.
\newblock {\em Stochastic integration and stochastic differential equations : a new approach},
\newblock{ Applications Math., Springer, Berlin }, 1990.

\bibitem{r:nirlp}
L. C. G. ROGERS.
\newblock {\em A new identity for real L\'evy processes},
\newblock { Ann. Inst. Henri Poincar\'e}, vol 20, 1, pp 21-34, 1984.

\bibitem{Sinai}
Ya. G. SINAI.
\newblock {\em A Random Walk with Random Potential},
\newblock { Theory Probab. Appl.}, vol 38, pp 382-385, 1985.

\bibitem{schumacher}
S. SCHUMACHER.
\newblock {\em Diffusions with random coefficients},
\newblock {Contemp. Math.}, vol 41, pp 351-356, 1985.

\bibitem{s:swvsc}
Z. SHI.
\newblock {\em Sinai's walk via stochastic calculus},
\newblock { Milieux Al\'eatoires, Soc. Math. France, Paris.}, pp 53-74, 2001.

\bibitem{s:ltcre}
Z. SHI.
\newblock {\em A Local time curiosity in random environment},
\newblock {Stoch. Proc. App. }, vol 76, pp 231-250, 1998.

\bibitem{s:rctdsnlp}
A. SINGH. 
\newblock {\em Rates of convergence of a transient diffusion in a spectrally negative potential},
\newblock { Ann. Probab.}, vol 36, pp 279-318, 2008.

\bibitem{t:atebmdbp}
M. TALET. 
\newblock {\em Annealed tail estimates for a Brownian motion in a drifted Brownian potential},
\newblock { Ann. Probab.}, vol 35, pp 32-67, 2007.

\bibitem{t:lpcspdre}
H. TANAKA. 
\newblock {\em L�vy processes conditioned to stay positive and diffusions in random environments},
\newblock { Advanced Studies in Pure Math.}, vol 39, pp 355-376, 2004.


\bibitem{Tanaka}
H. TANAKA. 
\newblock {\em Localization of a Diffusion Process in a One-Dimensional Brownian Environmement},
\newblock { Comm. Pure Appl. Math.}, vol 17, pp 755-766, 1994.

\bibitem{t:ltoddpbe}
H. TANAKA. 
\newblock {\em Limit theorem for one-dimensional diffusion process in Brownian Environmement},
\newblock { Lectures Notes in Math., Springer, Berlin et New-York }, 1322, pp 156-172, 1988.


\end{thebibliography}