  Let $(\{X_i(t)\}_{i\in \mathbb{Z}^d})_{t\geq 0}$ be the system of interacting
diffusions on $[0,\infty)$ defined by the following collection of coupled
stochastic differential equations: \begin{eqnarray}dX_i(t)=\sum\limits_{j\in
\mathbb{Z}^d}a(i,j)[X_j(t)-X_i(t)] dt+\sqrt{bX_i(t)^2} dW_i(t), \eqntext{i\in
\mathbb{Z}^d,t\geq 0.}\end{eqnarray} Here, $a(\cdot,\cdot)$ is an irreducible
random walk transition kernel on $\mathbb{Z}^d\times \mathbb{Z}^d$, $b\in
(0,\infty)$ is a diffusion parameter, and $(\{W_i(t)\}_{i\in
\mathbb{Z}^d})_{t\geq 0}$ is a collection of independent standard Brownian
motions on $\mathbb{R}$. The initial condition is chosen such that
$\{X_i(0)\}_{i\in \mathbb{Z}^d}$ is a shift-invariant and shift-ergodic random
field on $[0,\infty)$ with mean $\Theta\in (0,\infty)$ (the evolution preserves
the mean). We show that the long-time behavior of this system is the result of
a delicate interplay between $a(\cdot,\cdot)$ and $b$, in contrast to systems
where the diffusion function is subquadratic. In particular, let
$\hat{a}(i,j)={1/2}[a(i,j)+a(j,i)]$, $i,j\in \mathbb{Z}^d$, denote the
symmetrized transition kernel. We show that: (A) If $\hat{a}(\cdot,\cdot)$ is
recurrent, then for any $b>0$ the system locally dies out. (B) If
$\hat{a}(\cdot,\cdot)$ is transient, then there exist $b_*\geq b_2>0$ such
that: (B1)d The system converges to an equilibrium $\nu_{\Theta}$ (with mean
$\Theta$) if $0<b<b_*$. (B2) The system locally dies out if $b>b_*$. (B3)
$\nu_{\Theta}$ has a finite 2nd moment if and only if $0<b<b_2$. (B4) The 2nd
moment diverges exponentially fast if and only if $b>b_2$. The equilibrium
$\nu_{\Theta}$ is shown to be associated and mixing for all $0<b<b_*$. We argue
in favor of the conjecture that $b_*>b_2$. We further conjecture that the
system locally dies out at $b=b_*$. For the case where $a(\cdot,\cdot)$ is
symmetric and transient we further show that: (C) There exists a sequence
$b_2\geq b_3\geq b_4\geq ... >0$ such that: (C1) $\nu_{\Theta}$ has a finite
$m$th moment if and only if $0<b<b_m$. (C2) The $m$th moment diverges
exponentially fast if and only if $b>b_m$. (C3) $b_2\leq (m-1)b_m<2$. uad(C4)
$\lim_{m\to\infty}(m-1)b_m=c=\sup_{m\geq 2}(m-1)b_m$. The proof of these
results is based on self-duality and on a representation formula through which
the moments of the components are related to exponential moments of the
collision local time of random walks. Via large deviation theory, the latter
lead to variational expressions for $b_*$ and the $b_m$'s, from which sharp
bounds are deduced. The critical value $b_*$ arises from a stochastic
representation of the Palm distribution of the system. The special case where
$a(\cdot,\cdot)$ is simple random walk is commonly referred to as the parabolic
Anderson model with Brownian noise. This case was studied in the memoir by
Carmona and Molchanov [Parabolic Anderson Problem and Intermittency (1994)
Amer. Math. Soc., Providence, RI], where part of our results were already
established.
