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Phase transitions for the long-time behavior of interacting diffusions. (English) Zbl 1126.60085

Let \((\{X_ i(t)\}_{i\in\mathbb Z^ d})_{t\geq 0}\) be the system of interacting diffusions defined by the following collection of coupled stochastic differential equations: \[ d X_ i(t)=\sum_{j\in \mathbb Z^ d}a(i,j)[X_ j(t)-X_ i(t)]\,d t+\sqrt{b X_ i(t)^ 2}\,d W_ i(t),\qquad i\in\mathbb Z^ d, t\geq 0. \] 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 one-dimensional Brownian motions. 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).
The authors 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. Let \(\widehat a(i,j)=\frac 12[a(i,j)+a(j,i)]\) denote the symmetrized transition kernel. The authors show that: (A) If \(\widehat a\) is recurrent, then for any \(b>0\) the system locally dies out. (B) If \(\widehat a\) is transient, then there exist \(b_*\geq b_ 2>0\) such that: (B1) 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 second moment if and only if \(0<b<b_ 2\). (B4) The second moment diverges exponentially fast if and only if \(b>b_ 2\).
The equilibrium \(\nu_\Theta\) is shown to be associated and mixing for all \(b\in(0,b_ *)\). The authors argue in favor of the conjecture that \(b_ *>b_ 2\). They further conjecture that the system locally dies out at \(b=b_*\).
For the case where \(a\) is symmetric and transient the authors further show that: (C) There exists a sequence \(b_ 2\geq b_ 3\geq b_ 4\geq\dots>0\) 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) \(b_ 2\leq (m-1) b_ m<2\). (C4) \(\lim_{m\to\infty} (m-1)b_ m=\sup_{m\geq2}(m-1)b_ m\).
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 leads 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 \(a\) is simple random walk is commonly referred to as the parabolic Anderson model with Brownian noise. This case was studied by Carmona and Molchanov in 1994, where part of the results of the present paper were already established.

MSC:

60K35 Interacting random processes; statistical mechanics type models; percolation theory
60F10 Large deviations
60J60 Diffusion processes
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