Interacting diffusions in a random medium: comparison and longtime behavior.
Let \(S\) be an at most countable set, \(I\subseteq \mathbb R_ {+}\) either a closed interval, or \(\mathbb R_ {+}\) itself, \(\{W_ {t} (i),\, i\in S\}\) an independent family of standard Wiener processes. Let us consider a diffusion process on a state space \(E\subseteq I^ {S}\) defined by the stochastic differential equation \[ dX_ {t}(i) = \mathcal AX_ {t}(i) \,dt + \sqrt {2g _ {i,t}(X_ {t}(i))} \,dW_ {t}(i), \;i\in S, \quad X_ 0 = x_ 0\in E, \tag{1} \] where either \(\mathcal A\) or \(\mathcal A^ {\text{T}}\) is a \(q\)-matrix of a rate 1 Markov chain on \(S\) and for each \(i\in S\), \(t\geq 0\) the function \(g_ {i,t}: \mathbb R\to \mathbb R_ {+}\) is locally Lipschitz continuous, \(g_ {i,t}=0\) on the boundary of \(I\) and \(g_ {i,t} (x)\leq C_ {t}(i)(1+x^ 2)\) for a constant \(C_ {t}(i)<\infty \). Suppose moreover that the functions \(t\mapsto g_ {i,t}(x)\) belong to the Skorokhod space \(D(\mathbb R_ {+},\mathbb R)\) and are continuous except at isolated points, for all \(i\in S\) and \(x\in \mathbb R\). Set \(E =\{x\in I^ {S},\, \langle \gamma ,x\rangle <\infty \}\), where \(\gamma \) is a strictly positive measure on \(S\) satisfying \(\gamma \mathcal A\leq M\gamma \) for some finite constant \(M\). Under the above hypotheses it is proven that there exists a unique \(E\)-valued strong solution to (1), and this solution is a Markov process. Moreover, a comparison theorem is established. These general results are then used to study long-time behaviour of interacting diffusions in a random medium: Assume that the diffusion coefficients \(g_ {i,t}\) are of the form \(H_ {t}(i)g(x)\), \(g\) being a locally Lipschitz function of a quadratic growth, vanishing on the boundary of \(I\), and \(H\) an \(S\)-valued stochastic process which evolves autonomously. Several theorems on weak convergence of the pair \((H_ {t},X_ {t})\) as \(t\to \infty \) are proven for various choices of the process \(H\) (e.g.\ a voter process or a branching random walk).
scientific article; zbMATH DE number 2138945
scientific article