Dissipation and high disorder

Given a field $\{B(x)\}_{x\in\mathbf{Z}^d}$ of independent standard Brownian motions, indexed by $\mathbf{Z}^d$, the generator of a suitable Markov process on $\mathbf{Z}^d,\,\,\mathcal{G},$ and sufficiently nice function $σ:[0,\infty)\to[0,\infty),$ we consider the influence of the parameter $λ$ on the behavior of the system, \begin{align*} \rm{d} u_t(x) = & (\mathcal{G}u_t)(x)\,\rm{d} t + λσ(u_t(x))\rm{d} B_t(x) \qquad[t>0,\ x\in\mathbf{Z}^d], &u_0(x)=c_0δ_0(x). \end{align*} We show that for any $λ>0$ in dimensions one and two the total mass $\sum_{x\in\mathbf{Z}^d}u_t(x)\to 0$ as $t\to\infty$ while for dimensions greater than two there is a phase transition point $λ_c\in(0,\infty)$ such that for $λ>λ_c,\, \sum_{\mathbf{Z}^d}u_t(x)\to 0$ as $t\to\infty$ while for $λ<λ_c,\,\sum_{\mathbf{Z}^d}u_t(x)\not\to 0$ as $t\to\infty.$

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