Dissipation and high disorder

Abstract

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 $\sigma :[0,\infty )\to [0,\infty ),$ we consider the influence of the parameter $\lambda$ 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 $\lambda >0$ 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 ${\lambda }_c\in (0,\infty )$ such that for $\lambda >{\lambda }_c,{\sum }_{{\mathbf{Z}}^d}{u}_t(x)\to 0$ as $t\to \infty$ while for $\lambda <{\lambda }_c,{\sum }_{{\mathbf{Z}}^d}{u}_t(x)\nrightarrow 0$ as $t\to \infty .$

20 pages