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 $\sigma:[0,\infty)\mapsto [0,\infty)$, we consider the influence of the parameter $\lambda$ on the behavior of the system, \begin{eqnarray*}\mathrm{d}u_{t}(x)&=&(\mathcal{G}u_{t})(x)\,\mathrm{d}t+\lambda\sigma(u_{t}(x))\,\mathrm{d}B_{t}(x)\qquad [t>0,\ x\in\mathbf{Z}^{d}],\\u_{0}(x)&=&c_{0}\delta_{0}(x).\end{eqnarray*} We show that for any $\lambda>0$ in dimensions one and two the total mass $\sum_{x\in\mathbf{Z}^{d}}u_{t}(x)$ converges to zero as $t\to\infty$ while for dimensions greater than two there is a phase transition point $\lambda_{c}\in(0,\infty)$ such that for $\lambda>\lambda_{c},\sum_{x\in \mathbf{Z}^{d}}u_{t}(x)\to0$ as $t\to\infty$ while for $\lambda<\lambda_{c},\sum_{x\in \mathbf{Z}^{d}}u_{t}(x)\not\to0$ as $t\to\infty$.