Dissipation and high disorder

TL;DR

This paper investigates how the parameter mbda affects the long-term behavior of a stochastic system driven by Brownian motions on Z^d, revealing dimension-dependent phase transitions in dissipation.

Contribution

It establishes the existence of a phase transition for the total mass decay in high-dimensional systems influenced by Brownian noise.

Findings

In dimensions 1 and 2, total mass tends to zero for all mbda>0.

In dimensions >2, a critical mbda_c determines whether the mass vanishes or persists.

A phase transition occurs at mbda_c in higher dimensions.

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 + \lambda\sigma(u_t(x))\rm{d} B_t(x) \qquad[t>0,\ x\in\mathbf{Z}^d], &u_0(x)=c_0\delta_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 $\lambda<\lambda_c,\,\sum_{\mathbf{Z}^d}u_t(x)\not\to 0$ as $t\to\infty.$