Dissipation and high disorder

Abstract

Given a field {B(x)}x is an element of Z(d) of independent standard Brownian motions, indexed by Z(d), the generator of a suitable Markov process on Z(d), G, and sufficiently nice function sigma : [0, infinity) (bar right arrow) [0, infinity), we consider the influence of the parameter lambda on the behavior of the system,

du(t) (x) = (Gu(t))(x) dt + lambda sigma(u(t)(x)) dB(t)(x) [t > 0, x is an element of Z(d)],

u(0)(x) = c(0)delta(0)(x).

We show that for any lambda > 0 in dimensions one and two the total mass Sigma(x is an element of Zd) u(t) (x) converges to zero as t -> infinity while for dimensions greater than two there is a phase transition point lambda(c) is an element of (0, infinity) such that for lambda > lambda(c), Sigma(x is an element of Zd) u(t) (x) -> 0 as t -> infinity while for lambda < lambda(c), Sigma(x is an element of Zd) u(t) (x) negated right arrow 0 as t -> infinity.