Dissipation and high disorder

Given a field {B(x)}x.Zd of independent standard Brownian motions, indexed by Zd, the generator of a suitable Markov process on Zd,G, and sufficiently nice function .:[0,.).[0,.), we consider the influence of the parameter . on the behavior of the system, dut(x)=(Gut)(x)dt+..(ut(x))dBt(x)[t>0, x.Zd],u0(x)=c0.0(x). We show that for any .>0 in dimensions one and two the total mass .x.Zdut(x) converges to zero as t.. while for dimensions greater than two there is a phase transition point .c.(0,.) such that for .>.c,.x.Zdut(x).0 as t.. while for .<.c,.x.Zdut(x).0 as t...