Phase analysis for a family of stochastic reaction-diffusion equations

Abstract

We consider a reaction-diffusion equation of the type partial differential t0 = partial differential x20 + V (0) + & lambda;& sigma; (0)W on (0, & INFIN;) x T, subject to a "nice" initial value and periodic boundary, where T = [-1 , 1] and denotes space-time white noise. The reaction term V : l & RARR; l belongs to a large family of functions that includes Fisher-KPP nonlinearities [V (x) = x(1 - x)] as well as Allen-Cahn potentials [V (x) = x(1 - x)(1 + x)], the multiplicative nonlinearity & sigma; : l & RARR; l is non random and Lipschitz continuous, and & lambda; > 0 is a non-random number that measures the strength of the effect of the noise W. The principal finding of this paper is that: (i) When & lambda; is sufficiently large, the above equation has a unique invariant measure; and (ii) When & lambda; is sufficiently small, the collection of all invariant measures is a non-trivial line segment, in particular infinite. This proves an earlier prediction of Zimmerman et al. (2000). Our methods also say a great deal about the structure of these invariant measures. W