Phase analysis for a family of stochastic reaction-diffusion equations

Abstract

We consider a reaction-diffusion equation of the type

$${\partial }_t\mathit{\psi }={\partial }_x^2\mathit{\psi }+V(\mathit{\psi })+\mathit{\lambda }\mathit{\sigma }(\mathit{\psi })\dot{W}\text{on}(0,\mathrm{\infty })\times \mathbb{T},$$

subject to a “nice” initial value and periodic boundary, where $\mathbb{T}=[-1,1]$ and $\dot{W}$ denotes space-time white noise. The reaction term $V:\mathbb{R}\to \mathbb{R}$ 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 $\mathit{\sigma }:\mathbb{R}\to \mathbb{R}$ is non random and Lipschitz continuous, and $\mathit{\lambda }>0$ is a non-random number that measures the strength of the effect of the noise $\dot{W}$.

The principal finding of this paper is that: (i) When λ is sufficiently large, the above equation has a unique invariant measure; and (ii) When λ 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.