Phase Analysis for a family of Stochastic Reaction-Diffusion Equations
We consider a reaction-diffusion equation of the type \[ \partial_tψ= \partial^2_xψ+ V(ψ) + λσ(ψ)\dot{W} \qquad\text{on $(0\,,\infty)\times\mathbb{T}$}, \] 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 $σ:\mathbb{R}\to\mathbb{R}$ is non random and Lipschitz continuous, and $λ>0$ 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.