Phase analysis for a family of stochastic reaction-diffusion equations

We consider a reaction-diffusion equation of the type ∂tψ = ∂² _xψ + V (ψ) + λσ(ψ)W˙ on (0, ∞) × T, subject to a “nice” initial value and periodic boundary, where T = [−1, 1] and W˙ denotes space-time white noise. The reaction term V: R → 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: R → 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 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.