Phase analysis for a family of stochastic reaction-diffusion equations

We consider a reaction-diffusion equation of the type ∂tψ=∂x2ψ+V(ψ)+λσ(ψ)W˙on(0,∞)×T, subject to “nice” initial value and periodic boundary, where T=[−1,1] W˙ denotes space-time white noise. The reaction term V:R→R belongs large family functions that includes Fisher–KPP nonlinearities [V(x)=x(1−x)] as well Allen-Cahn potentials [V(x)=x(1−x)(1+x)], multiplicative nonlinearity σ:R→R is non random Lipschitz continuous, λ>0 non-random number measures strength effect noise W˙. principal finding this paper that: (i) When λ sufficiently large, above has unique invariant measure; (ii) small, collection all non-trivial line segment, in particular infinite. This proves an earlier prediction Zimmerman et al. (2000). Our methods also say great deal about structure these measures.