We consider a reaction-diffusion equation of the type ∂tψ=∂2xψ+V(ψ)+λσ(ψ)W˙on (0,∞)×?, subject to a "nice" initial value and periodic boundary, where ?=[−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.
This is based on joint work with Carl Mueller (Univ. Rochester) and Kunwoo Kim (POSTECH, S. Korea).