We consider a reaction-diffusion equation driven by a multiplicative space-time white noise, for a large class of reaction terms that include well-studied examples, such as the Fisher-KPP and the Allen-Cahn equations. We prove that, in the "intermittent regime": (1) If the equation is "sufficiently noisy,” then the resulting stochastic PDE has a unique invariant measure; and (2) If the equation is in a “low-noise regime,” then there are infinitely many invariant measures, and the collection of all invariant measures is a certain line segment in path space. This endeavor gives a proof of earlier predictions due to Zimmerman et al (2000), discovered first through experiments and computer simulations. The quoted terms will be defined carefully in the talk.
This is joint work with Carl Mueller (University of Rochester) and Kunwoo Kim (POSTECH).