Martin Boundary and Invariant Fields of Multiplicative SHE

Abstract

We study invariant measures of nonlinear multiplicative stochastic heat equations in the weak disorder regime. Under a natural second-moment condition, we show that positive invariant measures are in one-to-one correspondence with bounded positive harmonic functions of the underlying space. This implies that the space of invariant measures inherits the structure of the Martin boundary. We also show that whenever the deterministic semigroup flow converges to a bounded harmonic function, the stochastic evolution converges to the corresponding invariant measure. The results apply to many settings with nontrivial Martin boundary, such as negatively curved manifolds and trees.

arXiv commentTitle and Abstract Match submitted version, further typos corrected, major expositional and mathematical errors corrected