Stochastic heat equation with nondegenerate Hölder diffusion coefficient: uniqueness below the three-fourth threshold

Abstract

We consider stochastic heat equation (SHE) defined on 1-d torus $\mathbb{T}$ of the form $$\partial_t u=Δu+g(u)\dot{W},$$where $\dot{W}$ is a space-time white noise and $g$ is a real-valued function which is uniformly elliptic (i.e., $|g|$ is uniformly bounded away from 0), and is globally $β$-Holder continuous for some $β\in(0,1)$. We prove that weak uniqueness holds as long as $β>\frac{2}{3}$. The same uniqueness holds for vector-valued solutions where the coefficient $G$ has the same dimension as the white noise. Previously, uniqueness of solutions to the SHE with Holder diffusion coefficient was only established for $β>\frac{3}{4}$ via a Yamada Watanabe argument by Mytnik and Perkins (arxiv:0809.0248) without assuming $g$ is nonzero. And when $β<\frac{3}{4}$, Mueller, Mytnik and Perkins (arXiv:1201.2767) constructed a non-unique SPDE example satisfying $g(0)=0$. A later generalized coupling argument for nondegenerate $g$ also stopped at the same threshold $\frac{3}{4}$. Our result shows that uniform ellipticity of $g$ restores uniqueness to SHEs in the Holder regime where the same SHE with non-elliptic $g$ and the same Holder regularity are often non-unique in law. This constitutes the first general class of SHE weak uniqueness results in the $β\in(\frac{2}{3},\frac{3}{4}]$ regime.