2026/08/02 by Yi Han
Mathematics · #math.PR
arxiv created 2026/08/02 · arxiv updated 2026/08/04
We consider stochastic heat equation (SHE) defined on 1-d torus \mathbbT of the form ∂t u=Δu+g(u)W,where 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∈(0,1). We prove that weak uniqueness holds as long as β>(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>(3)/(4) via a Yamada Watanabe argument by Mytnik and Perkins (arxiv:0809.0248) without assuming g is nonzero. And when β<(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 (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∈((2)/(3),(3)/(4)] regime.