Weak uniqueness for the stochastic heat equation with nondegenerate diffusion: the full Hölder regime

Abstract

We prove weak well-posedness for the scalar stochastic heat equation on the one-dimensional torus with uniformly nondegenerate diffusion that is Holder continuous of any positive order. This resolves the weak uniqueness problem in the full positive Holder range, extending our earlier result for exponents above two thirds (arXiv:2608.01279). The proof couples a weak solution to a smooth companion and uses spatial cancellation to estimate the signed covariance difference in a negative Sobolev norm. A finite Gaussian expansion on each Littlewood-Paley block gives the decay needed to remove the coupling drift by a change of measure. A localized Girsanov argument extends the result to measurable drifts of linear growth.