PreprintCC BY 4.0arXiv
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.
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BibTeX
@misc{han2026weak,
title = {{Weak uniqueness for the stochastic heat equation with nondegenerate diffusion: the full Hölder regime}},
author = {Yi Han},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.07310},
eprint = {2611.07310},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.07310},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Weak uniqueness for the stochastic heat equation with nondegenerate diffusion: the full Hölder regime
AU - Yi Han
PY - 2026
DA - 2026/11/05
PB - arXiv
DO - 10.48550/arxiv.2611.07310
UR - https://arxiv.org/abs/2611.07310
AB - 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.
ER -