A remark on pathwise well-posedness of the 1-$d$ stochastic heat equation

Yufei Shao; Jiawei Li; Tadahiro Oh

A remark on pathwise well-posedness of the 1-$d$ stochastic heat equation

Analysis of PDEs 2026-07-10 v1 Probability

Authors: Yufei Shao , Jiawei Li , Tadahiro Oh

Abstract

We study pathwise well-posedness of the stochastic heat equation (SHE) with a multiplicative noise on the circle. By combining the convolution Young and rough integration theory, introduced by Gubinelli and Tindel (2010), with the random tensor estimate approach to pathwise well-posedness of stochastic dispersive PDEs with multiplicative noises, introduced by Chapouto and the second and third authors (2026), we establish pathwise well-posedness of SHE in both the Young and rough cases, improving the results in Gubinelli and Tindel (2010). In particular, in the rough case (= the white-in-time case), our result covers the case of almost space-time white noise, thus establishing an optimal result within the framework of one-parameter rough paths.

Keywords

stochastic heat equation pathwise well-posedness rough paths Young integration multiplicative noise random tensor estimates

Cite

@article{arxiv.2607.09033,
  title  = {A remark on pathwise well-posedness of the 1-$d$ stochastic heat equation},
  author = {Yufei Shao and Jiawei Li and Tadahiro Oh},
  journal= {arXiv preprint arXiv:2607.09033},
  year   = {2026}
}

Comments

35 pages, to appear in Proc. Roy. Soc. Edinburgh Sect. A. arXiv admin note: substantial text overlap with arXiv:2607.08385