Mathematical analysis demonstrates pathwise well-posedness in the one-dimensional stochastic heat equation on the circle, suggesting optimal bounds under one-parameter rough paths.
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.
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Shao et al. (2026) studied this question.
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