Well-posedness of stochastic heat equation with distributional drift and skew stochastic heat equation

We study stochastic reaction–diffusion equation (Formula presented.) where b is a generalized function in the Besov space (Formula presented.), (Formula presented.) and (Formula presented.) is a space-time white noise on (Formula presented.). We introduce a notion of a solution to this equation and obtain existence and uniqueness of a strong solution whenever (Formula presented.), (Formula presented.) and (Formula presented.). This class includes equations with b being measures, in particular, (Formula presented.) which corresponds to the skewed stochastic heat equation. For (Formula presented.), we obtain existence of a weak solution. Our results extend the work of Bass and Chen (2001) to the framework of stochastic partial differential equations and generalize the results of Gyöngy and Pardoux (1993) to distributional drifts. To establish these results, we exploit the regularization effect of the white noise through a new strategy based on the stochastic sewing lemma introduced in Lê (2020).