Analytically weak and mild solutions to stochastic heat equation with irregular drift

Siva Athreya, Oleg Butkovsky, Khoa Lê, Leonid Mytnik

Published 2024-10-09, updated 2024-12-17Version 2

Consider the stochastic heat equation \begin{equation*} \partial_t u_t(x)=\frac12 \partial^2_{xx}u_t(x) +b(u_t(x))+\dot{W}_{t}(x),\quad t\in(0,T],\, x\in D, \end{equation*} where $b$ is a generalized function, $D$ is either $[0,1]$ or $\mathbb{R}$, and $\dot W$ is space-time white noise on $\mathbb{R}_+\times D$. If the drift $b$ is a sufficiently regular function, then it is well-known that any analytically weak solution to this equation is also analytically mild, and vice versa. We extend this result to drifts that are generalized functions, with an appropriate adaptation of the notions of mild and weak solutions. As a corollary of our results, we show that for $b\in L_p(\mathbb{R})$, $p\ge1$, this equation has a unique analytically weak and mild solution, thus extending the classical results of Gy\"ongy and Pardoux (1993).