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

TL;DR

This paper extends the equivalence of weak and mild solutions for the stochastic heat equation to cases where the drift term is a generalized function, establishing existence and uniqueness under broader conditions.

Contribution

It generalizes the known equivalence of weak and mild solutions to stochastic heat equations with irregular drifts, including those in $L_p$ spaces.

Findings

Established equivalence of weak and mild solutions for generalized drifts.

Proved existence and uniqueness of solutions for drifts in $L_p$ spaces.

Extended classical results to irregular drift functions.

Abstract

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).