On well-posedness of semilinear stochastic evolution equations on $L_p$ spaces

TL;DR

This paper proves the well-posedness of a class of stochastic semilinear evolution equations on $L_p$ spaces with multiplicative noise, using advanced stochastic calculus and monotonicity methods.

Contribution

It introduces a mild solution framework for equations with quasi-monotone, polynomially growing drift operators that are not necessarily continuous.

Findings

Established existence and uniqueness of solutions.

Developed a minimal integrability condition framework.

Applied stochastic calculus in Banach spaces.

Abstract

We establish well-posedness in the mild sense for a class of stochastic semilinear evolution equations on $L_p$ spaces, driven by multiplicative Wiener noise, with a drift term given by an evaluation operator that is assumed to be quasi-monotone and polynomially growing, but not necessarily continuous. In particular, we consider a notion of mild solution ensuring that the evaluation operator applied to the solution is still function-valued, but satisfies only minimal integrability conditions. The proofs rely on stochastic calculus in Banach spaces, monotonicity and convexity techniques, and weak compactness in $L_1$ spaces.