Existence, uniqueness, and regularity for stochastic evolution equations with irregular initial values

TL;DR

This paper develops a framework for analyzing parabolic semilinear stochastic evolution equations with irregular initial data and coefficients, establishing fundamental existence, uniqueness, and regularity results, supported by counterexamples.

Contribution

It introduces a novel framework to handle singularities in initial conditions and coefficients for stochastic evolution equations, advancing theoretical understanding.

Findings

Proved existence and uniqueness of mild solutions with irregular initial data.

Established regularity properties of solutions under singular conditions.

Provided counterexamples demonstrating the optimality of the results.

Abstract

In this article we develop a framework for studying parabolic semilinear stochastic evolution equations (SEEs) with singularities in the initial condition and singularities at the initial time of the time-dependent coefficients of the considered SEE. We use this framework to establish existence, uniqueness, and regularity results for mild solutions of parabolic semilinear SEEs with singularities at the initial time. We also provide several counterexample SEEs that illustrate the optimality of our results.