Heat equation with general stochastic measure colored in time

TL;DR

This paper studies a stochastic heat equation driven by a general stochastic measure, establishing existence, uniqueness, and regularity of solutions under minimal assumptions on the measure.

Contribution

It introduces a framework for analyzing stochastic heat equations with general stochastic measures, extending previous models to broader classes of integrators.

Findings

Existence and uniqueness of solutions proved.

Hölder regularity of solutions established.

Applicable to stochastic measures with only σ-additivity in probability.

Abstract

A stochastic heat equation on $[0,T]\times{\mathbb{R}}$ driven by a general stochastic measure $d\mu(t)$ is investigated in this paper. For the integrator $\mu$, we assume the $\sigma$-additivity in probability only. The existence, uniqueness, and H\"{o}lder regularity of the solution are proved.