Existence and uniqueness of mild solution to stochastic heat equation with white and fractional noises

TL;DR

This paper establishes the existence and uniqueness of mild solutions for a class of stochastic heat equations driven by both white and fractional noises, expanding understanding of such complex stochastic PDEs.

Contribution

It provides the first proof of existence and uniqueness for non-autonomous stochastic heat equations with mixed white and fractional noises under specified conditions.

Findings

Proved existence and uniqueness of mild solutions.

Handled equations with fractional Brownian motion.

Extended results to non-autonomous parabolic SPDEs.

Abstract

We prove the existence and uniqueness of a mild solution for a class of non-autonomous parabolic mixed stochastic partial differential equations defined on a bounded open subset $D \subset \mathbb{R}^d$ and involving standard and fractional $L^2(D)$-valued Brownian motions. We assume that the coefficients are homogeneous, Lipschitz continuous and the coefficient at the fractional Brownian motion is an affine function.