Mild Solution of Semilinear SPDEs with Young Drifts

TL;DR

This paper develops a framework for solving semilinear stochastic partial differential equations with Young drift terms, establishing existence and uniqueness of mild solutions using fixed-point methods.

Contribution

It introduces a novel approach to define and analyze mild solutions for SPDEs with Young drifts, combining two methods for Young convolution integrals.

Findings

Established existence and uniqueness of mild solutions

Developed two approaches for Young convolution integrals

Extended the theory to semilinear SPDEs with Young drifts

Abstract

In this paper, we study a semilinear SPDE with a linear Young drift $du_{t}=Lu_{t}dt+f\left(t, u_{t}\right)dt+\left(G_{t}u_{t}+g_{t}\right)d\eta_{t}+h\left(t, u_{t}\right)dW_{t}$, where $L$ is the generator of an analytical semigroup, $\eta$ is an $\alpha$-H\"older continuous path with $\alpha \in \left(1/2, 1\right)$ and $W$ is a Brownian motion. After establishing through two different approaches the Young convolution integrals for stochastic integrands, we introduce the corresponding definition of mild solutions and continuous mild solutions, and give via a fixed-point argument the existence and uniqueness of the (continuous) mild solution under suitable conditions.