Local mild solutions for rough stochastic partial differential equations

TL;DR

This paper develops a framework for solving stochastic evolution equations driven by fractional Brownian motion with Hurst parameter between 1/3 and 1/2 in infinite-dimensional spaces, using rough paths theory to define integrals and establish solutions.

Contribution

It introduces a novel integral with respect to fractional Brownian motion and demonstrates the existence of mild solutions in infinite-dimensional Banach spaces.

Findings

Established a pathwise solution framework for fBm-driven equations

Extended rough paths theory to infinite-dimensional stochastic PDEs

Provided conditions for the existence of mild solutions

Abstract

We investigate mild solutions for stochastic evolution equations driven by a fractional Brownian motion (fBm) with Hurst parameter H in (1/3, 1/2] in infinite-dimensional Banach spaces. Using elements from rough paths theory we introduce an appropriate integral with respect to the fBm. This allows us to solve pathwise our stochastic evolution equation in a suitable function space.