Pathwise solutions to stochastic partial differential equations driven by fractional Brownian motions with Hurst parameters in $(1/3,1/2]$

TL;DR

This paper develops a framework combining fractional calculus and rough path theory to establish existence and uniqueness of solutions for stochastic PDEs driven by fractional Brownian motion with Hurst parameters in (1/3, 1/2], extending classical stochastic integration methods.

Contribution

It introduces a novel pathwise integral and a second variable tensor approach to analyze SPDEs driven by fractional Brownian motion in the specified Hurst range.

Findings

Proves existence and uniqueness of solutions under smoothness conditions.

Constructs a new stochastic integral generalizing Young's integral.

Develops a tensor-based second variable for rough path analysis.

Abstract

Combining fractional calculus and the Rough Path Theory we study the existence and uniqueness of mild solutions to evolutions equations driven by a H\"older continuous function with H\"older exponent in $(1/3,1/2)$. Our stochastic integral is in some sense a generalization of the well-known Young integral and can be defined independently of the initial condition. Similar to the Rough Path Theory we establish a second variable which is given, roughly speaking, by a tensor product. It is then necessary to formulate a second equation for this new variable, and we do in a mild sense. The crucial point in order to get this new equation is to construct a tensor depending on the noise path but also on the semigroup. We then prove the existence of a unique H\"older continuous solution of the system of equations, consisting of the path and the area components, if the nonlinear term and the initial condition are sufficiently smooth. Once the abstract theory is developed, we can present a pathwise nonlinear SPDE driven by a fractional Brownian motion with Hurst parameter in $(1/3,1/2]$.