Pathwise solutions of SPDEs driven by H\"older-continuous integrators with exponent larger than 1/2 and random dynamical systems

TL;DR

This paper establishes the existence and uniqueness of pathwise solutions to certain stochastic evolution equations driven by H"older continuous integrators with exponent greater than 1/2, including fractional Brownian motion, and constructs associated random dynamical systems.

Contribution

It extends the theory of pathwise solutions for SPDEs to H"older continuous integrators with exponent > 1/2, including fractional Brownian motion, and develops the framework for associated random dynamical systems.

Findings

Proves existence and uniqueness of solutions in H"older spaces.

Constructs non-autonomous dynamical systems from solutions.

Shows the generated system is a random dynamical system for fractional Brownian motion.

Abstract

This article is devoted to the existence and uniqueness of pathwise solutions to stochastic evolution equations, driven by a H\"older continuous function with H\"older exponent in $(1/2,1)$, and with nontrivial multiplicative noise. As a particular situation, we shall consider the case where the equation is driven by a fractional Brownian motion $B^H$ with Hurst parameter $H>1/2$. In contrast to the article by Maslowski and Nualart, we present here an existence and uniqueness result in the space of H\"older continuous functions with values in a Hilbert space $V$. If the initial condition is in the latter space this forces us to consider solutions in a different space, which is a generalization of the H\"older continuous functions. That space of functions is appropriate to introduce a non-autonomous dynamical system generated by the corresponding solution to the equation. In fact, when choosing $B^H$ as the driving process, we shall prove that the dynamical system will turn out to be a random dynamical system, defined over the ergodic metric dynamical system generated by the infinite dimensional fractional Brownian motion