Setvalued dynamical systems for stochastic evolution equations driven by fractional noise

TL;DR

This paper develops a framework for analyzing set-valued dynamical systems arising from stochastic evolution equations driven by fractional noise with H"older index greater than 1/2, addressing non-uniqueness issues.

Contribution

It introduces a multivalued dynamical system approach for fractional noise-driven equations and constructs a separable metric dynamical system to establish measurability.

Findings

Established existence of multivalued nonautonomous dynamical systems

Proved measurability of the solution set mapping

Extended analysis to fractional noises with Hurst parameters in (1/2,1)

Abstract

We consider Hilbert-valued evolution equations driven by H\"{o}lder paths with H\"{o}lder index greater than 1/2, which includes the case of fractional noises with Hurst parameters in (1/2,1). The assumptions of the drift term will not be enough to ensure the uniqueness of solutions. Nevertheless, adopting a multivalued setting, we will prove that the set of all solutions corresponding to the same initial condition generates a (multivalued) nonautonomous dynamical system $\Phi$. Finally, to prove that $\Phi$ is measurable (and hence a (multivalued) random dynamical system), we need to construct a new metric dynamical system that models the noise with the property that the set space is separable