Stochastic evolution equations driven by Liouville fractional Brownian motion

TL;DR

This paper develops a theory for stochastic integration with Liouville fractional Brownian motion in Hilbert and Banach spaces, and applies it to establish regularity of solutions for certain stochastic partial differential equations driven by fractional noise.

Contribution

It introduces a new stochastic integration framework for Liouville fractional Brownian motion and applies it to prove existence and regularity of solutions to SPDEs with fractional temporal noise.

Findings

Established equivalence of stochastic integrability for different Hurst parameters.

Proved existence of mild solutions for SPDEs driven by Liouville fractional noise.

Showed solutions are Hölder continuous in space and time.

Abstract

Let H be a Hilbert space and E a Banach space. We set up a theory of stochastic integration of L(H,E)-valued functions with respect to H-cylindrical Liouville fractional Brownian motions (fBm) with arbitrary Hurst parameter in the interval (0,1). For Hurst parameters in (0,1/2) we show that a function F:(0,T)\to L(H,E) is stochastically integrable with respect to an H-cylindrical Liouville fBm if and only if it is stochastically integrable with respect to an H-cylindrical fBm with the same Hurst parameter. As an application we show that second-order parabolic SPDEs on bounded domains in \mathbb{R}^d, driven by space-time noise which is white in space and Liouville fractional in time with Hurst parameter in (d/4,1) admit mild solution which are H\"older continuous both and space.