Bilinear equations in Hilbert space driven by paths of low regularity

TL;DR

This paper studies bilinear evolution equations in Hilbert space driven by paths with low regularity, providing explicit solutions and applying results to stochastic PDEs with fractional noise.

Contribution

It introduces explicit solutions for bilinear equations driven by low-regularity paths and links solutions to inhomogeneous initial value problems, with applications to stochastic PDEs.

Findings

Explicit solutions for bilinear equations with low regularity paths

Conditions for existence of solutions to associated initial value problems

Pathwise solutions to stochastic PDEs with fractional noise

Abstract

In the article, some bilinear evolution equations in Hilbert space driven by paths of low regularity are considered and solved explicitly. The driving paths are scalar-valued and continuous, and they are assumed to have a finite $p$-th variation along a given sequence of partitions in the sense given by Cont and Perkowski \cite{ConPer18} ($p$ being an even positive integer). Typical functions that satisfy this condition are trajectories of the fractional Brownian motion of the Hurst parameter $H=\sfrac{1}{p}$. A strong solution to the bilinear problem is shown to exist if there is a solution to a certain temporally inhomogeneous initial value problem. Subsequently, sufficient conditions for the existence of the solution to this initial value problem are given. The abstract results are applied to several stochastic partial differential equations with multiplicative fractional noise, both of the parabolic and hyperbolic type, that are solved explicitly in a pathwise sense.