Mild Solutions for a Class of Fractional SPDEs and Their Sample Paths

TL;DR

This paper introduces a notion of mild solution for fractional SPDEs driven by infinite-dimensional fractional noise, proves existence, regularity, and uniqueness under certain conditions, and relates it to variational solutions.

Contribution

It defines and analyzes mild solutions for a class of fractional SPDEs driven by fractional noise, establishing existence, regularity, and connection with variational solutions.

Findings

Existence of mild solutions under specified conditions.

Hölder continuity of sample paths of solutions.

Equivalence of mild and variational solutions when unique.

Abstract

In this article we introduce and analyze a notion of mild solution for a class of non-autonomous parabolic stochastic partial differential equations defined on a bounded open subset $D\subset\mathbb{R}^{d}$ and driven by an infinite-dimensional fractional noise. The noise is derived from an $L^{2}(D)$-valued fractional Wiener process $W^{H}$ whose covariance operator satisfies appropriate restrictions; moreover, the Hurst parameter $H$ is subjected to constraints formulated in terms of $d$ and the H\"{o}lder exponent of the derivative $h^\prime$ of the noise nonlinearity in the equations. We prove the existence of such solution, establish its relation with the variational solution introduced in \cite{nuavu} and also prove the H\"{o}lder continuity of its sample paths when we consider it as an $L^{2}(D)$--valued stochastic processes. When $h$ is an affine function, we also prove uniqueness. The proofs are based on a relation between the notions of mild and variational solution established in Sanz-Sol\'e and Vuillermot 2003, and adapted to our problem, and on a fine analysis of the singularities of Green's function associated with the class of parabolic problems we investigate. An immediate consequence of our results is the indistinguishability of mild and variational solutions in the case of uniqueness.