Global variational solutions to a class of fractional spde's on unbounded domains

TL;DR

This paper establishes the existence and uniqueness of global variational solutions for a class of non-autonomous stochastic parabolic PDEs driven by fractional noise on unbounded domains, using compactness methods.

Contribution

It provides new theoretical results on well-posedness for fractional SPDEs on unbounded domains with multiplicative noise, extending previous work in the field.

Findings

Proved existence of solutions under certain geometric conditions.

Established uniqueness of solutions for the class of equations.

Analyzed the influence of fractional noise parameters on solution properties.

Abstract

In this article we prove new results regarding the existence and the uniqueness of global variational solutions to Neumann initial-boundary value problems for a class of non-autonomous stochastic parabolic partial differential equations. The equations we consider are defined on unbounded open domains in Euclidean space satisfying certain geometric conditions, and are driven by a multiplicative noise derived from an infinite-dimensional fractional Wiener process characterized by a sequence of Hurst parameters $\mathsf{H} = (\mathsf{H}_i)_{i\in \mathbb{N}^+} \subset (\frac{1}{2},1)$. These parameters are in fact subject to further constraints that are intimately tied up with the nature of the nonlinearity in the stochastic term of the equations, and with the choice of the functional spaces in which the problem at hand is well-posed. Our method of proof rests on compactness arguments in an essential way.