SPDEs with fractional noise in space: continuity in law with respect to the Hurst index

TL;DR

This paper studies stochastic wave and heat equations driven by fractional noise in space, proving that their solutions depend continuously on the Hurst index in the sense of convergence in law.

Contribution

It establishes the continuity in law of solutions to SPDEs with fractional spatial noise as the Hurst index varies, extending understanding of their probabilistic behavior.

Findings

Solutions are continuous in law with respect to the Hurst index H.

The results apply to quasi-linear stochastic wave and heat equations.

The approach covers equations with additive Gaussian noise that is white in time and fractional in space.

Abstract

In this article, we consider the quasi-linear stochastic wave and heat equations on the real line and with an additive Gaussian noise which is white in time and behaves in space like a fractional Brownian motion with Hurst index $H\in (0,1)$. The drift term is assumed to be globally Lipschitz. We prove that the solution of each of the above equations is continuous in terms of the index $H$, with respect to the convergence in law in the space of continuous functions.