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

TL;DR

This paper studies one-dimensional stochastic wave and heat equations driven by fractional Gaussian noise, showing that their solutions vary continuously with the Hurst index in law, using tightness criteria and Malliavin calculus.

Contribution

It establishes the continuity in law of solutions to SPDEs with respect to the Hurst index, a novel result in the analysis of fractional noise-driven equations.

Findings

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

The proof employs tightness criteria and Malliavin calculus techniques.

Results apply to both wave and heat equations driven by fractional noise.

Abstract

In this article, we consider the one-dimensional stochastic wave and heat equations driven by a linear multiplicative Gaussian noise which is white in time and behaves in space like a fractional Brownian motion with Hurst index $H\in (\frac 14,1)$. 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. The proof is based on a tightness criterion on the plane and Malliavin calculus techniques in order to identify the limit law.