Some linear SPDEs driven by a fractional noise with Hurst index greater than 1/2

TL;DR

This paper establishes conditions for the existence of solutions to certain linear stochastic partial differential equations driven by fractional Gaussian noise with Hurst index greater than 1/2, linking solutions to potential theory and intersection local times.

Contribution

It provides necessary and sufficient conditions for solutions of linear SPDEs driven by fractional noise, connecting existence to potential theory and intersection local times of Lévy processes.

Findings

Existence of solutions characterized by potential theory of associated Markov processes.

Connection between solution existence and weighted intersection local times.

Application to stable Lévy processes with specific Hurst index.

Abstract

In this article, we identify the necessary and sufficient conditions for the existence of a random field solution for some linear s.p.d.e.'s of parabolic and hyperbolic type. These equations rely on a spatial operator $\cL$ given by the $L^2$-generator of a $d$-dimensional L\'evy process $X=(X_t)_{t \geq 0}$, and are driven by a spatially-homogeneous Gaussian noise, which is fractional in time with Hurst index $H>1/2$. As an application, we consider the case when $X$ is a $\beta$-stable process, with $\beta \in (0,2]$. In the parabolic case, we develop a connection with the potential theory of the Markov process $\bar{X}$ (defined as the symmetrization of $X$), and we show that the existence of the solution is related to the existence of a "weighted" intersection local time of two independent copies of $\bar{X}$.