Pathwise regularity of solutions for a class of elliptic SPDEs with symmetric L\'evy noise

TL;DR

This paper studies the existence, uniqueness, and regularity of solutions to elliptic SPDEs driven by symmetric Lévy noise, establishing conditions under which solutions are continuous, especially for spectral fractional Laplacians with order exceeding half the spatial dimension.

Contribution

It provides new existence and regularity results for elliptic SPDEs with Lévy noise, including the case of spectral fractional Laplacians, under general coefficient conditions.

Findings

Existence and uniqueness of solutions under broad conditions.

Solutions have continuous modifications when the fractional order exceeds half the dimension.

Framework applies to spectral fractional Laplacians with order gamma > d/2.

Abstract

In this article, we investigate the existence and uniqueness of random-field solutions to the elliptic SPDE $-\mathcal{L}u=\dot{\xi}$ on a bounded domain $D$ with Dirichlet boundary conditions $u=0$ on $\partial D$, driven by symmetric L\'evy noise $\dot{\xi}$. Under general sufficient conditions on the coefficients of the second-order operator $\mathcal{L}$, we prove the existence of a mild solution via the corresponding Green's function and show that the same framework applies to the spectral fractional Laplacian of power $\gamma \in (0,\infty)$. In particular, whenever $\gamma>\tfrac{d}{2}$, the solution admits a continuous modification.