Pathwise regularity of solutions for a class of elliptic SPDEs with symmetric Lévy noise

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 Lévy 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 $\gamma >\frac{d}{2}$, the solution admits a continuous modification.

18 pages, minor typo corrections