Regularity of solutions to anisotropic nonlocal equations

TL;DR

This paper establishes Hölder regularity for bounded harmonic functions associated with systems of stochastic differential equations driven by independent symmetric stable processes with different stability indices.

Contribution

It provides the first regularity results for harmonic functions related to anisotropic nonlocal operators from such stochastic systems.

Findings

Proves Hölder continuity of harmonic functions for these systems.

Extends regularity theory to anisotropic, nonlocal operators with different stability indices.

Provides a foundation for further analysis of anisotropic nonlocal PDEs.

Abstract

We study harmonic functions associated to systems of stochastic differential equations of the form $dX_t^i=A_{i1}(X_{t-})dZ_t^1+\cdots+A_{id}(X_{t-})dZ_t^d$, $i\in\{1,\dots,d\}$, where $Z_t^j$ are independent one-dimensional symmetric stable processes with indices $\alpha_j\in(0,2)$, $j\in\{1,\dots,d\}$. In this article we prove H\"older regularity of bounded harmonic functions with respect to solutions to such systems.