Harnack Inequality and Regularity for a Product of Symmetric Stable Process and Brownian Motion

TL;DR

This paper establishes Harnack inequalities and regularity results for harmonic functions related to a combined process of a symmetric stable process and Brownian motion, extending classical potential theory to this new setting.

Contribution

It introduces a class of harmonic functions for the product process and proves Harnack inequality and Hölder continuity for these functions, a novel extension in stochastic analysis.

Findings

Bounded non-negative harmonic functions satisfy Harnack inequality.

Harmonic functions are locally Hölder continuous.

Results connect harmonic extension and Littlewood-Paley functions.

Abstract

In this paper, we consider a product of a symmetric stable process in $\mathbb{R}^d$ and a one-dimensional Brownian motion in $\mathbb{R}^+$. Then we define a class of harmonic functions with respect to this product process. We show that bounded non-negative harmonic functions in the upper-half space satisfy Harnack inequality and prove that they are locally H\"older continuous. We also argue a result on Littlewood-Paley functions which are obtained by the $\alpha$-harmonic extension of an $L^p(\mathbb{R}^d)$ function.