Regularity of harmonic functions for a class of singular stable-like processes

TL;DR

This paper studies the regularity of harmonic functions for a class of singular stable-like processes defined by stochastic differential equations driven by independent symmetric stable processes, showing Holder continuity but not Harnack inequality.

Contribution

It establishes Holder continuity of bounded harmonic functions for these processes despite the Levy measure's high singularity, highlighting new regularity properties.

Findings

Bounded harmonic functions are Holder continuous.

Harnack inequality may not hold for these processes.

The Levy measure is highly singular.

Abstract

We consider the system of stochastic differential equations dX_t=A(X_{t-}) dZ_t, where Z_t^1, ..., Z^d_t are independent one-dimensional symmetric stable processes of order \alpha, and the matrix-valued function A is bounded, continuous and everywhere non-degenerate. We show that bounded harmonic functions associated with X are Holder continuous, but a Harnack inequality need not hold. The Levy measure associated with the vector-valued process Z is highly singular.