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

2009/04/22 by Richard F. Bass, Zhen-Qing Chen, Bass, Richard F. +1 · 1 citation
Economics, Econometrics and Finance · Mathematics · #60H10 #FOS: Mathematics #Probability (math.PR) #Spectral Theory in Mathematical Physics #Stochastic processes and financial applications #advanced mathematical theories

paper · pdf · doi:10.48550/arxiv.0904.3518

openalex publication_date 2009/04/22 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We consider the system of stochastic differential equations dX_t=A(X_t-) dZ_t, where Z_t¹, ..., Z^d_t are independent one-dimensional symmetric stable processes of order α, 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.

Cited by

• Regularity theory for mixed local-nonlocal problem involving general stable operators

Related

• On harmonic functions of symmetric Levy processes
• Pathwise uniqueness for singular SDEs driven by stable processes
• Potential Theory of Truncated Stable Processes
• Weak Harnack Inequality and Hölder Regularity for Symmetric Stable Lévy Processes
• Uniqueness of stable-like processes