Uniform H\"{o}lder Estimates on Semigroups Generated by Non-Local Operators of Variable Order

TL;DR

This paper proves Hölder regularity estimates for semigroups generated by variable order non-local operators, using probabilistic coupling methods applicable to stable-like and time-changed stable processes.

Contribution

It establishes Hölder estimates for semigroups of variable order non-local operators under mild conditions, extending regularity results to a broader class of processes.

Findings

Hölder regularity for semigroups of variable order operators.

Applicable to stable-like processes and time-changed stable processes.

Uses probabilistic coupling method for proofs.

Abstract

We consider the non-local operator of variable order as follows $$Lf(x)= \int_{\R^d\setminus\{0\}}\big(f(x+z)-f(x)-\<\nabla f(x),z\> \I_{\{|z|\le 1\}}\big)\frac{n(x,z)}{|z|^{d+\alpha(x)}}\,dz.$$ Under mild conditions on $\alpha(x)$ and $n(x,z)$, we establish the H\"{o}lder regularity for the associated semigroups. The proof is based on the probabilistic coupling method, and it successfully applies to both stable-like processes in the sense of Bass and time-change of symmetric stable processes.