Uniform Hölder Estimates on Semigroups Generated by Non-Local Operators of Variable Order

Abstract

We consider the non-local operator of variable order as follows $$Lf(x)= \int_{\R^d\setminus\{0\}}\big(f(x+z)-f(x)-\<\nabla f(x),z\> \I_{\{|z|\le 1\}}\big)\frac{n(x,z)}{|z|^{d+α(x)}}\,dz.$$ Under mild conditions on $\alpha (x)$ and $n(x,z)$, we establish the Hölder 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.

25 pages

References in corpus (1)

• Uniqueness for the martingale problem associated with pure jump processes of variable order

Cited by in corpus (1)

• Gradient Estimates and Ergodicity for SDEs Driven by Multiplicative Lévy Noises via Coupling