Kernel estimates for a class of fractional Kolmogorov operators

TL;DR

This paper establishes kernel estimates for fractional powers of Kolmogorov operators using weighted Nash inequalities, providing bounds on the associated Markov transition kernels.

Contribution

It introduces a method to derive kernel bounds for fractional Kolmogorov operators based on weighted Nash inequalities, advancing understanding of their transition behaviors.

Findings

Derived weighted Nash inequalities for fractional operators

Established non-uniform bounds on transition kernels

Extended kernel estimates to fractional Kolmogorov operators

Abstract

Assuming a weighted Nash type inequality for the generator $-A$ of a Markov semigroup, we prove a weighted Nash type inequality for its fractional power and deduce non-uniform bounds on the transition kernel corresponding to the Markov semigroup generated by $-A^\alpha$.