Perturbations of Functional Inequalities for L\'evy Type Dirichlet Forms

TL;DR

This paper investigates how perturbations affect functional inequalities for Lévý type Dirichlet forms, highlighting differences between finite and infinite jump ranges and providing examples to demonstrate the optimality of the results.

Contribution

It extends existing results on perturbations of functional inequalities from diffusion processes to Lévý processes with finite jumps and explores the more complex infinite jump case.

Findings

Results for finite jump range are natural extensions of diffusion cases.

Infinite jump range cases are fundamentally different.

Examples illustrate the optimality of the derived inequalities.

Abstract

Perturbations of super Poincar\'e and weak Poincar\'e inequalities for L\'evy type Dirichlet forms are studied. When the range of jumps is finite our results are natural extensions to the corresponding ones derived earlier for diffusion processes; and we show that the study for the situation with infinite range of jumps is essentially different. Some examples are presented to illustrate the optimality of our results.