Coupling property and gradient estimates of L\'{e}vy processes via the symbol

TL;DR

This paper establishes coupling properties and gradient estimates for Lévy processes using their symbols, applicable to various classes including stable and tempered stable processes, based on asymptotic behavior of the characteristic exponent.

Contribution

It provides explicit coupling and gradient estimates for Lévy processes derived from the asymptotic behavior of their symbols, broadening understanding of their transition semigroups.

Findings

Explicit coupling property derived for Lévy processes.

Gradient estimates for transition semigroups established.

Applicable to a wide class of Lévy processes, including stable and tempered stable types.

Abstract

We derive explicitly the coupling property for the transition semigroup of a L\'{e}vy process and gradient estimates for the associated semigroup of transition operators. This is based on the asymptotic behaviour of the symbol or the characteristic exponent near zero and infinity, respectively. Our results can be applied to a large class of L\'{e}vy processes, including stable L\'{e}vy processes, layered stable processes, tempered stable processes and relativistic stable processes.