On Transience of L\'evy-Type Processes

TL;DR

This paper investigates the transience properties of Lévý-type processes, deriving sharp conditions based on pseudo-differential operator symbols, and generalizes classical results for diffusion and stable Lévý processes.

Contribution

It provides Chung-Fuchs type criteria for transience of Lévý-type processes, extending known results to more general pseudo-differential operators and dimensions.

Findings

Derived sharp transience conditions using the symbol of the operator

Generalized classical results to Lévý-type processes with variable symbols

Provided criteria for transience based on Lévý measures in radial cases

Abstract

In this paper, we study weak and strong transience of a class of Feller processes associated with pseudo-differential operators, the so-called L\'evy-type processes. As a main result, we derive Chung-Fuchs type conditions (in terms of the symbol of the corresponding pseudo-differential operator) for these properties, which are sharp for L\'evy processes. Also, as a consequence, we discuss the weak and strong transience with respect to the dimension of the state space and Pruitt indices, thus generalizing some well-known results related to elliptic diffusion and stable L\'evy processes. Finally, in the case when the symbol is radial (in the co-variable) we provide conditions for the weak and strong transience in terms of the L\'evy measures.