A transience condition for a class of one-dimensional symmetric L\'evy processes

TL;DR

This paper establishes a sufficient transience condition for a class of one-dimensional symmetric Lévy processes, based on integrability criteria involving the Lévy measure's density or jump probabilities.

Contribution

It provides new criteria for transience of symmetric Lévy processes and random walks with continuous or discrete jumps, extending previous understanding.

Findings

Transience occurs if ^{} dy / y^3 f(y) <  or  1 / n^3 p_n < .

Derived analogous conditions for symmetric random walks with jumps.

Offers a unified criterion for transience in continuous and discrete cases.

Abstract

In this paper, we give a sufficient condition for transience for a class of one-dimensional symmetric L\'evy processes. More precisely, we prove that a one-dimensional symmetric L\'evy process with the L\'evy measure $\nu(dy)=f(y)dy$ or $\nu(\{n\})=p_n$, where the density function $f(y)$ is such that $f(y)>0$ a.e. and the sequence $\{p_n\}_{n\geq1}$ is such that $p_n>0$ for all $n\geq1$, is transient if $$\int_1^{\infty}\frac{dy}{y^{3}f(y)}<\infty\quad\textrm{or}\quad \sum_{n=1}^{\infty}\frac{1}{n^{3}p_n}<\infty.$$ Similarly, we derive an analogous transience condition for one-dimensional symmetric random walks with continuous and discrete jumps.