A transience condition for a class of one-dimensional symmetric Lévy processes

Abstract

In this paper, we give a sufficient condition for transience for a class of one-dimensional symmetric Lévy processes. More precisely, we prove that a one-dimensional symmetric Lévy process with the Lévy 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 $\{p_n{\}}_{n\ge 1}$ is such that $p_n>0$ for all $n\ge 1$, is transient if $${\int }_1^{\infty }\frac{dy}{y^{3}f(y)}<\infty \text{or}\sum _{n=1}^{\infty }\frac{1}{n^3{p}_n}<\infty .$$ Similarly, we derive an analogous transience condition for one-dimensional symmetric random walks with continuous and discrete jumps.

arXiv comment13 pages