engleski

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

In this paper, we give a sufficient condition for the 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.

characteristics of a semimartingale; electrical network; Lévy measure; Lévy process; random walk; recurrence; transience

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