# An Overshoot Approach to Recurrence and Transience of Markov Processes

Bj\"orn B\"ottcher

Arxiv ID: 1007.2055•Last updated: 4/17/2020

We develop criteria for recurrence and transience of one-dimensional Markov
processes which have jumps and oscillate between $+\infty$ and $-\infty$. The
conditions are based on a Markov chain which only consists of jumps
(overshoots) of the process into complementary parts of the state space. In
particular we show that a stable-like process with generator
$-(-\Delta)^{\alpha(x)/2}$ such that $\alpha(x)=\alpha$ for $x<-R$ and
$\alpha(x)=\beta$ for $x>R$ for some $R>0$ and $\alpha,\beta\in(0,2)$ is
transient if and only if $\alpha+\beta<2$, otherwise it is recurrent. As a
special case this yields a new proof for the recurrence, point recurrence and
transience of symmetric $\alpha$-stable processes.

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