An Overshoot Approach to Recurrence and Transience of Markov Processes

Details An Overshoot Approach to Recurrence and Transience of Markov Processes An Overshoot Approach to Recurrence and Transience of Markov Processes

2010-07-13

arxiv.org/abs/1007.2055v1

Mathematics

Probability

Scientific paper

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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