An Overshoot Approach to Recurrence and Transience of Markov Processes

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 $-(-Δ)^{α(x)/2}$ such that $α(x)=α$ for $x<-R$ and $α(x)=β$ for $x>R$ for some $R>0$ and $α,β\in(0,2)$ is transient if and only if $α+β<2$, otherwise it is recurrent. As a special case this yields a new proof for the recurrence, point recurrence and transience of symmetric $α$-stable processes.

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