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 $-(-\Delta)^{\alpha(x)/2}$ 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.