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 +[infinity] and -[infinity]. 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][set membership, variant](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.