An overshoot approach to recurrence and transience of Markov processes

AbstractWe develop criteria for recurrence and transience of one-dimensional Markov processes which have jumps and oscillate between +∞ and −∞. 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∈(0,2) is transient if and only if α+β