An overshoot approach to recurrence and transience of Markov processes

Abstract

We 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 < 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.