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...

We construct optimal Markov couplings of Lévy processes, whose Lévy (jump) measure has an absolutely continuous component. The construction is based on properties of subordinate Brownian motions and the coupling of Brownian motions by reflection.