Recurrence and transience of diffusions in a half-space

Abstract

For non-degenerate diffusions in the half-space with oblique reflection, a dichotomy between recurrence and transience is established; convenient characterizations of recurrence and transience are given. Verifiable criteria for recurrence/transience are derived in terms of the generator and the boundary operator. Using these criteria, 'real variables proofs' of some results due to Rogers, concerning reflecting Brownian motion in a half-plane, are obtained. The problem of transience down a side in the case of diffusions in the half-plane is dealt with. Positive recurrence of diffusions in half-space is also considered; it is shown that the hitting time of any open set has finite expectation if there is just one positive recurrent point.