Abstract: Weak solutions to stochastic differential equations in Rd ,d≥2 are continuous-time Markov processes. We show that under very general conditions such solutions possess irreducibility and continuity properties which enable criteria for Harris recurrence and transience, developed in Meyn and Tweedie (1993b), Down, Meyn and Tweedie (1995) and Stramer and Tweedie (1994), to be applied to them. All of our criteria are in terms of the second-order differential operator, and hence a unified approach to the stability classification of weak solutions is obtained which generalises that of Khas'minskii (1980); we also develop explicit forms of the stationary measure for many such processes. The results are applicable in continuous-time time series analysis (see Stramer, Tweedie and Brockwell (1996) and Stramer, Brockwell and Tweedie (1996)) and we consider a multi-dimensional threshold model as one such application.
Key words and phrases: Degenerate diffusions, exponential ergodicity, irreducible Markov processes, multi-dimensional non-linear diffusions, recurrence, stationary measures, stochastic differential equations, transience.