Quasistationary Theorems for Diffusions in a Bounded Open Set

Let X be the minimal diffusion associated with a uniformly elliptic differential operator L on a bounded subdomain of Rd, with C2 boundary. Under the only assumption that the coefficients of L be Hölder continuous, we prove all the standard quasistationary limit theorems (cf. Markov chain theory). Moreover, we show that the laws of X, conditioned on explosion occuring after time s, converge in total variation, as s tends to infinity, to the law of a positive recurrent diffusion X∞, which is related to X by the addition of the drift a∇ logφ, where φ is the ground state of L. Previously, such results were shown only for symmetrically reversible diffusions.