Killed diffusions and their conditioning

Abstract

Let X=(Xt)t≧0be a diffusion determined by an elliptic differential operator L in R n(n≧1). For any bounded C 1,1 domain D, we define the conditional killed diffusion X ϕon D by the semigroup: $$T_t^\phi f(x) = \phi _0 (x)^{ - 1} E_x \left[ {f(X_t )\phi _0 (X_t ),\tau _D > } \right.\left. t \right]e^{\lambda _0 t} (t > 0)$$ where λ0 and φ0 are the principle eigenvalue and eigenfunction respectively of L on D with the Dirichlet boundary condition. In this paper, we prove that X ϕis a strong Feller process on D and that {T φ } is strongly continuous on C(D). For any T>0 we consider the conditioned process X Ti.e. the process X in D conditioned on {τ D >T}, and prove that X Tconverges weakly to X ϕas T→∞ without any additional hypotheses.