Summary
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:
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 tφ } 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.
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Gong, G., Qian, M. & Zhao, Z. Killed diffusions and their conditioning. Probab. Th. Rel. Fields 80, 151–167 (1988). https://doi.org/10.1007/BF00348757
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DOI: https://doi.org/10.1007/BF00348757