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Killed diffusions and their conditioning
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  • Published: December 1988

Killed diffusions and their conditioning

  • Guanglu Gong1,
  • Minping Qian2 nAff3 &
  • Zhongxin Zhao2 nAff4 

Probability Theory and Related Fields volume 80, pages 151–167 (1988)Cite this article

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  • 22 Citations

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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:

$$T_t^\phi f(x) = \phi _0 (x)^{ - 1} E_x \left[ {f(X_t )\phi _0 (X_t ),\tau _D > } \right.\left. t \right]e^{\lambda _0 t} (t > 0)$$

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 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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Author information

Author notes
  1. Minping Qian

    Present address: Department of Probability and Statistics, Peking University, Beijing

  2. Zhongxin Zhao

    Present address: Department of Mathematics, University of Missouri, 65211, Columbia, MO, USA

Authors and Affiliations

  1. Department of Probability and Statistics, Peking University, Beijing, People's Republic of China

    Guanglu Gong

  2. IMA, University of Minnesota, 55455, MI, USA

    Minping Qian & Zhongxin Zhao

Authors
  1. Guanglu Gong
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  2. Minping Qian
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  3. Zhongxin Zhao
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Cite this article

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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  • Received: 24 January 1986

  • Revised: 25 February 1988

  • Issue Date: December 1988

  • DOI: https://doi.org/10.1007/BF00348757

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Keywords

  • Boundary Condition
  • Stochastic Process
  • Probability Theory
  • Differential Operator
  • Statistical Theory
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