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Non-symmetric stable processes: Dirichlet heat kernel, Martin kernel and Yaglom limit

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  • Leżaj, Łukasz

Abstract

We study a d-dimensional non-symmetric strictly α-stable Lévy process X, whose spherical density is bounded and bounded away from the origin. First, we give sharp two-sided estimates on the transition density of X killed when leaving an arbitrary κ-fat set. We apply these results to get the existence of the Yaglom limit for arbitrary κ-fat cone. In the meantime we also obtain the spacial asymptotics of the survival probability at the vertex of the cone expressed by means of the Martin kernel for Γ and its homogeneity exponent. Our results hold for the dual process X̂, too.

Suggested Citation

  • Leżaj, Łukasz, 2024. "Non-symmetric stable processes: Dirichlet heat kernel, Martin kernel and Yaglom limit," Stochastic Processes and their Applications, Elsevier, vol. 174(C).
    • Handle: RePEc:eee:spapps:v:174:y:2024:i:c:s0304414924000681
    DOI: 10.1016/j.spa.2024.104362
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    References listed on IDEAS

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    1. Chaumont, L., 1996. "Conditionings and path decompositions for Lévy processes," Stochastic Processes and their Applications, Elsevier, vol. 64(1), pages 39-54, November.
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    5. Grzywny, Tomasz & Kim, Kyung-Youn & Kim, Panki, 2020. "Estimates of Dirichlet heat kernel for symmetric Markov processes," Stochastic Processes and their Applications, Elsevier, vol. 130(1), pages 431-470.
    6. Haas, Bénédicte & Rivero, Víctor, 2012. "Quasi-stationary distributions and Yaglom limits of self-similar Markov processes," Stochastic Processes and their Applications, Elsevier, vol. 122(12), pages 4054-4095.
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