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Lévy measure of superprocesses; absorption processes

In: Itô’s Stochastic Calculus and Probability Theory

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  • E. B. Dynkin

    (Cornell University, Department of Mathematics)

Abstract

Summary We prove that the (modified) Lévy measure of a superprocess X is the lifting of its discontinuous branching characteristic. We also investigate, for a wide class of functions f, the continuous martingale part of process . (For time-homogeneous processes with local branching, similar results have been obtained earlier by El Karoui and Roelly.) We use the expression for the Lévy measure to establish the predictability of absorption processes. For a particle system, an absorption process can be obtained by freezing every particle at the first exit from a subset Q in the product of time interval and state space. The mass distribution X t ’ of particles frozen during time interval [0, t] is a measure on the complement of Q. The monotone increasing measure-valued process Xt is called an absorption process. A process of this kind can be constructed for every superprocess X in E and every finely open Borel subset Q of ℝ+ × E. In a recent paper by Dynkin and Kuznetsov, absorption processes have been used to describe linear additive functionals of superprocesses.

Suggested Citation

  • E. B. Dynkin, 1996. "Lévy measure of superprocesses; absorption processes," Springer Books, in: Nobuyuki Ikeda & Shinzo Watanabe & Masatoshi Fukushima & Hiroshi Kunita (ed.), Itô’s Stochastic Calculus and Probability Theory, pages 1-14, Springer.
    • Handle: RePEc:spr:sprchp:978-4-431-68532-6_1
    DOI: 10.1007/978-4-431-68532-6_1
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