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Semi-stable Markov processes in rn

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  • Kiu, Sun-Wah

Abstract

A Markov process in Rn{xt} with transition function Pt is called semi-stable of order [alpha]>0 if for every a>0, Pt(x, E) = Pat(aax, aaE). Let [phi]t([omega])=[integral operator]t0xs([omega])-1/[alpha] ds, T(t) be its inverse and {yt}={xT(t)}. Theorem 1: {Yt} is a multiplicative invariant process; i.e., it has transition function qt satisfying qt(x,E)=qt(ax,aE) for all a > 0. Theorem 2: If {xt} is Feller, right continuous and uniformly stochastic continuous on a neighborhood of the origin, then {yt} is Feller.

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  • Kiu, Sun-Wah, 1980. "Semi-stable Markov processes in rn," Stochastic Processes and their Applications, Elsevier, vol. 10(2), pages 183-191, September.
    • Handle: RePEc:eee:spapps:v:10:y:1980:i:2:p:183-191
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    Cited by:

    1. Liao, Ming & Wang, Longmin, 2011. "Isotropic self-similar Markov processes," Stochastic Processes and their Applications, Elsevier, vol. 121(9), pages 2064-2071, September.
    2. Chybiryakov, Oleksandr, 2006. "The Lamperti correspondence extended to Lévy processes and semi-stable Markov processes in locally compact groups," Stochastic Processes and their Applications, Elsevier, vol. 116(5), pages 857-872, May.
    3. Grégoire Véchambre, 2022. "General Self-Similarity Properties for Markov Processes and Exponential Functionals of Lévy Processes," Journal of Theoretical Probability, Springer, vol. 35(4), pages 2083-2144, December.

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