42 research outputs found

    On the continuity of local times of Borel right Markov processes

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    The problem of finding a necessary and sufficient condition for the continuity of the local times for a general Markov process is still open. Barlow and Hawkes have completely treated the case of the L\'{e}vy processes, and Marcus and Rosen have solved the case of the strongly symmetric Markov processes. We treat here the continuity of the local times of Borel right processes. Our approach unifies that of Barlow and Hawkes and of Marcus and Rosen, by using an associated Gaussian process, that appears as a limit in a CLT involving the local time process.Comment: Published at http://dx.doi.org/10.1214/009117906000000980 in the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org

    A characterization of the infinitely divisible squared Gaussian processes

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    We show that, up to multiplication by constants, a Gaussian process has an infinitely divisible square if and only if its covariance is the Green function of a transient Markov process.Comment: Published at http://dx.doi.org/10.1214/009117905000000684 in the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org

    Decompositions of infinitely divisible nonnegative processes

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    International audienceWe establish decomposition formulas for nonnegative infinitely divisible processes. They allow to give an explicit expression of their LĂ©vy measure. In the special case of infinitely divisible permanental processes, one of these decompositions represents a new isomorphism theorem involving the local time process of a transient Markov process. We obtain in this case the expression of the LĂ©vy measure of the total local time process which is in itself a new result on the local time process. Finally, we identify a determining property of the local times for their connection with permanental processes

    A family of integral representations for the brownian variables

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    ABSTRACT. – The natural filtration of a real Brownian motion and its excursion filtration are sharing a fundamental property: the property of integral representation. As a consequence, every Brownian variable admits two distinct integral representations. We show here that there are other integral representations of the Brownian variables. They make use of a stochastic flow studied by Bass and Burdzy. Our arguments are inspired by Rogers and Walsh’s results on stochastic integration with respect to the Brownian local times. 2003 Éditions scientifiques et médicales Elsevier SAS MSC: 60H05; 60G4

    Dynkin's isomorphism theorem and the stochastic heat equation

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    Consider the stochastic heat equation \partial_t u = \sL u + \dot{W}, where \sL is the generator of a [Borel right] Markov process in duality. We show that the solution is locally mutually absolutely continuous with respect to a smooth perturbation of the Gaussian process that is associated, via Dynkin's isomorphism theorem, to the local times of the replica-symmetric process that corresponds to \sL.In the case that \sL is the generator of a L\'evy process on Rd\R^d, our result gives a probabilistic explanation of the recent findings of Foondun et al

    Temps locaux, excursions et lieu le plus visite par un mouvement brownien reel

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    SIGLEINIST T 72933 / INIST-CNRS - Institut de l'Information Scientifique et TechniqueFRFranc