Comptes Rendus
Probability Theory/Potential Theory
Markov processes associated with Lp-resolvents, applications to quasi-regular Dirichlet forms and stochastic differential equations
Comptes Rendus. Mathématique, Volume 346 (2008) no. 5-6, pp. 323-328.

We show that every C0-resolvent on Lp(E,μ), where (E,B) is a Lusin measurable space and μ is a σ-finite measure on B, has an associate sufficiently regular Markov process on a (larger) Lusin topological space containing E as a Borel subset. We give general conditions on the resolvent's generator such that the above process lives on E. We present two applications: (i) we settle a question of G. Mokobodzki on the existence of a (Lusin) topology on E having B as Borel σ-algebra such that a given Dirichlet form on L2(E,μ) becomes quasi-regular; (ii) we solve stochastic differential equations on Hilbert spaces in the sense of a martingale problem.

Nous montrons que à toute résolvante sous-markovienne continue sur Lp(E,μ), où (E,B) est un espace mesurable de Lusin et μ est une mesure σ-finie sur B, on peut associer un processus droit sur un espace topologique de Lusin contenant E comme un sousensemble borélien finement dense. Nous donnons des conditions sufficientes sur le générateur infinitesimal de la résolvante tel que l'espace d'états du processus soit E. Nous obtenons deux applications : (i) une réponse à une question posée par G. Mokobodzki sur l'existence d'une topologie de Lusin sur E ayant B comme tribue borélienne, telle que une forme de Dirichlet donnée sur L2(E,μ) devienne quasi-régulière ; (ii) on résoudre des équations différentielles stochastiques sur des espaces de Hilbert, dans le sense du problème de martingale.

Received:
Accepted:
Published online:
DOI: 10.1016/j.crma.2007.12.005
Lucian Beznea 1, 2; Nicu Boboc 3; Michael Röckner 4, 5

1 Institute of Mathematics “Simion Stoilow” of the Romanian Academy, P.O. Box 1-764, RO-014700 Bucharest, Romania
2 University of Piteşti, Romania
3 Faculty of Mathematics and Informatics, University of Bucharest, str. Academiei 14, RO-010014 Bucharest, Romania
4 Fakultät für Mathematik, Universität Bielefeld, Postfach 100 131, 33501 Bielefeld, Germany
5 Departments of Mathematics and Statistics, Purdue University, 150 N. University St. West Lafayette, IN 47907-2067, USA
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     author = {Lucian Beznea and Nicu Boboc and Michael R\"ockner},
     title = {Markov processes associated with $ {L}^{p}$-resolvents, applications to quasi-regular {Dirichlet} forms and stochastic differential equations},
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Lucian Beznea; Nicu Boboc; Michael Röckner. Markov processes associated with $ {L}^{p}$-resolvents, applications to quasi-regular Dirichlet forms and stochastic differential equations. Comptes Rendus. Mathématique, Volume 346 (2008) no. 5-6, pp. 323-328. doi : 10.1016/j.crma.2007.12.005. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2007.12.005/

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