Markov processes associated with $ {L}^{p}$-resolvents, applications to quasi-regular Dirichlet forms and stochastic differential equations

Lucian Beznea; Nicu Boboc; Michael Röckner

doi:10.1016/j.crma.2007.12.005

Probability Theory/Potential Theory

Markov processes associated with

L^{p}

-resolvents, applications to quasi-regular Dirichlet forms and stochastic differential equations
[Processus de Markov associés aux résolvants sur

L^{p}

, applications aux formes de Dirichlet qusi-régulières et aux équations différentielles stochastiques]

Présenté par : Paul Malliavin

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

Comptes Rendus. Mathématique, Volume 346 (2008) no. 5-6, pp. 323-328.

Résumé
Abstract

Nous montrons que à toute résolvante sous-markovienne continue sur $L^{p} (E, μ)$ , 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 $L^{2} (E, μ)$ devienne quasi-régulière ; (ii) on résoudre des équations différentielles stochastiques sur des espaces de Hilbert, dans le sense du problème de martingale.

We show that every $C_{0}$ -resolvent on $L^{p} (E, μ)$ , 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 $L^{2} (E, μ)$ becomes quasi-regular; (ii) we solve stochastic differential equations on Hilbert spaces in the sense of a martingale problem.

Reçu le : 2007-04-20
Accepté le : 2007-12-10
Publié le : 2008-02-07

DOI : 10.1016/j.crma.2007.12.005

Affiliations des auteurs :

Lucian Beznea ^{1, 2} ; Nicu Boboc ³ ; Michael Röckner ^{4, 5}

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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},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {323--328},
     publisher = {Elsevier},
     volume = {346},
     number = {5-6},
     year = {2008},
     doi = {10.1016/j.crma.2007.12.005},
     language = {en},
}

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%A Nicu Boboc
%A Michael Röckner
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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/

Bibliographie
Cité par

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[4] L. Beznea; N. Boboc; M. Röckner Markov processes associated with $L^{p}$ -resolvents and applications to stochastic differential equations on Hilbert space, J. Evol. Equ., Volume 6 (2006), pp. 745-772

[5] S. Carillo-Menendez Processus de Markov associé à une forme de Dirichlet non symétrique, Z. Wahrsch. Verw. Geb., Volume 33 (1975), pp. 139-154

[6] G. Da Prato; M. Röckner Singular dissipative stochastic equations in Hilbert spaces, Probab. Theory Rel. Fields, Volume 124 (2002), pp. 261-303

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