BV functions in a Gelfand triple and the stochastic reflection problem on a convex set of a Hilbert space

Röckner, Michael; Zhu, Rongchan; Zhu, Xiangchan

doi:10.1016/j.crma.2010.10.018

Functional Analysis/Probability Theory

BV functions in a Gelfand triple and the stochastic reflection problem on a convex set of a Hilbert space
[Fonctions BV dans triplet de Gelfand et le problème de réflexion sur un ensemble convexe d'un espace de Hilbert]

Présenté par : the Editorial Board

Röckner, Michael ¹ ; Zhu, Rongchan ² ; Zhu, Xiangchan ³

¹ Department of Mathematics, University of Bielefeld, D-33615 Bielefeld, Germany
² Institute of Applied Mathematics, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, China
³ School of Mathematical Sciences, Peking University, Beijing 100871, China

Comptes Rendus. Mathématique, Tome 348 (2010) no. 21-22, pp. 1175-1178.

Résumé
Abstract

Dans cette Note, on introduit des fonctions BV dans un triplet de Gelfand qui est une extension de fonctions BV dans Ambrosio et al., preprint [1] en utilizant la forme de Dirichlet. Par cette définition, on peut considérer le problème de réflexion stochastique associé à un opérateur auto-adjoint A et un processus de Wiener cylindrique sur un ensemble convexe Γ. Nous démontrons l'existence et l'unicité d'une solution forte de ce problème si Γ et un ensemble convexe régulier. Le résultat est aussi étendu au cas non symétrique. Finalement, nous utilisons les fonctions BV dans le cas $Γ = K_{α}$ , où $K_{α} = {f \in L^{2} (0, 1) | f ⩾ - α}$ , $α ⩾ 0$ .

In this Note we introduce BV functions in a Gelfand triple, which is an extension of BV functions in Ambrosio et al., preprint [1], by using Dirichlet form theory. By this definition, we can consider the stochastic reflection problem associated with a self-adjoint operator A and a cylindrical Wiener process on a convex set Γ. We prove the existence and uniqueness of a strong solution of this problem when Γ is a regular convex set. The result is also extended to the non-symmetric case. Finally, we extend our results to the case when $Γ = K_{α}$ , where $K_{α} = {f \in L^{2} (0, 1) | f ⩾ - α}$ , $α ⩾ 0$ .

Reçu le : 2010-06-04
Accepté le : 2010-10-08
Publié le : 2010-10-27

DOI : 10.1016/j.crma.2010.10.018

Affiliations des auteurs :

Röckner, Michael ¹ ; Zhu, Rongchan ² ; Zhu, Xiangchan ³

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     title = {BV functions in a {Gelfand} triple and the stochastic reflection problem on a convex set of a {Hilbert} space},
     journal = {Comptes Rendus. Math\'ematique},
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Röckner, Michael; Zhu, Rongchan; Zhu, Xiangchan. BV functions in a Gelfand triple and the stochastic reflection problem on a convex set of a Hilbert space. Comptes Rendus. Mathématique, Tome 348 (2010) no. 21-22, pp. 1175-1178. doi : 10.1016/j.crma.2010.10.018. http://www.numdam.org/articles/10.1016/j.crma.2010.10.018/

Bibliographie
Cité par

[1] L. Ambrosio, G. Da Prato, D. Pallara, BV functions in a Hilbert space with respect to a Gaussian measure, preprint.

[2] Barbu, V.; Da Prato, G.; Tubaro, L. Kolmogorov equation associated to the stochastic reflection problem on a smooth convex set of a Hilbert spaces, The Annals of Probability, Volume 4 (2009), pp. 1427-1458

[3] Fukushima, M. BV functions and distorted Ornstein–Uhlenbecl processes over the abstract Wiener space, Journals of Functional Analysis, Volume 174 (2000), pp. 227-249

[4] Fukushima, M.; Hino, Masanori On the space of BV functions and a related stochastic calculus in infinite dimensions, Journals of Functional Analysis, Volume 183 (2001), pp. 245-268

[5] Malliavin, P. Stochastic Analysis, Springer, Berlin, 1997

[6] Ma, Z.M.; Röckner, M. Introduction to the Theory of (Non-symmetric) Dirichlet Forms, Springer-Verlag, Berlin/Heidelberg/New York, 1992

[7] Rao, M.M.; Ren, Z.D. Theory of Orlicz Spaces, Monographs and Textbooks in Pure and Applied Mathematics, vol. 146, Dekker, New York, 1991

[8] Zambotti, L. Integration by parts formulae on convex sets of paths and applications to SPDEs with reflection, Probability Theory Related Fields, Volume 123 (2002), pp. 579-600

Cité par Sources :

^☆ Research supported by 973 project, NSFC, key Lab of CAS, the DFG through IRTG 1132 and CRC 701 and the I. Newton Institute, Cambridge, UK.