[Fonctions BV dans triplet de Gelfand et le problème de réflexion sur un ensemble convexe d'un espace de Hilbert]
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
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
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Publié le :
Michael Röckner 1 ; Rongchan Zhu 2 ; Xiangchan Zhu 3
@article{CRMATH_2010__348_21-22_1175_0, author = {Michael R\"ockner and Rongchan Zhu and Xiangchan Zhu}, 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}, pages = {1175--1178}, publisher = {Elsevier}, volume = {348}, number = {21-22}, year = {2010}, doi = {10.1016/j.crma.2010.10.018}, language = {en}, }
TY - JOUR AU - Michael Röckner AU - Rongchan Zhu AU - Xiangchan Zhu TI - BV functions in a Gelfand triple and the stochastic reflection problem on a convex set of a Hilbert space JO - Comptes Rendus. Mathématique PY - 2010 SP - 1175 EP - 1178 VL - 348 IS - 21-22 PB - Elsevier DO - 10.1016/j.crma.2010.10.018 LA - en ID - CRMATH_2010__348_21-22_1175_0 ER -
%0 Journal Article %A Michael Röckner %A Rongchan Zhu %A Xiangchan Zhu %T BV functions in a Gelfand triple and the stochastic reflection problem on a convex set of a Hilbert space %J Comptes Rendus. Mathématique %D 2010 %P 1175-1178 %V 348 %N 21-22 %I Elsevier %R 10.1016/j.crma.2010.10.018 %G en %F CRMATH_2010__348_21-22_1175_0
Michael Röckner; Rongchan Zhu; Xiangchan Zhu. BV functions in a Gelfand triple and the stochastic reflection problem on a convex set of a Hilbert space. Comptes Rendus. Mathématique, Volume 348 (2010) no. 21-22, pp. 1175-1178. doi : 10.1016/j.crma.2010.10.018. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2010.10.018/
[1] L. Ambrosio, G. Da Prato, D. Pallara, BV functions in a Hilbert space with respect to a Gaussian measure, preprint.
[2] 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] BV functions and distorted Ornstein–Uhlenbecl processes over the abstract Wiener space, Journals of Functional Analysis, Volume 174 (2000), pp. 227-249
[4] 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] Stochastic Analysis, Springer, Berlin, 1997
[6] Introduction to the Theory of (Non-symmetric) Dirichlet Forms, Springer-Verlag, Berlin/Heidelberg/New York, 1992
[7] Theory of Orlicz Spaces, Monographs and Textbooks in Pure and Applied Mathematics, vol. 146, Dekker, New York, 1991
[8] 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
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☆ 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.
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