Relatively compact criteria for Hilbert valued random fields on abstract Wiener space

Xicheng Zhang

doi:10.1016/j.crma.2006.01.004

Probability Theory

Relatively compact criteria for Hilbert valued random fields on abstract Wiener space

Presented by: Paul Malliavin

Xicheng Zhang ¹

¹ Department of Mathematics, Huazhong University of Science and Technology, Wuhan, Hubei 430074, PR China

Comptes Rendus. Mathématique, Volume 342 (2006) no. 6, pp. 437-440.

Abstract
Résumé

In terms of the compact embedding theorems in finite dimensional Sobolev spaces, conditions are given under which Hilbert valued random fields on abstract Wiener space are relatively compact in some $L^{p}$ -space.

Nous obtenons un nouveau critère pour qu'une famille de l'espace $L^{p} (X, B)$ , définie sur un espace de Wiener et à valeurs dans un espace de Banach B, soit compacte. La démonstration utilise l'approximation de dimension finie et l'hypercontractivité du semi-groupe d'Ornstein–Uhlenbeck. Notre résultat est différent d'un résultat récent de Bally–Saussereau dans le sens où nous travaillons dans $L^{p}$ pour tout $p > 1$ tandis que le résultat de Bally–Saussereau est limité à $p = 2$ .

Received: 2005-09-20
Accepted: 2005-12-20
Published online: 2006-01-26

DOI: 10.1016/j.crma.2006.01.004

Author's affiliations:

Xicheng Zhang ¹

¹ Department of Mathematics, Huazhong University of Science and Technology, Wuhan, Hubei 430074, PR China

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     author = {Xicheng Zhang},
     title = {Relatively compact criteria for {Hilbert} valued random fields on abstract {Wiener} space},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {437--440},
     publisher = {Elsevier},
     volume = {342},
     number = {6},
     year = {2006},
     doi = {10.1016/j.crma.2006.01.004},
     language = {en},
}

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Xicheng Zhang. Relatively compact criteria for Hilbert valued random fields on abstract Wiener space. Comptes Rendus. Mathématique, Volume 342 (2006) no. 6, pp. 437-440. doi : 10.1016/j.crma.2006.01.004. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2006.01.004/

References
Cited by

[1] R.A. Adams Sobolev Spaces, Academic Press, New York, 1978

[2] V. Bally; B. Saussereau A relative compactness criterion in Wiener–Sobolev spaces and application to semi-linear stochastic PDEs, J. Funct. Anal., Volume 210 (2004) no. 2, pp. 465-515

[3] G. Da Prato; P. Malliavin; D. Nualart Compact families of Wiener functionals, C. R. Acad. Sci. Paris, Sér. I, Volume 315 (1992), pp. 1287-1291

[4] P. Malliavin Stochastic Analysis, Grundlehren Math. Wiss., Springer-Verlag, Berlin, 1997

[5] A. Pazy Semi-Groups of Linear Operators and Applications, Springer-Verlag, Berlin, 1985

[6] X. Zhang Relatively compact sets on abstract Wiener space, Acta Math. Sinica, Volume 21 (2005) no. 4, pp. 819-822

[7] X. Zhang, Relatively compact families of functionals on abstract Wiener space and applications, J. Func. Anal., in press

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