A Note on time regularity of generalized Ornstein–Uhlenbeck processes with cylindrical stable noise

Yong Liu; Jianliang Zhai

doi:10.1016/j.crma.2011.11.017

Probability Theory

A Note on time regularity of generalized Ornstein–Uhlenbeck processes with cylindrical stable noise
[Une Note sur la régularité du temps des processus Ornstein–Uhlenbeck généralisée avec le bruit stables cylindriques]

Présenté par : the Editorial Board

Yong Liu ^1,² ; Jianliang Zhai ¹

¹ LMAM, School of Mathematical Science, Peking University, Beijing 100871, PR China
² Center for Statistical Science, Peking University, Beijing 100871, PR China

Comptes Rendus. Mathématique, Volume 350 (2012) no. 1-2, pp. 97-100.

Résumé
Abstract

Une condition nécessaire et suffisante de la modification càdlàg des processus dʼOrnstein–Uhlenbeck avec bruit stables cylindriques dans un éspace Hilbert est montré dans cette Note. En appliquant ce résultat, quelques questions dans Time irregularity of generalized Ornstein–Uhlenbeck processes [C. R. Acad. Sci. Paris, Ser. I 348 (2010) 273–276] et Structural properties of semilinear SPDEs driven by cylindrical stable process [Probab. Theory Related Fields 149 (2011) 97–137] sont répondues.

A necessary and sufficient condition of càdlàg modification of Ornstein–Uhlenbeck process with cylindrical stable noise in a Hilbert space is given in this Note. Applying this result, some questions in Time irregularity of generalized Ornstein–Uhlenbeck processes [C. R. Acad. Sci. Paris, Ser. I 348 (2010) 273–276] and Structural properties of semilinear SPDEs driven by cylindrical stable process [Probab. Theory Related Fields 149 (2011) 97–137] are answered.

Reçu le : 2011-09-27
Accepté le : 2011-11-29
Publié le : 2011-12-12

DOI : 10.1016/j.crma.2011.11.017

Affiliations des auteurs :

Yong Liu ^{1, 2} ; Jianliang Zhai ¹

¹ LMAM, School of Mathematical Science, Peking University, Beijing 100871, PR China
² Center for Statistical Science, Peking University, Beijing 100871, PR China

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     author = {Yong Liu and Jianliang Zhai},
     title = {A {Note} on time regularity of generalized {Ornstein{\textendash}Uhlenbeck} processes with cylindrical stable noise},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {97--100},
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     year = {2012},
     doi = {10.1016/j.crma.2011.11.017},
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Yong Liu; Jianliang Zhai. A Note on time regularity of generalized Ornstein–Uhlenbeck processes with cylindrical stable noise. Comptes Rendus. Mathématique, Volume 350 (2012) no. 1-2, pp. 97-100. doi : 10.1016/j.crma.2011.11.017. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2011.11.017/

Bibliographie
Cité par

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[2] Z. Brzeźniak; J. Zabczyk Regularity of Ornstein–Uhlenbeck processes driven by Lévy white noise, Potential Anal., Volume 32 (2010), pp. 153-188

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[5] A. Jakubowski On the Skorohod topology, Ann. Inst. Henri Poincaré, Volume 22 (1986), pp. 263-285

[6] S. Peszat; J. Zabczyk Stochastic Partial Differential Equations with Lévy Noise, Cambridge University Press, Cambridge, 2007

[7] E. Priola, J. Zabczyk, On linear evolution with cylindrical Lévy noise, in: G. Da Prato, L. Tubaro (Eds.), Stochastic Partial Differential Equations and Applications VIII, Proceedings of the Levico Conference, 2008.

[8] E. Priola; J. Zabczyk Structural properties of semilinear SPDEs driven by cylindrical stable process, Probab. Theory Related Fields, Volume 149 (2011), pp. 97-137

[9] K.I. Sato Lévy Processes and Infinite Divisible Distributions, Cambridge University Press, London, 1999

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