Comptes Rendus
Functional Analysis/Probability Theory
Time irregularity of generalized Ornstein–Uhlenbeck processes
[Irrégularité en temps des processus Ornstein–Uhlenbeck généralisés]
Comptes Rendus. Mathématique, Volume 348 (2010) no. 5-6, pp. 273-276.

Dans cette Note on traite les propriétés de solutions d'équations d'évolution linéaires perturbées par des processus de Lévy cylindriques. Sous des conditions assez faibles, on trouve que les solutions ne possèdent pas de modifications càdlàg. On énonce quelques questions naturelles s'en déduisant.

This Note is concerned with the properties of solutions to a linear evolution equation perturbed by a cylindrical Lévy process. It turns out that solutions, under rather weak requirements, do not have a càdlàg modification. Some natural open questions are also stated.

Reçu le :
Accepté le :
Publié le :
DOI : 10.1016/j.crma.2010.01.022

Zdzisław Brzeźniak 1 ; Ben Goldys 2 ; Peter Imkeller 3 ; Szymon Peszat 4 ; Enrico Priola 5 ; Jerzy Zabczyk 6

1 Department of Mathematics, The University of York, Heslington, York YO10 5DD, UK
2 School of Mathematics, The University of New South Wales, Sydney 2052, Australia
3 Institut für Mathematik, Humboldt-Universität zu Berlin, 10099 Berlin, Germany
4 Institute of Mathematics, Polish Academy of Sciences, Św. Tomasza 30/7, 31-027 Kraków, Poland
5 Dipartimento di Matematica, Universita di Torino, via Carlo Alberto 10, 10-123 Torino, Italy
6 Institute of Mathematics, Polish Academy of Sciences, 00-950 Warszawa, Poland
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Zdzisław Brzeźniak; Ben Goldys; Peter Imkeller; Szymon Peszat; Enrico Priola; Jerzy Zabczyk. Time irregularity of generalized Ornstein–Uhlenbeck processes. Comptes Rendus. Mathématique, Volume 348 (2010) no. 5-6, pp. 273-276. doi : 10.1016/j.crma.2010.01.022. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2010.01.022/

[1] D. Applebaum Lévy Processes and Stochastic Calculus, Cambridge Studies in Advanced Mathematics, vol. 93, Cambridge University Press, Cambridge, 2004

[2] Z. Brzeźniak; J. Zabczyk Regularity of Ornstein–Uhlenbeck processes driven by a Lévy white noise, Potential Anal., Volume 32 (2010) no. 2, pp. 153-188

[3] G. Da Prato; J. Zabczyk Stochastic Equations in Infinite Dimensions, Cambridge University Press, Cambridge, 1992

[4] I. Iscoe; M.B. Marcus; D. McDonald; M. Talagrand; J. Zinn Continuity of l2-valued Ornstein–Uhlenbeck processes, Ann. Probab., Volume 18 (1990), pp. 68-84

[5] P. Kotelenez A maximal inequality for stochastic convolution integrals on Hilbert spaces and space-time regularity of linear stochastic partial differential equations, Stochastics, Volume 21 (1987), pp. 345-458

[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 2008 Conference

[8] E. Priola, J. Zabczyk, Structural properties of semilinear SPDEs driven by cylindrical stable processes, Probab. Theory Related Fields, in press

Cité par Sources :

Supported by the Polish Ministry of Science and Education project 1PO 3A 034 29 “Stochastic evolution equations with Lévy noise” and by EC FP6 Marie Curie ToK programme SPADE2.

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