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 Probability Surveys > Vol. 12 (2015) open journal systems 

Infinite dimensional Ornstein-Uhlenbeck processes driven by Lévy processes

David Applebaum, University of Sheffield, UK

We review the probabilistic properties of Ornstein-Uhlenbeck processes in Hilbert spaces driven by Lévy processes. The emphasis is on the different contexts in which these processes arise, such as stochastic partial differential equations, continuous-state branching processes, generalised Mehler semigroups and operator self-decomposable distributions. We also examine generalisations to the case where the driving noise is cylindrical.

AMS 2000 subject classifications: Primary 60G51; secondary 60H15, 60H10, 60E07, 60J80.

Keywords: Lévy process, Ornstein-Uhlenbeck process, Mehler semigroup, skew–convolution semigroup, branching property, invariant measure, operator self–decomposability, Urbanik semigroup, cylindrical process.

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Applebaum, David, Infinite dimensional Ornstein-Uhlenbeck processes driven by Lévy processes, Probability Surveys, 12, (2015), 33-54 (electronic). DOI: 10.1214/14-PS249.


[1]    Albeverio, S. and Rüdiger, B. (2005), Stochastic integrals and the Lévy-Itô decomposition theorem on separable Banach spaces, Stoch. Anal. Appl. 23, 217–253. MR2130348

[2]    Albeverio, S., di Persio, L., Mastrogiamoco, E. and Smii, B. (2013), Explicit invariant measures for infinite dimensional SDE driven by Lévy noise with dissipative nonlinear drift I, arXiv:1312.2398v1.

[3]    Applebaum, D. (2006), Martingale-valued measures, Ornstein-Uhlenbeck processes with jumps and operator self-decomposability in Hilbert space, In Memoriam Paul-André Meyer, Séminaire de Probabilités 39, eds. M. Emery and M. Yor, Lecture Notes in Math, Vol. 1874, pp. 173–198, Springer-Verlag. MR2276896

[4]    Applebaum, D. (2007), On the infinitesimal generators of Ornstein-Uhlenbeck processes with jumps in Hilbert Space, Potential Anal. 26, 79–100. MR2276526

[5]    Applebaum, D. (2009), Lévy Processes and Stochastic Calculus (second edition), Cambridge University Press. MR2512800

[6]    Applebaum, D. and Riedle, M. (2010), Cylindrical Lévy processes in Banach spaces, Proc. London Math. Soc. 101 697–726. MR2734958

[7]    Applebaum, D. and van Neerven, J.M.A.M. (2014), Second quantisation for skew convolution products of measures in Banach spaces, Electronic J. Prob. 19, paper 11 (17 pages). MR3164764

[8]    Applebaum, D. and van Neerven, J.M.A.M. (2015), Second quantisation for skew convolution products of infinitely divisible measures, Infinite Dimensional Anal. and Quantum Prob. 18, 155003 (12 pages). MR3324719

[9]    Applebaum, D. and Wu, J.L. (2000), Stochastic partial differential equations driven by Lévy space-time white noise, Random Operators and Stochastic Equations 8, 245–260. MR1796675

[10]    Barndorff-Nielsen, O.E. and Shephard, N. (2001), Non-Gaussian Ornstein-Uhlenbeck-based models and some of their uses in financial economics, J.R. Statis. Soc. B 63, 167–241. MR1841412

[11]    Bogachev, V.I., Röckner, M. and Schmuland, B. (1996), Generalized Mehler semigroups and applications, Prob. Th. Rel. Fields 105, 193–225. MR1392452

[12]    Bradley, R.C. and Jurek, Z.J. (2014), The strong mixing and the self-decomposability properties, Stat. Prob. Lett. 84 64–71. MR3131257

[13]    Brzeźniak, Z., Goldys, B., Imkeller, P., Peszat, S., Priola, E. and Zabczyk, J. (2010), Time irregularity of generalised Ornstein-Uhlenbeck processes, C. R. Math. Acad. Sci. Paris, Ser. 1 348, 273–276. MR2600121

[14]    Brzeźniak, Z. and Zabczyk, J. (2010), Regularity of Ornstein-Uhlenbeck processes driven by a Lévy white noise, Potential Anal. 32, 153–188. MR2584982

[15]    Chojnowska-Michalik, A. (1987), On processes of Ornstein-Uhlenbeck type in Hilbert space, Stochastics 21, 251–286. MR0900115

[16]    Dalang, R. and Quer-Sardanyons, L. (2011), Stochastic integral for spde’s: A comparison, Exp. Math. 29, 67–109. MR2785545

[17]    Dawson, D.A. and Li, Z. (2004), Non-differentiable skew-convolution semigroups and related Ornstein-Uhlenbeck processes, Potential Anal. 20, 285–302. MR2032499

[18]    Dawson, D.A. and Li, Z. (2006), Skew convolution semigroups and affine processes, Ann. Prob. 34, 1103–1142. MR2243880

[19]    Dawson, D.A., Li, Z., Schmuland, B. and Sun, W. (2004), Generalized Mehler semigroups and catalytic branching processes with immigration, Potential Anal. 21, 75–97. MR2048508

[20]    Da Prato, G. and Zabczyk, J. (1992), Stochastic Equations in Infinite Dimensions, Cambridge University Press. MR1207136

[21]    Da Prato, G. and Zabczyk, J. (1996), Ergodicity for Infinite Dimensional Systems, Cambridge University Press. MR1417491

[22]    Da Prato, G. and Zabczyk, J. (2002), Second Order Partial Differential Equations in Hilbert Spaces, Cambridge University Press. MR1985790

[23]    Doob, J.L. (1942), The Brownian movement and stochastic equations, Ann. Math. 43, 351–369. MR0006634

[24]    Duffie, D., Filipović, D. and Schachermeyer, W. (2003), Affine processes and applications in finance, Ann. Appl. Prob. 13, 984–1053. MR1994043

[25]    Fuhrman, M. and Röckner, M. (2000), Generalized Mehler semigroups: The non-Gaussian case, Potential Anal. 12, 1–47. MR1745332

[26]    Jakubowski, A., Kwapień, S., de Fitte, P.R. and Rosińki, J. (2002), Radonifaction of cylindrical semimartingales by a single Hilbert-Schmidt operator, Infinite Dimensional Anal. and Quantum Prob. 5, 429–440. MR1930962

[27]    Jespersen, S., Metzler, R. and Fogedby, H. (1999), Lévy flights in external force fields: Langevin and fractional Fokker-Planck equations and their solutions, Phys. Rev. E 59 2736–2745.

[28]    Jurek, Z.J. (1982), An integral representation of operator-self-decomposable random variables, Bull. Acad. Pol. Sci. 30, 385–393. MR0707753

[29]    Jurek, Z.J. (2009), On relations between Urbanik and Mehler semigroups, Prob. Math. Stat. 29, 297–308. MR2792545

[30]    Jurek, Z.J. and Vervaat, W. (1983), An integral representation for selfdecomposable Banach space valued random variables, Z. Wahrscheinlichkeitstheorie verw. Gebiete 62, 247–262. MR0688989

[31]     Klüppelberg, C., Lindner, A. and Maller, R. (2006), Continuous time volatility modelling: COGARCH versus Ornstein-Uhlenbeck models, in: From Stochastic Calculus to Mathematical Finance, pp. 393–419, Springer, Berlin. MR2234284

[32]    Lescot, P. and Röckner, M. (2002), Generators of Mehler-type semigroups as pseudo-differential operators, Infinite Dimensional Anal. and Quantum Prob. 5, 297–315. MR1930955

[33]    Lescot, P. and Röckner, M. (2004), Perturbations of generalized Mehler semigroups and applications to stochastic heat equations with Lévy noise and singular drift, Potential Anal. 20, 317–344. MR2032114

[34]    Li, Z., (2011), Measure-Valued Branching Markov Processes, Springer-Verlag, Berlin, Heidelberg. MR2760602

[35]    Liu, Y. and Zhai, J. (2012), A note on time regularity of generalized Ornstein-Uhlenbeck processes with cylindrical stable noise, C. R. Math. Acad. Sci. Paris, Ser. 1 350, 97–100. MR2887844

[36]    Metivier, M. and Pellaumail, J. (1980), Stochastic Integration, Academic Press. MR0578177

[37]    Ornstein, L.S. and Uhlenbeck, G.E. (1930), On the theory of Brownian motion, Phys. Rev. 36, 823–841.

[38]    Ouyang, S. and Röckner, M. (2012), Time inhomogeneous generalized Mehler semigroups and skew-convolution equations, to appear in Forum Math., arXiv:1009.5314v3.

[39]    Ouyang, S., Röckner, M., Ouyang, S. and Wang, F.-Y. (2012), Harnack inequalities and applications for Ornstein-Uhlenbeck semigroups with jumps, Potential Anal. 36, 301–315. MR2886463

[40]    Parthasarthy, K.R. (1967), Probability Measures on Metric Spaces, Academic Press, New York. MR0226684

[41]    Peszat, S. (2011), Lévy-Ornstein-Uhlenbeck transition operator as second quantised operator, J. Funct. Anal. 260, 3457–3473. MR2781967

[42]    Peszat, S. and Zabczyk, J. (2007), Stochastic Partial Differential Equations with Lévy Noise, Encyclopedia of Mathematics and Its Applications, Vol. 113, Cambridge University Press. MR2356959

[43]    Priola, E. and Zabczyk, J. (2010), On linear evolution equations for a class of cylindrical Lévy noises. Stochastic partial differential equations and applications, Quad. Mat., 25 223–242, Dept. Math., Seconda Univ. Napoli, Caserta. MR2985090

[44]    Priola, E. and Zabczyk, J. (2011), Structural properties of semilinear SPDEs driven by cylindrical stable processes, Probab. Th. Rel. Fields 149, 97–137. MR2773026

[45]    Peszat, S. and Zabczyk, J. (2013), Time regularity of solutions to linear equations with Lévy noise in infinite dimensions, Stoch. Proc. Appl. 123, 719–751. MR3005003

[46]    Riedle, M. (2014), Stochastic integration with respect to cylindrical Lévy processes in Hilbert space: An L2 approach, Infinite Dimensional Anal. and Quantum Prob. 17 1450008 (19 pages). MR3189652

[47]    Riedle, M. (2015), Ornstein-Uhlenbeck processes driven by cylindrical Lévy processes, Potential Anal. 42, 809–838. MR3339223

[48]    Riedle, M. and van Gaans, O. (2009), Stochastic integration for Lévy processes with values in Banach spaces, Stoch. Proc. App. 119, 1952–1974. MR2519352

[49]    Röckner, M. and Wang, F.-Y. (2003), Harnack and functional inequalities for generalized Mehler semigroups, J.Funct.Anal. 203, 237–261. MR1996872

[50]    Sato, K.-I. (1999), Lévy Processes and Infinite Divisibility, Cambridge University Press. MR1739520

[51]    Schmuland, B. and Sun, E. (2001), On the equation μs+t = μs Tsμt, Stat. Prob. Lett. 52, 183–188. MR1841407

[52]    Schwartz, L. (1973), Radon Measures on Arbitrary Topological Spaces and Cylindrical Measures, Oxford University Press. MR0426084

[53]    Urbanik, K. (1978), Lévy’s probability measures on Banach spaces, Studia Math. 63, 283–308. MR0515497

[54]    Walsh, J.B. (1986), An introduction to stochastic partial differential equations, in: Ecole d’Eté de Probabilités de St Flour XIV, Lecture Notes in Mathematics, Vol. 1180, pp. 266–439, Springer-Verlag. MR0876085

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