Abstract
We study Hilbert space valued Ornstein–Uhlenbeck processes (Y(t), t ≥ 0) which arise as weak solutions of stochastic differential equations of the type dY = JY + CdX(t) where J generates a C 0 semigroup in the Hilbert space H, C is a bounded operator and (X(t), t ≥ 0) is an H-valued Lévy process. The associated Markov semigroup is of generalised Mehler type. We discuss an analogue of the Feller property for this semigroup and explicitly compute the action of its generator on a suitable space of twice-differentiable functions. We also compare the properties of the semigroup and its generator with respect to the mixed topology and the topology of uniform convergence on compacta.
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Albeverio, S., Rüdiger, B.: Stochastic integrals and the Lévy–Itô decomposition theorem on separable Banach spaces. Stochastic Anal. Appl. 23, 217–253 (2005)
Applebaum, D.: Lévy Processes and Stochastic Calculus. Cambridge University Press, Cambridge, UK (2004)
Applebaum, D.: Martingale-valued measures, Ornstein–Uhlenbeck processes with jumps and operator self-decomposability in Hilbert space. In: Emery, M., Yor, M. (eds.) Memoriam Paul-André Meyer, Séminaire de Probabilités, vol. 39. Lecture Notes in Math, vol. 1874, pp. 173–198. Springer, Berlin Heidelberg New York (2006)
Bogachev, V.I., Röckner, M., Schmuland, B.: Generalized Mehler semigroups and applications. Probab. Theory Related Fields 105, 193–225 (1996)
Cerrai, S.: A Hille–Yosida theorem for weakly continuous semigroups. Semigroup Forum 49, 349–367 (1994)
Chojnowska-Michalik, A.: On processes of Ornstein–Uhlenbeck type in Hilbert space. Stochastics 21, 251–286 (1987)
Da Prato, G., Lunardi, A.: On the Ornstein–Uhlenbeck operator in spaces of continuous functions. J. Funct. Anal. 131, 94–114 (1995)
Da Prato, G., Zabczyk, J.: Stochastic Equations in Infinite Dimensions. Cambridge University Press, Cambridge, UK (1992)
Da Prato, G., Zabczyk, J.: Second Order Partial Differential Equations in Hilbert Space. Cambridge University Press, Cambridge, UK (2002)
Davies, E.B.: One-Parameter Semigroups. Academic, London (1980)
Dawson, D.A., Li, Z., Schmuland, B., Sun, W.: Generalized Mehler semigroups and catalytic branching processes with immigration. Potential Anal. 21, 75–97 (2004)
Dettweiler, E.: Banach space valued processes with independent increments and stochastic integration. In: Beck, A., Jacobs, K. (eds.) Probability in Banach Spaces IV, Proceedings Oberwolfach 1982. Lecture Notes in Mathematics, vol. 990, pp. 54–84. Springer, Berlin, Heidelberg, New York (1983)
Fuhrman, M., Röckner, M.: Generalized Mehler semigroups: the non-Gaussian case. Potential Anal. 12, 1–47 (2000)
Goldys, B.: Diffusion semigroups on spaces of continuous functions. (In preparation)
Goldys, B., Kocan, M.: Diffusion semigroups in spaces of continuous functions with mixed topology. J. Differential Equations 173, 17–39 (2001)
Goldys, B., van Neerven, J.M.A.M.: Transition semigroups of Banach space valued Ornstein–Uhlenbeck processes. Acta Appl. Math. 76, 283–330 (2003)
Jurek, Z.J., Vervaat, W.: An integral representation for selfdecomposable Banach space valued random variables. Z. Wahrscheinlichkeitstheor. Verw. Geb. 62, 247–262 (1983)
Kolmogorov, A.N.: Zufällige Bewegungen. Ann. of Math. 116, 116–117 (1934)
Kōmura, T.: Semigroups of operators in locally convex spaces. J. Funct. Anal. 2, 258–296 (1968)
Kühnemund, F.: A Hille–Yosida theorem for bi-continuous semigroups. Semigroup Forum 68, 87–107 (2003)
Kühnemund, F.: Bi-continuous semigroups on spaces with two topologies. Ph.D thesis, Eberhard Karls Universität Tübingen (2001)
Leha, G., Ritter, G.: On diffusion processes and their semigroups in Hilbert spaces with an application to interacting stochastic systems. Ann. Probab. 12, 1077–1112 (1984)
Lescot, P., Röckner, M.: Generators of Mehler-type semigroups as pseudo-differential operators. Infin. Dimens. Anal. Quantum Probab. Relat. Top. 5, 297–315 (2002)
Lescot, P., Röckner, M.: Perturbations of generalized Mehler semigroups and applications to stochastic heat equations with Lévy noise and singular drift. Potential Anal. 20, 317–344 (2004)
Metafune, G., Pallara, D., Priola, E.: Spectrum of Ornstein–Uhlenbeck operators in \(L^{p}\)-spaces with respect to invariant measures. J. Funct. Anal. 196, 40–60 (2002)
Métivier, M.: Semimartingales, A Course on Stochastic Processes. W. de Gruyter, Berlin (1982)
Ornstein, L.S., Uhlenbeck, G.E.: On the theory of Brownian motion. Phys. Rev. 36, 823–841 (1930)
Parthasarathy, K.R.: Probability Measures on Metric Spaces. Academic, New York (1967)
Priola, E.: On a class of Markov type semigroups in spaces of uniformly continuous and bounded functions. Studia Math. 136, 271–295 (1999)
Priola, E., Zabczyk, J.: Liouville theorems for non-local operators. J. Funct. Anal. 216, 455–490 (2004)
Röckner, M., Wang, F.-Y.: Harnack and functional inequalities for generalized Mehler semigroups. J. Funct. Anal. 203, 237–261 (2003)
Sato, K.-I., Yamazoto, M.: Operator-selfdecomposable distributions as limit distributions of processes of Ornstein–Uhlenbeck type. Stochastic Process. Appl. 17, 73–100 (1984)
Schmuland, B., Sun, W.: On the equation \(\mu_{s+t} = \mu_{s}*T_{s}\mu_{t}\). Statist. Probab. Lett. 52, 183–188 (2001)
Tessitore, G., Zabczyk, J.: Trotter’s formula for transition semigroups. Semigroup Forum 63, 114–126 (2001)
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Applebaum, D. On the Infinitesimal Generators of Ornstein–Uhlenbeck Processes with Jumps in Hilbert Space. Potential Anal 26, 79–100 (2007). https://doi.org/10.1007/s11118-006-9028-y
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DOI: https://doi.org/10.1007/s11118-006-9028-y
Key words
- H-valued Lévy process
- Ornstein–Uhlenbeck process
- generalised Mehler semigroup
- auxiliary semigroup
- operator-selfdecomposability
- quasi-locally equicontinuous semigroup
- pseudo-Feller property
- mixed topology
- cylinder function
- Kolmogorov–Lévy operator