Abstract
In this paper, we are attempting to study the uniqueness of invariant measures of a stochastic differential equation driven by a Lévy type noise in a real separable Hilbert space. To investigate this problem, we study the strong Feller property and irreducibility of the corresponding Markov transition semigroup respectively. To show the strong Feller property, we generalize a Bismut–Elworthy–Li type formula to our Markov transition semigroup under a non-degeneracy condition of the coefficient of the Wiener process.
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Albeverio, S., Mandrekarc, V., Rüdiger, B.: Existence of mild solutions for stochastic differential equations and semilinear equations with non-Gaussian Le’vy noise. Stoch. Process. Their Appl. 119(3), 835–863 (2009)
Albeverio, S., Rüdiger, B.: Stochastic integrals and the Lévy–Itô decomposition theorem on separable Banach spaces. Stoch. Anal. Appl. 23(2), 217–253 (2005)
Albeverio, S., Wu, J.L., Zhang, T.S.: Parabolic SPDEs driven by Poisson white noise. Stoch. Process. Their Appl. 74(1), 21–36 (1998)
Applebaum, D.: Lévy Processes and Stochastic Calculus. Cambridge Studies in Advanced Mathematics, 93, xxiv+384 pp. Cambridge University Press, Cambridge (2004)
Applebaum, D.: Martingale-valued Measures, Ornstein–Uhlenbeck Processes with Jumps and Operator Self-decomposability in Hilbert Space. Lecture Notes in Math., 2006 (1874)
Applebaum, D.: On the infinitesimal generators of Ornstein–Uhlenbeck processes with jumps in Hilbert space. Potential Anal. 26(1), 79–100 (2007)
Chojnowska-Michalik, A.: On processes of Ornstein–Uhlenbeck type in Hilbert space. Stoch. Stoch. Rep. 21, 251–286 (1987)
Da Prato, G., K.D. Elworthy, Zabczyk, J.: Strong Feller property for stochastic semilinear equations. Stoch. Anal. Appl. 13(1), 35–45 (1995)
Da Prato, G., Zabczyk, J.: Stochastic Equations in Infinite Dimensions. Encyclopedia of Mathematics and its Applications, 44. Cambridge University Press, Cambridge (1992)
Da Prato, G., Zabczyk, J.: Ergodicity for Infinite-dimensional Systems. London Mathematical Society Lecture Note Series, 229. Cambridge University Press, Cambridge (1996)
Deuschel, J.D., Zambotti, L.: Bismut–Elworthy’s formula and random walk representation for SDEs with reflection. Stoch. Process. Their Appl. 115(6), 907–925 (2005)
Elworthy, K.D.: Stochastic flows on Riemannian manifolds. In: Diffusion Processes and Related Problems in Analysis, (Charlotte, NC, 1990), vol. II, pp. 37–72, Progr. Probab., 27. Birkha”user Boston, Boston, MA (1992)
Fuhrman, M., Röckner, M.: Generalized Mehler semigroups: the non-Gaussian case. Potential Anal. 12, 1–47 (2000)
Funaki, T.: Random motion of strings and related stochastic evolution equations. Nagoya Math. J. 89, 129–193 (1983)
Funaki, T., Xie, B.: Stochastic heat equation with the distributions of Le’vy processes as its invariant measures. Stoch. Process. Their Appl. 119(2), 307–326 (2009)
Goldys, B., Maslowski, B.: Lower estimates of transition densities and bounds on exponential ergodicity for stochastic PDE’s. Ann. Probab. 34(4), 1451–1496 (2006)
Goldys, B., Maslowski, B.: Exponential ergodicity for stochastic reaction-diffusion equations. In: Stochastic Partial Differential Equations and Applications-VII, pp. 115–131. Lect. Notes Pure Appl. Math., 245. Chapman & Hall/CRC, Boca Raton, FL (2006)
Hausenblas, E.: Existence, uniqueness and regularity of parabolic SPDEs driven by Poisson random measure. Electron. J. Probab. 10, 1496–1546 (2005) (electronic)
Hausenblas, E.: A note on the Itô formula of stochastic integrals in Banach spaces. Random Oper. Stoch. Equ. 14(1), 45–58 (2006)
Hausenblas, E.: SPDEs driven by Poisson random measure with non Lipschitz coefficients: existence results. Probab. Theory Relat. Fields 137(1–2), 161–200 (2007)
Hausenblas, E., Seidler, J.: A note on maximal inequality for stochastic convolutions. Czechoslovak Math. J. 51(126)(4), 785–790 (2001)
Jurek, Z.J., Vervaat, W.: An integral representation for self-decomposable Banach space valued random variables. Z. Wahrscheinlichkeitstheor. Verw. Geb. 62(2), 247–262 (1983)
Lescot, P., Röckner, M.: Perturbations of generalized Mehler semigroups and applications to stochastic heat equations with Lévy noise and singular drift. Potential Anal. 20(4), 317–344 (2004)
Lasota, A., Szarek, T.: Lower bound technique in the theory of a stochastic differential equation. J. Diff. Equ. 231(2), 513–533 (2006)
Mandrekar, V., Rüdiger, B.: Existence and uniqueness of path wise solutions for stochastic integral equations driven by Lev́y noise on separable Banach spaces. Stochastics 78(4), 189–212 (2006)
Maslowski, B., Seidler, J.: Probabilistic approach to the strong Feller property. Probab. Theory Relat. Fields 118(2), 187–210 (2000)
Peszat, S., Zabczyk, J.: Strong Feller property and irreducibility for diffusions on Hilbert spaces. Ann. Probab. 23(1), 157–172 (1995)
Peszat, S., Zabczyk, J.: Stochastic Partial Differential Equations with Lévy Noise. An Evolution Equation Approach. Encyclopedia of Mathematics and its Applications, 113, xii+419. Cambridge University Press, Cambridge (2007)
Röckner, M., Wang, F.Y.: Harnack and functional inequalities for generalized Mehler semigroups. J. Funct. Anal. 203(1), 237–261 (2003)
Walsh, J.B.: An introduction to stochastic partial differential equations. Ecole déte de probabilites de Saint-Flour, XIV—1984. Lecture Notes in Math., 1180, pp. 265–439. Springer, Berlin (1986)
Wang, F.Y.: Logarithmic Sobolev inequalities on noncompact Riemannian manifolds. Probab. Theory Relat. Fields 109(3), 417–424 (1997)
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Xie, B. Uniqueness of Invariant Measures of Infinite Dimensional Stochastic Differential Equations Driven by Lévy Noises. Potential Anal 36, 35–66 (2012). https://doi.org/10.1007/s11118-011-9220-6
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DOI: https://doi.org/10.1007/s11118-011-9220-6