Gauss-Markov processes on Hilbert spaces

Goldys, Ben; Peszat, Szymon; Zabczyk, Jerzy

doi:10.1090/tran/6329

Gauss-Markov processes on Hilbert spaces
HTML articles powered by AMS MathViewer

by Ben Goldys, Szymon Peszat and Jerzy Zabczyk PDF

Trans. Amer. Math. Soc. 368 (2016), 89-108 Request permission

Abstract:

K. Itô characterised in 1984 zero-mean stationary Gauss–Markov processes evolving on a class of infinite-dimensional spaces. In this work we extend the work of Itô in the case of Hilbert spaces: Gauss–Markov families that are time-homogenous are identified as solutions to linear stochastic differential equations with singular coefficients. Choosing an appropriate locally convex topology on the space of weakly sequentially continuous functions we also characterize the transition semigroup, the generator and its core, thus providing an infinite-dimensional extension of the classical result of Courrège in the case of Gauss–Markov semigroups.

References

Elisa Alòs and Stefano Bonaccorsi, Stochastic partial differential equations with Dirichlet white-noise boundary conditions, Ann. Inst. H. Poincaré Probab. Statist. 38 (2002), no. 2, 125–154 (English, with English and French summaries). MR 1899108, DOI 10.1016/S0246-0203(01)01097-4
Zdzisław Brzeźniak, Ben Goldys, Szymon Peszat, and Francesco Russo, Second order PDEs with Dirichlet white noise boundary conditions, J. Evol. Equ. 15 (2015), no. 1, 1–26. MR 3315663, DOI 10.1007/s00028-014-0246-2
Z. Brzeźniak, M. Ondrejat, and J. Seidler, Invariant measures for stochastic nonlinear beam and wave equations (2012), preprint.
Ph. Courrège, Sur la forme integro-différentielle des opérateurs de $C_K^\infty$ dans $C_0$ satisfaisant au principe du maximum, Sém. Théorie du Potentiel, Exposé 2, 1965/1966.
Anna Chojnowska-Michalik, Stochastic differential equations in Hilbert spaces, Probability theory (Papers, VIIth Semester, Stefan Banach Internat. Math. Center, Warsaw, 1976) Banach Center Publ., vol. 5, PWN, Warsaw, 1979, pp. 53–74. MR 561468
Sandra Cerrai, Weakly continuous semigroups in the space of functions with polynomial growth, Dynam. Systems Appl. 4 (1995), no. 3, 351–371. MR 1348505
G. Da Prato and J. Zabczyk, Evolution equations with white-noise boundary conditions, Stochastics Stochastics Rep. 42 (1993), no. 3-4, 167–182. MR 1291187, DOI 10.1080/17442509308833817
Giuseppe Da Prato and Jerzy Zabczyk, Stochastic equations in infinite dimensions, Encyclopedia of Mathematics and its Applications, vol. 44, Cambridge University Press, Cambridge, 1992. MR 1207136, DOI 10.1017/CBO9780511666223
G. Da Prato and J. Zabczyk, Ergodicity for infinite-dimensional systems, London Mathematical Society Lecture Note Series, vol. 229, Cambridge University Press, Cambridge, 1996. MR 1417491, DOI 10.1017/CBO9780511662829
Edward Brian Davies, One-parameter semigroups, London Mathematical Society Monographs, vol. 15, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], London-New York, 1980. MR 591851
Andreas Defant and Klaus Floret, Tensor norms and operator ideals, North-Holland Mathematics Studies, vol. 176, North-Holland Publishing Co., Amsterdam, 1993. MR 1209438
D. Duffie, D. Filipović, and W. Schachermayer, Affine processes and applications in finance, Ann. Appl. Probab. 13 (2003), no. 3, 984–1053. MR 1994043, DOI 10.1214/aoap/1060202833
Nelson Dunford and Jacob T. Schwartz, Linear operators. Part I, Wiley Classics Library, John Wiley & Sons, Inc., New York, 1988. General theory; With the assistance of William G. Bade and Robert G. Bartle; Reprint of the 1958 original; A Wiley-Interscience Publication. MR 1009162
G. Fabbri and B. Goldys, An LQ problem for the heat equation on the halfline with Dirichlet boundary control and noise, SIAM J. Control Optim. 48 (2009), no. 3, 1473–1488. MR 2496985, DOI 10.1137/070711529
B. Goldys and M. Kocan, Diffusion semigroups in spaces of continuous functions with mixed topology, J. Differential Equations 173 (2001), no. 1, 17–39. MR 1836243, DOI 10.1006/jdeq.2000.3918
Kiyosi It\B{o}, Infinite-dimensional Ornstein-Uhlenbeck processes, Stochastic analysis (Katata/Kyoto, 1982) North-Holland Math. Library, vol. 32, North-Holland, Amsterdam, 1984, pp. 197–224. MR 780759, DOI 10.1016/S0924-6509(08)70394-5
Olav Kallenberg, Foundations of modern probability, 2nd ed., Probability and its Applications (New York), Springer-Verlag, New York, 2002. MR 1876169, DOI 10.1007/978-1-4757-4015-8
N. V. Krylov, The heat equation in $L_q((0,T),L_p)$-spaces with weights, SIAM J. Math. Anal. 32 (2001), no. 5, 1117–1141. MR 1828321, DOI 10.1137/S0036141000372039
Avi Mandelbaum, Linear estimators and measurable linear transformations on a Hilbert space, Z. Wahrsch. Verw. Gebiete 65 (1984), no. 3, 385–397. MR 731228, DOI 10.1007/BF00533743
Bohdan Maslowski and Jan Seidler, Strong Feller solutions to SPDE’s are strong Feller in the weak topology, Studia Math. 148 (2001), no. 2, 111–129. MR 1881256, DOI 10.4064/sm148-2-2
A. Pazy, Semigroups of linear operators and applications to partial differential equations, Applied Mathematical Sciences, vol. 44, Springer-Verlag, New York, 1983. MR 710486, DOI 10.1007/978-1-4612-5561-1
Byron Schmuland and Wei Sun, On the equation $\mu _{t+s}=\mu _s*T_s\mu _t$, Statist. Probab. Lett. 52 (2001), no. 2, 183–188. MR 1841407, DOI 10.1016/S0167-7152(00)00235-2
N. N. Vakhania, V. I. Tarieladze, and S. A. Chobanyan, Probability distributions on Banach spaces, Mathematics and its Applications (Soviet Series), vol. 14, D. Reidel Publishing Co., Dordrecht, 1987. Translated from the Russian and with a preface by Wojbor A. Woyczynski. MR 1435288, DOI 10.1007/978-94-009-3873-1