Stochastic convolution in separable Banach spaces and the stochastic linear Cauchy problem

Brzeźniak, Zdzisław ; van Neerven, Jan

Studia Mathematica, Tome 141 (2000), p. 43-74 / Harvested from The Polish Digital Mathematics Library

Résumé

Let H be a separable real Hilbert space and let E be a separable real Banach space. We develop a general theory of stochastic convolution of ℒ(H,E)-valued functions with respect to a cylindrical Wiener process ${W_{t}^{H}}_{t ∈ [0,T]}$ with Cameron-Martin space H. This theory is applied to obtain necessary and sufficient conditions for the existence of a weak solution of the stochastic abstract Cauchy problem (ACP) $dX_t = AX_tdt + BdW_t^H$ (t∈ [0,T]), $X_0 = 0$ almost surely, where A is the generator of a $C_0$-semigroup ${S(t)}_{t ≥ 0}$ of bounded linear operators on E and B ∈ ℒ(H,E) is a bounded linear operator. We further show that whenever a weak solution exists, it is unique, and given by a stochastic convolution $X_t = ∫^{t}_{0} S(t-s)BdW_{s}^{H}$.

Détail

Publié le : 2000-01-01

Zbl 0964.60043

EUDML-ID : urn:eudml:doc:216809

Bibliographie

[000] [ABB] S. Albeverio, A. M. Boutet de Monvel-Berthier and Z. Brzeźniak, The trace formula for Schrödinger operators from infinite dimensional oscillatory integrals, Math. Nachr. 182 (1996), 21-65. | Zbl 0866.58020

[001] [AC] A. Antoniadis and R. Carmona, Eigenfunction expansions for infinite dimensional Ornstein-Uhlenbeck processes, Probab. Theory Related Fields 74 (1987), 31-54. | Zbl 0586.60073

[002] [Bax] P. Baxendale, Gaussian measures on function spaces, Amer. J. Math. 98 (1976), 891-952. | Zbl 0384.28011

[003] [BRS] V. I. Bogachev, M. Röckner and B. Schmuland, Generalized Mehler semigroups and applications, Probab. Theory Related Fields 105 (1996), 193-225. | Zbl 0849.60066

[004] [Br1] Z. Brzeźniak, Stochastic partial differential equations in M-type 2 Banach spaces, Potential Anal. 4 (1995), 1-45. | Zbl 0831.35161

[005] [Br2] Z. Brzeźniak, On Sobolev and Besov spaces regularity of Brownian paths, Stochastics Stochastics Rep. 56 (1996), 1-15. | Zbl 0890.60077

[006] [Br3] Z. Brzeźniak, On stochastic convolutions in Banach spaces and applications, ibid. 61 (1997), 245-295.

[007] [BGN] Z. Brzeźniak, B. Goldys and J. M. A. M. van Neerven, Mean square continuity of Ornstein-Uhlenbeck processes in Banach spaces, in preparation. | Zbl 1037.60054

[008] [BN] Z. Brzeźniak and J. M. A. M. van Neerven, Equivalence of Banach space-valued Ornstein-Uhlenbeck processes, Stochastics Stochastics Rep. 69 (2000), 77-94. | Zbl 0956.60029

[009] [BP] Z. Brzeźniak and S. Peszat, Space-time continuous solutions to SPDE's driven by a homogeneous Wiener process, Studia Math. 137 (1999), 261-299. | Zbl 0944.60075

[010] [Ca] R. Carmona, Tensor products of Gaussian measures, in: Proc. Conf. Vector Space Measures and Applications I (Dublin, 1977), Lecture Notes in Math. 644, Springer, 1978, 96-124.

[011] [DZ] G. Da Prato and J. Zabczyk, Stochastic Equations in Infinite Dimensions, Encyclopedia Math. Appl. 44, Cambridge Univ. Press, Cambridge, 1992.

[012] [DS] D. A. Dawson and H. Salehi, Spatially homogeneous random evolutions, J. Multivariate Anal. 10 (1980), 141-180. | Zbl 0439.60051

[013] [DU] J. Diestel and J. J. Uhl, Vector Measures, Math. Surveys 15, Amer. Math. Soc., Providence, R.I., 1977.

[014] [Di] J. Dixmier, Sur un théorème de Banach, Duke Math. J. 15 (1948), 1057-1071.

[015] [DFL] R. M. Dudley, J. Feldman and L. Le Cam, On seminorms and probabilities, and abstract Wiener spaces, Ann. of Math. 93 (1971), 390-408. | Zbl 0193.44603

[016] [DS] N. Dunford and J. T. Schwartz, Linear Operators, Part I: General Theory, Interscience, New York, 1958.

[017] [Kuo] H. H. Kuo, Gaussian Measures in Banach Spaces, Lecture Notes in Math. 463, Springer, 1975. | Zbl 0306.28010

[018] [MS] A. Millet and W. Smole/nski, On the continuity of Ornstein-Uhlenbeck processes in infinite dimensions, Probab. Theory Related Fields 92 (1992), 531-547.

[019] [Ne1] J. M. A. M. van Neerven, Non-symmetric Ornstein-Uhlenbeck semigroups in Banach spaces, J. Funct. Anal. 155 (1998), 495-535. | Zbl 0928.47031

[020] [Ne2] J. M. A. M. van Neerven, Sandwiching $C_0$-semigroups, J. London Math. Soc. 60 (1999), 581-588.

[021] [Nh] A. L. Neidhardt, Stochastic integrals in $2$-uniformly smooth Banach spaces, Ph.D. thesis, Univ. of Wisconsin, 1978.

[022] [Nv] J. Neveu, Processus Aléatoires Gaussiens, Les Presses de l'Univ. Montréal, 1968. | Zbl 0192.54701

[023] [PZ] S. Peszat and J. Zabczyk, Stochastic evolution equations with a spatially homogeneous Wiener process, Stochastic Process. Appl. 72 (1997), 187-204. | Zbl 0943.60048

[024] [Ram] R. Ramer, On nonlinear transformations of Gaussian measures, J. Funct. Anal. 15 (1974), 166-187. | Zbl 0288.28011

[025] [Rö1] H. Röckle, Abstract Wiener spaces, infinite-dimensional Gaussian processes and applications, Ph.D. thesis, Ruhr-Universität Bochum, 1993.

[026] [Rö2] H. Röckle, Banach space valued Ornstein-Uhlenbeck processes with general drift coefficients, Acta Appl. Math. 47 (1997), 323-349. | Zbl 0884.60005

[027] [Schw1] L. Schwartz, Sous-espaces hilbertiens d'espaces vectoriels topologiques et noyaux associés, J. Anal. Math. 13 (1964), 115-256. | Zbl 0124.06504

[028] [Schw2] L. Schwartz, Radon Measures on Arbitrary Topological Vector Spaces, Oxford Univ. Press, Oxford, 1973.

[029] [VTC] N. N. Vakhania, V. I. Tarieladze and S. A. Chobanyan, Probability Distributions on Banach Spaces, D. Reidel, Dordrecht, 1987.

[030] [Wa] J. B. Walsh, An introduction to stochastic partial differential equations, in: P. L. Hennequin (ed.), École d'Été de Probabilités de Saint-Flour, Lecture Notes in Math. 1180, Springer, 1986, 265-439.