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

Zdzisław Brzeźniak; Jan van Neerven

Studia Mathematica (2000)

  • Volume: 143, Issue: 1, page 43-74
  • ISSN: 0039-3223

Abstract

top
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) d X t = A X t d t + B d W 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 = ∫ 0 t S ( t - s ) B d W s H .

How to cite

top

Brzeźniak, Zdzisław, and van Neerven, Jan. "Stochastic convolution in separable Banach spaces and the stochastic linear Cauchy problem." Studia Mathematica 143.1 (2000): 43-74. <http://eudml.org/doc/216809>.

@article{Brzeźniak2000,
abstract = {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 $X_t = ∫^\{t\}_\{0\} S(t-s)BdW_\{s\}^\{H\}$.},
author = {Brzeźniak, Zdzisław, van Neerven, Jan},
journal = {Studia Mathematica},
language = {eng},
number = {1},
pages = {43-74},
title = {Stochastic convolution in separable Banach spaces and the stochastic linear Cauchy problem},
url = {http://eudml.org/doc/216809},
volume = {143},
year = {2000},
}

TY - JOUR
AU - Brzeźniak, Zdzisław
AU - van Neerven, Jan
TI - Stochastic convolution in separable Banach spaces and the stochastic linear Cauchy problem
JO - Studia Mathematica
PY - 2000
VL - 143
IS - 1
SP - 43
EP - 74
AB - 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 $X_t = ∫^{t}_{0} S(t-s)BdW_{s}^{H}$.
LA - eng
UR - http://eudml.org/doc/216809
ER -

References

top
  1. [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. Zbl0866.58020
  2. [AC] A. Antoniadis and R. Carmona, Eigenfunction expansions for infinite dimensional Ornstein-Uhlenbeck processes, Probab. Theory Related Fields 74 (1987), 31-54. Zbl0586.60073
  3. [Bax] P. Baxendale, Gaussian measures on function spaces, Amer. J. Math. 98 (1976), 891-952. Zbl0384.28011
  4. [BRS] V. I. Bogachev, M. Röckner and B. Schmuland, Generalized Mehler semigroups and applications, Probab. Theory Related Fields 105 (1996), 193-225. Zbl0849.60066
  5. [Br1] Z. Brzeźniak, Stochastic partial differential equations in M-type 2 Banach spaces, Potential Anal. 4 (1995), 1-45. Zbl0831.35161
  6. [Br2] Z. Brzeźniak, On Sobolev and Besov spaces regularity of Brownian paths, Stochastics Stochastics Rep. 56 (1996), 1-15. Zbl0890.60077
  7. [Br3] Z. Brzeźniak, On stochastic convolutions in Banach spaces and applications, ibid. 61 (1997), 245-295. 
  8. [BGN] Z. Brzeźniak, B. Goldys and J. M. A. M. van Neerven, Mean square continuity of Ornstein-Uhlenbeck processes in Banach spaces, in preparation. Zbl1037.60054
  9. [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. Zbl0956.60029
  10. [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. Zbl0944.60075
  11. [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. 
  12. [DZ] G. Da Prato and J. Zabczyk, Stochastic Equations in Infinite Dimensions, Encyclopedia Math. Appl. 44, Cambridge Univ. Press, Cambridge, 1992. 
  13. [DS] D. A. Dawson and H. Salehi, Spatially homogeneous random evolutions, J. Multivariate Anal. 10 (1980), 141-180. Zbl0439.60051
  14. [DU] J. Diestel and J. J. Uhl, Vector Measures, Math. Surveys 15, Amer. Math. Soc., Providence, R.I., 1977. 
  15. [Di] J. Dixmier, Sur un théorème de Banach, Duke Math. J. 15 (1948), 1057-1071. 
  16. [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. Zbl0193.44603
  17. [DS] N. Dunford and J. T. Schwartz, Linear Operators, Part I: General Theory, Interscience, New York, 1958. 
  18. [Kuo] H. H. Kuo, Gaussian Measures in Banach Spaces, Lecture Notes in Math. 463, Springer, 1975. Zbl0306.28010
  19. [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. 
  20. [Ne1] J. M. A. M. van Neerven, Non-symmetric Ornstein-Uhlenbeck semigroups in Banach spaces, J. Funct. Anal. 155 (1998), 495-535. Zbl0928.47031
  21. [Ne2] J. M. A. M. van Neerven, Sandwiching C 0 -semigroups, J. London Math. Soc. 60 (1999), 581-588. 
  22. [Nh] A. L. Neidhardt, Stochastic integrals in 2 -uniformly smooth Banach spaces, Ph.D. thesis, Univ. of Wisconsin, 1978. 
  23. [Nv] J. Neveu, Processus Aléatoires Gaussiens, Les Presses de l'Univ. Montréal, 1968. Zbl0192.54701
  24. [PZ] S. Peszat and J. Zabczyk, Stochastic evolution equations with a spatially homogeneous Wiener process, Stochastic Process. Appl. 72 (1997), 187-204. Zbl0943.60048
  25. [Ram] R. Ramer, On nonlinear transformations of Gaussian measures, J. Funct. Anal. 15 (1974), 166-187. Zbl0288.28011
  26. [Rö1] H. Röckle, Abstract Wiener spaces, infinite-dimensional Gaussian processes and applications, Ph.D. thesis, Ruhr-Universität Bochum, 1993. 
  27. [Rö2] H. Röckle, Banach space valued Ornstein-Uhlenbeck processes with general drift coefficients, Acta Appl. Math. 47 (1997), 323-349. Zbl0884.60005
  28. [Schw1] L. Schwartz, Sous-espaces hilbertiens d'espaces vectoriels topologiques et noyaux associés, J. Anal. Math. 13 (1964), 115-256. Zbl0124.06504
  29. [Schw2] L. Schwartz, Radon Measures on Arbitrary Topological Vector Spaces, Oxford Univ. Press, Oxford, 1973. 
  30. [VTC] N. N. Vakhania, V. I. Tarieladze and S. A. Chobanyan, Probability Distributions on Banach Spaces, D. Reidel, Dordrecht, 1987. 
  31. [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. 

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.