Space-time continuous solutions to SPDE's driven by a homogeneous Wiener process

Zdzisław Brzeźniak; Szymon Peszat

Studia Mathematica (1999)

  • Volume: 137, Issue: 3, page 261-299
  • ISSN: 0039-3223

Abstract

top
Stochastic partial differential equations on d are considered. The noise is supposed to be a spatially homogeneous Wiener process. Using the theory of stochastic integration in Banach spaces we show the existence of a Markovian solution in a certain weighted L q -space. Then we obtain the existence of a space continuous solution by means of the Da Prato, Kwapień and Zabczyk factorization identity for stochastic convolutions.

How to cite

top

Brzeźniak, Zdzisław, and Peszat, Szymon. "Space-time continuous solutions to SPDE's driven by a homogeneous Wiener process." Studia Mathematica 137.3 (1999): 261-299. <http://eudml.org/doc/216686>.

@article{Brzeźniak1999,
abstract = {Stochastic partial differential equations on $ℝ^d$ are considered. The noise is supposed to be a spatially homogeneous Wiener process. Using the theory of stochastic integration in Banach spaces we show the existence of a Markovian solution in a certain weighted $L^q$-space. Then we obtain the existence of a space continuous solution by means of the Da Prato, Kwapień and Zabczyk factorization identity for stochastic convolutions.},
author = {Brzeźniak, Zdzisław, Peszat, Szymon},
journal = {Studia Mathematica},
keywords = {stochastic partial differential equations in $L^q$-spaces; homogeneous Wiener process; random environment; stochastic integration in Banach spaces; stochastic partial differential equations; continuous solutions},
language = {eng},
number = {3},
pages = {261-299},
title = {Space-time continuous solutions to SPDE's driven by a homogeneous Wiener process},
url = {http://eudml.org/doc/216686},
volume = {137},
year = {1999},
}

TY - JOUR
AU - Brzeźniak, Zdzisław
AU - Peszat, Szymon
TI - Space-time continuous solutions to SPDE's driven by a homogeneous Wiener process
JO - Studia Mathematica
PY - 1999
VL - 137
IS - 3
SP - 261
EP - 299
AB - Stochastic partial differential equations on $ℝ^d$ are considered. The noise is supposed to be a spatially homogeneous Wiener process. Using the theory of stochastic integration in Banach spaces we show the existence of a Markovian solution in a certain weighted $L^q$-space. Then we obtain the existence of a space continuous solution by means of the Da Prato, Kwapień and Zabczyk factorization identity for stochastic convolutions.
LA - eng
KW - stochastic partial differential equations in $L^q$-spaces; homogeneous Wiener process; random environment; stochastic integration in Banach spaces; stochastic partial differential equations; continuous solutions
UR - http://eudml.org/doc/216686
ER -

References

top
  1. [1] V. Bally, I. Gyöngy and E. Pardoux, White noise driven parabolic SPDE'S with measurable drift, J. Funct. Anal. 120 (1994), 484-510. Zbl0801.60049
  2. [2] P. Baxendale, Gaussian measures on function spaces, Amer. J. Math. 98 (1976), 891-952. Zbl0384.28011
  3. [3] B. Beauzamy, Introduction to Banach Spaces and their Geometry, North-Holland, Amsterdam, 1985. 
  4. [4] Z. Brzeźniak, Stochastic partial differential equations in M-type 2 Banach spaces, Potential Anal. 4 (1995), 1-45. Zbl0831.35161
  5. [5] Z. Brzeźniak, On stochastic convolutions in Banach spaces and applications, Stochastics Stochastics Rep. 61 (1997), 245-295. Zbl0891.60056
  6. [6] Z. Brzeźniak and D. Gątarek, Martingale solutions and invariant measures for stochastic evolution equations in Banach spaces, Stochastic Process. Appl., to appear. Zbl0996.60074
  7. [7] Z. Brzeźniak and S. Peszat, Stochastic two dimensional Euler equations, Preprint 2, School of Mathematics, University of Hull, Hull, 1999. Zbl1032.60055
  8. [8] D. L. Burkholder, Martingales and Fourier analysis in Banach spaces, in: Probability and Analysis (Varenna, 1985), Lecture Notes in Math. 1206, Springer, Berlin, 1986, 61-108. 
  9. [9] M. Capiński and S. Peszat, On the existence of a solution to stochastic Navier-Stokes equations, Nonlinear Anal., to appear. 
  10. [10] G. Da Prato, S. Kwapień and J. Zabczyk, Regularity of solutions of linear stochastic equations in Hilbert spaces, Stochastics 23 (1987), 1-23. Zbl0634.60053
  11. [11] G. Da Prato and J. Zabczyk, Stochastic Equations in Infinite Dimensions, Cambridge Univ. Press, Cambridge, 1992. Zbl0761.60052
  12. [12] G. Da Prato and J. Zabczyk, Ergodicity for Infinite Dimensional Systems, Cambridge Univ. Press, Cambridge, 1996. Zbl0849.60052
  13. [13] E. B. Davies, One-Parameter Semigroups, Academic Press, London, 1980. Zbl0457.47030
  14. [14] D. A. Dawson and H. Salehi, Spatially homogeneous random evolutions, J. Multivariate Anal. 10 (1980), 141-180. Zbl0439.60051
  15. [15] E. Dettweiler, Stochastic integration relative to Brownian motion on a general Banach space, Doǧa Mat. 15 (1991), 6-44. Zbl0970.60517
  16. [16] C. Donati-Martin and E. Pardoux, White noise driven SPDE'S with reflection, Probab. Theory Related Fields 95 (1993), 1-24. Zbl0794.60059
  17. [17] S. D. Eidel'man, Parabolic Systems, North-Holland, Amsterdam, 1969. 
  18. [18] T. Funaki, Regularity properties for stochastic partial differential equations of parabolic type, Osaka J. Math. 28 (1991), 495-516. Zbl0770.60062
  19. [19] N. Yu. Goncharuk and P. Kotelenez, Fractional step method for stochastic evolution equations, Stochastic Process. Appl. 73 (1998), 1-45. Zbl0942.60057
  20. [20] D. Henry, Geometric Theory of Semilinear Parabolic Equations, Lecture Notes in Math. 840, Springer, Berlin, 1981. Zbl0456.35001
  21. [21] K. Itô, Foundations of Stochastic Differential Equations in Infinite Dimensional Spaces, SIAM, Philadelphia, 1984. 
  22. [22] P. Kotelenez, Existence, uniqueness and smoothness for a class of function valued stochastic partial differential equations, Stochastics Stochastics Rep. 41 (1992), 177-199. Zbl0766.60078
  23. [23] P. Kotelenez, Comparison methods for a class of function valued stochastic differential equations, Probab. Theory Related Fields 93 (1992), 1-19. Zbl0767.60053
  24. [24] H. H. Kuo, Gaussian Measures in Banach Spaces, Lecture Notes in Math. 463, Springer, Berlin, 1975. Zbl0306.28010
  25. [25] O. A. Ladyženskaja, V. A. Solonnikov and N. N. Ural'ceva, Linear and Quasilinear Equations of Parabolic Type, Transl. Math. Monographs 23, Amer. Math. Soc., Providence, RI, 1968. 
  26. [26] R. Manthey and T. Zausinger, Stochastic evolution equations in L 2 ν ϱ , Stochastics Stochastics Rep. 66 (1999), 37-85. Zbl0926.60051
  27. [27] M. Marcus and G. Pisier, Random Fourier Series, with Applications to Harmonic Analysis, Princeton Univ. Press, Princeton, 1981. Zbl0474.43004
  28. [28] A. L. Neidhardt, Stochastic integrals in 2-uniformly smooth Banach spaces, Ph.D. Thesis, University of Wisconsin, 1978. 
  29. [29] J. Nobel, Evolution equation with Gaussian potential, Nonlinear Anal. 28 (1997), 103-135. 
  30. [30] A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer, New York, 1983. 
  31. [31] S. Peszat, Existence and uniqueness of the solution for stochastic equations on Banach spaces, Stochastics Stochastics Rep. 55 (1995), 167-193. Zbl0886.60064
  32. [32] S. Peszat and J. Seidler, Maximal inequalities and space-time regularity of stochastic convolutions, Math. Bohem. 123 (1998), 7-32. Zbl0903.60047
  33. [33] S. Peszat and J. Zabczyk, Strong Feller property and irreducibility for diffusions on Hilbert spaces, Ann. Probab. 23 (1995), 157-172. Zbl0831.60083
  34. [34] S. Peszat and J. Zabczyk, Stochastic evolution equations with a spatially homogeneous Wiener process, Stochastic Process. Appl. 72 (1997), 187-204. Zbl0943.60048
  35. [35] S. Peszat and J. Zabczyk, Nonlinear stochastic wave and heat equations, Probab. Theory Related Fields, to appear. Zbl0959.60044
  36. [36] G. Pisier, Martingales with values in uniformly convex spaces, Israel J. Math. 20 (1976), 326-350. Zbl0344.46030
  37. [37] E. Sinestrari, Accretive differential operators, Boll. Un. Mat. Ital. A 13 (1976), 19-31. Zbl0343.35016
  38. [38] G. Tessitore and J. Zabczyk, Invariant measures for stochastic heat equations, Probab. Math. Statist. 18 (1999), 271-287. Zbl0986.60057
  39. [39] G. Tessitore and J. Zabczyk, Strict positivity for stochastic heat equations, Stochastic Process. Appl. 77 (1998), 83-98. Zbl0933.60071
  40. [40] N. N. Vakhania, V. I. Tarieladze and S. A. Chobanyan, Probability Distributions on Banach Spaces, Reidel, Dordrecht, 1987. 
  41. [41] J. B. Walsh, An introduction to stochastic partial differential equations, in: École d'été de probabilités de Saint-Flour XIV-1984, Lecture Notes in Math. 1180, Springer, Berlin, 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.