# 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

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topBrzeźniak, 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] 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] P. Baxendale, Gaussian measures on function spaces, Amer. J. Math. 98 (1976), 891-952. Zbl0384.28011
- [3] B. Beauzamy, Introduction to Banach Spaces and their Geometry, North-Holland, Amsterdam, 1985.
- [4] Z. Brzeźniak, Stochastic partial differential equations in M-type 2 Banach spaces, Potential Anal. 4 (1995), 1-45. Zbl0831.35161
- [5] Z. Brzeźniak, On stochastic convolutions in Banach spaces and applications, Stochastics Stochastics Rep. 61 (1997), 245-295. Zbl0891.60056
- [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] Z. Brzeźniak and S. Peszat, Stochastic two dimensional Euler equations, Preprint 2, School of Mathematics, University of Hull, Hull, 1999. Zbl1032.60055
- [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] M. Capiński and S. Peszat, On the existence of a solution to stochastic Navier-Stokes equations, Nonlinear Anal., to appear.
- [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] G. Da Prato and J. Zabczyk, Stochastic Equations in Infinite Dimensions, Cambridge Univ. Press, Cambridge, 1992. Zbl0761.60052
- [12] G. Da Prato and J. Zabczyk, Ergodicity for Infinite Dimensional Systems, Cambridge Univ. Press, Cambridge, 1996. Zbl0849.60052
- [13] E. B. Davies, One-Parameter Semigroups, Academic Press, London, 1980. Zbl0457.47030
- [14] D. A. Dawson and H. Salehi, Spatially homogeneous random evolutions, J. Multivariate Anal. 10 (1980), 141-180. Zbl0439.60051
- [15] E. Dettweiler, Stochastic integration relative to Brownian motion on a general Banach space, Doǧa Mat. 15 (1991), 6-44. Zbl0970.60517
- [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] S. D. Eidel'man, Parabolic Systems, North-Holland, Amsterdam, 1969.
- [18] T. Funaki, Regularity properties for stochastic partial differential equations of parabolic type, Osaka J. Math. 28 (1991), 495-516. Zbl0770.60062
- [19] N. Yu. Goncharuk and P. Kotelenez, Fractional step method for stochastic evolution equations, Stochastic Process. Appl. 73 (1998), 1-45. Zbl0942.60057
- [20] D. Henry, Geometric Theory of Semilinear Parabolic Equations, Lecture Notes in Math. 840, Springer, Berlin, 1981. Zbl0456.35001
- [21] K. Itô, Foundations of Stochastic Differential Equations in Infinite Dimensional Spaces, SIAM, Philadelphia, 1984.
- [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] P. Kotelenez, Comparison methods for a class of function valued stochastic differential equations, Probab. Theory Related Fields 93 (1992), 1-19. Zbl0767.60053
- [24] H. H. Kuo, Gaussian Measures in Banach Spaces, Lecture Notes in Math. 463, Springer, Berlin, 1975. Zbl0306.28010
- [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] R. Manthey and T. Zausinger, Stochastic evolution equations in ${L}^{2}{\nu}_{\varrho}$, Stochastics Stochastics Rep. 66 (1999), 37-85. Zbl0926.60051
- [27] M. Marcus and G. Pisier, Random Fourier Series, with Applications to Harmonic Analysis, Princeton Univ. Press, Princeton, 1981. Zbl0474.43004
- [28] A. L. Neidhardt, Stochastic integrals in 2-uniformly smooth Banach spaces, Ph.D. Thesis, University of Wisconsin, 1978.
- [29] J. Nobel, Evolution equation with Gaussian potential, Nonlinear Anal. 28 (1997), 103-135.
- [30] A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer, New York, 1983.
- [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] S. Peszat and J. Seidler, Maximal inequalities and space-time regularity of stochastic convolutions, Math. Bohem. 123 (1998), 7-32. Zbl0903.60047
- [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] S. Peszat and J. Zabczyk, Stochastic evolution equations with a spatially homogeneous Wiener process, Stochastic Process. Appl. 72 (1997), 187-204. Zbl0943.60048
- [35] S. Peszat and J. Zabczyk, Nonlinear stochastic wave and heat equations, Probab. Theory Related Fields, to appear. Zbl0959.60044
- [36] G. Pisier, Martingales with values in uniformly convex spaces, Israel J. Math. 20 (1976), 326-350. Zbl0344.46030
- [37] E. Sinestrari, Accretive differential operators, Boll. Un. Mat. Ital. A 13 (1976), 19-31. Zbl0343.35016
- [38] G. Tessitore and J. Zabczyk, Invariant measures for stochastic heat equations, Probab. Math. Statist. 18 (1999), 271-287. Zbl0986.60057
- [39] G. Tessitore and J. Zabczyk, Strict positivity for stochastic heat equations, Stochastic Process. Appl. 77 (1998), 83-98. Zbl0933.60071
- [40] N. N. Vakhania, V. I. Tarieladze and S. A. Chobanyan, Probability Distributions on Banach Spaces, Reidel, Dordrecht, 1987.
- [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.

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