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## Studia Mathematica

1999 | 137 | 3 | 261-299
Tytuł artykułu

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

Autorzy
Treść / Zawartość
Warianty tytułu
Języki publikacji
EN
Abstrakty
EN
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.
Słowa kluczowe
EN
Kategorie tematyczne
Czasopismo
Rocznik
Tom
Numer
Strony
261-299
Opis fizyczny
Daty
wydano
1999
otrzymano
1998-03-03
poprawiono
1999-01-11
Twórcy
autor
autor
• Institute of Mathematics, Polish Academy of Sciences, Św. Tomasza 30/7, 31-027 Kraków, Poland, napeszat@cyf-kr.edu.pl
Bibliografia
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• [2] P. Baxendale, Gaussian measures on function spaces, Amer. J. Math. 98 (1976), 891-952.
• [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.
• [5] Z. Brzeźniak, On stochastic convolutions in Banach spaces and applications, Stochastics Stochastics Rep. 61 (1997), 245-295.
• [6] Z. Brzeźniak and D. Gątarek, Martingale solutions and invariant measures for stochastic evolution equations in Banach spaces, Stochastic Process. Appl., to appear.
• [7] Z. Brzeźniak and S. Peszat, Stochastic two dimensional Euler equations, Preprint 2, School of Mathematics, University of Hull, Hull, 1999.
• [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.
• [11] G. Da Prato and J. Zabczyk, Stochastic Equations in Infinite Dimensions, Cambridge Univ. Press, Cambridge, 1992.
• [12] G. Da Prato and J. Zabczyk, Ergodicity for Infinite Dimensional Systems, Cambridge Univ. Press, Cambridge, 1996.
• [13] E. B. Davies, One-Parameter Semigroups, Academic Press, London, 1980.
• [14] D. A. Dawson and H. Salehi, Spatially homogeneous random evolutions, J. Multivariate Anal. 10 (1980), 141-180.
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• [16] C. Donati-Martin and E. Pardoux, White noise driven SPDE'S with reflection, Probab. Theory Related Fields 95 (1993), 1-24.
• [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.
• [19] N. Yu. Goncharuk and P. Kotelenez, Fractional step method for stochastic evolution equations, Stochastic Process. Appl. 73 (1998), 1-45.
• [20] D. Henry, Geometric Theory of Semilinear Parabolic Equations, Lecture Notes in Math. 840, Springer, Berlin, 1981.
• [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.
• [23] P. Kotelenez, Comparison methods for a class of function valued stochastic differential equations, Probab. Theory Related Fields 93 (1992), 1-19.
• [24] H. H. Kuo, Gaussian Measures in Banach Spaces, Lecture Notes in Math. 463, Springer, Berlin, 1975.
• [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 $L^2ν_ϱ$, Stochastics Stochastics Rep. 66 (1999), 37-85.
• [27] M. Marcus and G. Pisier, Random Fourier Series, with Applications to Harmonic Analysis, Princeton Univ. Press, Princeton, 1981.
• [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.
• [32] S. Peszat and J. Seidler, Maximal inequalities and space-time regularity of stochastic convolutions, Math. Bohem. 123 (1998), 7-32.
• [33] S. Peszat and J. Zabczyk, Strong Feller property and irreducibility for diffusions on Hilbert spaces, Ann. Probab. 23 (1995), 157-172.
• [34] S. Peszat and J. Zabczyk, Stochastic evolution equations with a spatially homogeneous Wiener process, Stochastic Process. Appl. 72 (1997), 187-204.
• [35] S. Peszat and J. Zabczyk, Nonlinear stochastic wave and heat equations, Probab. Theory Related Fields, to appear.
• [36] G. Pisier, Martingales with values in uniformly convex spaces, Israel J. Math. 20 (1976), 326-350.
• [37] E. Sinestrari, Accretive differential operators, Boll. Un. Mat. Ital. A 13 (1976), 19-31.
• [38] G. Tessitore and J. Zabczyk, Invariant measures for stochastic heat equations, Probab. Math. Statist. 18 (1999), 271-287.
• [39] G. Tessitore and J. Zabczyk, Strict positivity for stochastic heat equations, Stochastic Process. Appl. 77 (1998), 83-98.
• [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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Bibliografia
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