Equivalence and Hölder-Sobolev regularity of solutions for a class of non-autonomous stochastic partial differential equations
Annales de l'I.H.P. Probabilités et statistiques, Tome 39 (2003) no. 4, pp. 703-742.
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     author = {Sanz-Sol\'e, Marta and Vuillermot, Pierre-A.},
     title = {Equivalence and {H\"older-Sobolev} regularity of solutions for a class of non-autonomous stochastic partial differential equations},
     journal = {Annales de l'I.H.P. Probabilit\'es et statistiques},
     pages = {703--742},
     publisher = {Elsevier},
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     year = {2003},
     doi = {10.1016/S0246-0203(03)00015-3},
     zbl = {1026.60080},
     language = {en},
     url = {http://www.numdam.org/articles/10.1016/S0246-0203(03)00015-3/}
}
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Sanz-Solé, Marta; Vuillermot, Pierre-A. Equivalence and Hölder-Sobolev regularity of solutions for a class of non-autonomous stochastic partial differential equations. Annales de l'I.H.P. Probabilités et statistiques, Tome 39 (2003) no. 4, pp. 703-742. doi : 10.1016/S0246-0203(03)00015-3. http://www.numdam.org/articles/10.1016/S0246-0203(03)00015-3/

[1] H. Amann, Linear and Quasilinear Parabolic Problems, I: Abstract Linear Theory, Monographs in Mathematics, 89, Birkhäuser, Basel, 1995. | MR | Zbl

[2] J.M. Ball, Strongly continuous semigroups, weak solutions, and the variation of constants formula, Proc. Amer. Math. Soc. 63 (1977) 370-373. | MR | Zbl

[3] V. Bally, A. Millet, M. Sanz-Solé, Approximation and support theorem in Hölder norm for parabolic stochastic partial differential equations, Ann. Probab. 23 (1) (1995) 178-222. | MR | Zbl

[4] B. Bergé, I.D. Chueshov, P.-A. Vuillermot, On the behavior of solutions to certain parabolic SPDE's driven by Wiener processes, Stochastic Process. Appl. 92 (2001) 237-263. | MR | Zbl

[5] S.R. Bernfeld, Y.Y. Hu, P.-A. Vuillermot, Large-time asymptotic equivalence for a class of non-autonomous semilinear parabolic equations, Bull. Sci. Math. 122 (5) (1998) 337-368. | MR | Zbl

[6] Z. Brzeźniak, S. Peszat, Space-time continuous solutions to SPDE's driven by a homogeneous Wiener process, Studia Math. 137 (3) (1999) 261-299. | MR | Zbl

[7] C. Cardon-Weber, A. Millet, On strongly Petrovskiĭ's parabolic SPDE's in arbitrary dimension, Preprint 685, Laboratoire de Probabilités et des Modèles Aléatoires, Université Paris 6, 2001. | MR

[8] A. Chojnowska-Michalik, Stochastic differential equations in Hilbert spaces, Banach Center Publications 5 (1979) 53-73. | MR | Zbl

[9] I.D. Chueshov, P.-A. Vuillermot, Long-time behavior of solutions to a class of quasilinear parabolic equations with random coefficients, Ann. Inst. Henri Poincaré AN 15 (2) (1998) 191-232. | Numdam | MR | Zbl

[10] I.D. Chueshov, P.-A. Vuillermot, Long-time behavior of solutions to a class of stochastic parabolic equations with homogeneous white noise: Stratonovitch's case, Probab. Theory Related Fields 112 (1998) 149-202. | MR | Zbl

[11] I.D. Chueshov, P.-A. Vuillermot, Long-time behavior of solutions to a class of stochastic parabolic equations with homogeneous white noise: Itô's case, Stochastic Anal. Appl. 18 (4) (2000) 581-615. | MR | Zbl

[12] R.C. Dalang, Extending martingale measure stochastic integral with applications to spatially homogeneous SPDE's, Electron. J. Probab. 4 (1999) 1-29. | EuDML | MR | Zbl

[13] R.C. Dalang, N.E. Frangos, The stochastic wave equation in two spatial dimensions, Ann. Probab. 26 (1) (1998) 187-212. | MR | Zbl

[14] G. Da Prato, J. Zabczyk, Stochastic Equations in Infinite Dimensions, Encyclopedia of Mathematics and its Applications, 44, Cambridge University Press, Cambridge, 1992. | MR | Zbl

[15] G. Da Prato, S. Kwapien, J. Zabczyk, Regularity of solutions of linear stochastic equations in Hilbert spaces, Stochastics 23 (1987) 1-23. | MR | Zbl

[16] D.A. Dawson, L.G. Gorostiza, Solutions of evolution equations in Hilbert space, J. Differential Equations 68 (1987) 299-319. | MR | Zbl

[17] S.D. Eidelman, S.D. Ivasis̆En, Investigation of the Green matrix for a homogeneous parabolic boundary value problem, Trans. Moscow Math. Soc. 23 (1970) 179-242. | Zbl

[18] S.D. Eidelman, N.V. Zhitarashu, Parabolic Boundary Value Problems, Operator Theory, Advances and Applications, 101, Birkhäuser, Basel, 1998. | MR | Zbl

[19] A. Friedman, Partial Differential Equations of Parabolic Type, Prentice Hall, Englewood Cliffs, NJ, 1964. | MR | Zbl

[20] T. Funaki, Regularity properties for stochastic partial differential equations of parabolic type, Osaka J. Math. 28 (1991) 495-516. | MR | Zbl

[21] I.M. Gelfand, N.Y. Vilenkin, Les Distributions, Collection Universitaire de Mathématiques, 4, Dunod, Paris, 1967. | MR

[22] I. Gyöngy, C. Rovira, On Lp-solutions of semilinear stochastic partial differential equations, Stochastic Process. Appl. 90 (2000) 83-108. | MR | Zbl

[23] P. Hess, Periodic-Parabolic Boundary-Value Problems and Positivity, Pitman Research Notes in Mathematics Series, 247, Langman, Harlow, 1991. | MR | Zbl

[24] I. Karatzas, S.E. Shreve, Brownian Motion and Stochastic Calculus, Graduate Texts in Mathematics, 113, Springer, New York, 1991. | MR | Zbl

[25] A. Karczeswka, J. Zabczyk, Stochastic PDE's with function-valued solutions, in: Infinite-Dimensional Stochastic Analysis, Proceedings of the Colloquium of the Royal Netherlands Academy of Arts and Science, North-Holland, Amsterdam, 1999, pp. 197-216. | Zbl

[26] T. Kato, Abstract evolution equations of parabolic type in Banach and Hilbert spaces, Nagoya Math. J. 19 (1961) 93-125. | MR | Zbl

[27] T. Kato, Perturbation Theory for Linear Operators, Grundlehren der Mathematischen Wissenschaften, 132, Springer, New York, 1984. | Zbl

[28] A.A. Kirillov, A.D. Gvishiani, Theorems and Problems in Functional Analysis, Problem Books in Mathematics, 9, Springer, New York, 1982. | MR | Zbl

[29] N.V. Krylov, An analytic approach to SPDE's, in: Stochastic Partial Differential Equations: Six Perspectives, AMS-Mathematical Surveys and Monographs, 64, American Mathematical Society, Providence, RI, 1999, pp. 185-242. | MR | Zbl

[30] N.V. Krylov, B.L. Rozovskii, Stochastic evolution equations, J. Sov. Math. 16 (1981) 1233-1277. | Zbl

[31] O. Ladyzenskaya, N. Uraltceva, V.A. Solonnikov, Linear and Quasilinear Equations of Parabolic Type, AMS-Transl. of Math. Monographs, 23, American Mathematical Society, Providence, RI, 1968. | Zbl

[32] J.A. León, Stochastic evolution equations with respect to semimartingales in Hilbert space, Stochastics 27 (1989) 1-21. | MR | Zbl

[33] O. Lévêque, Hyperbolic stochastic partial differential equations driven by boundary noises, Thèse 2452, EPFL, Lausanne, 2001.

[34] J.L. Lions, Équations Différentielles Opérationnelles et Problèmes aux Limites, Die Grundlehren der Mathematischen Wissenschaften in Einzeldarstellungen, 111, Springer, New York, 1961. | MR | Zbl

[35] S.V. Lototsky, Dirichlet problem for stochastic parabolic equations in smooth domains, Stochastics Stochastics Rep. 68 (1999) 145-175. | MR | Zbl

[36] S.V. Lototsky, Linear stochastic parabolic equations, degenerating on the boundary of a domain, Electron. J. Probab. 6 (2001) 1-14. | EuDML | MR | Zbl

[37] D. Marquez-Carreras, M. Mellouk, M. Sarrà, On stochastic partial differential equations with spatially correlated noise: smoothness of the law, Stochastic Process. Appl. 93 (2001) 269-284. | MR | Zbl

[38] V. Mikhaïlov, Équations aux Dérivées Partielles, Mir, Moscow, 1980. | MR | Zbl

[39] R. Mikulevicius, On the Cauchy problem for parabolic SPDE's in Hölder classes, Ann. Probab. 28 (1) (2000) 74-103. | MR | Zbl

[40] A. Millet, M. Sanz-Solé, A stochastic wave equation in two space dimension: smoothness of the law, Ann. Probab. 27 (2) (1999) 803-844. | MR | Zbl

[41] A. Millet, M. Sanz-Solé, Approximation and support theorem for a wave equation in two space dimensions, Bernoulli 6 (5) (2000) 887-915. | MR | Zbl

[42] E. Pardoux, Équations aux dérivées partielles stochastiques nonlinéaires monotones: étude de solutions fortes de type Itô, Thèse 1556, Université Paris-Orsay, Paris, 1975.

[43] E. Pardoux, Stochastic partial differential equations, a review, Bull. Sci. Math. 117 (1993) 29-47. | MR | Zbl

[44] S. Peszat, J. Zabczyk, Nonlinear stochastic wave and heat equations, Probab. Theory Related Fields 116 (2000) 421-443. | MR | Zbl

[45] M. Sanz-Solé, P.-A. Vuillermot, Hölder-Sobolev regularity of solutions to a class of SPDE's driven by a spatially colored noise, C. R. Acad. Sci. Paris, Sér. I 334 (2002) 869-874. | MR | Zbl

[46] M. Sanz-Solé, M. Sarrà, Hölder continuity for the stochastic heat equation with spatially correlated noise, in: Progress in Probability, Stochastic Analysis, Random Fields and Applications, Birkhäuser, Basel, 2002, to appear. | MR

[47] V.A. Solonnikov, On boundary-value problems for linear parabolic systems of differential equations of general form, Proc. Steklov Inst. Math. 83 (1965). | MR | Zbl

[48] H. Tanabe, Equations of Evolution, Monographs and Studies in Mathematics, 6, Pitman, London, 1979. | MR | Zbl

[49] P.-A. Vuillermot, Global exponential attractors for a class of nonautonomous reaction-diffusion equations on RN, Proc. Amer. Math. Soc. 116 (3) (1992) 775-782. | MR | Zbl

[50] J.B. Walsh, An introduction to stochastic partial differential equations, in: École d'Été de Probabilités de Saint-Flour XIV, Lecture Notes in Mathematics, 1180, Springer, New York, 1986, pp. 265-439. | MR | Zbl

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