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2000 | 141 | 2 | 109-142
Tytuł artykułu

Dirichlet problem for parabolic equations on Hilbert spaces

Autorzy
Treść / Zawartość
Warianty tytułu
Języki publikacji
EN
Abstrakty
EN
We study a linear second order parabolic equation in an open subset of a separable Hilbert space, with the Dirichlet boundary condition. We prove that a probabilistic formula, analogous to one obtained in the finite-dimensional case, gives a solution to this equation. We also give a uniqueness result.
Słowa kluczowe
Czasopismo
Rocznik
Tom
141
Numer
2
Strony
109-142
Opis fizyczny
Daty
wydano
2000
otrzymano
1999-08-19
poprawiono
2000-02-02
Twórcy
  • Faculty of Mathematics, Computer Science and Mechanics, University of Warsaw, Banacha 2, 02-097 Warszawa, Poland
Bibliografia
  • [1] R. M. Blumenthal and R. K. Getoor, Markov Processes and Potential Theory, Academic Press, 1968.
  • [2] P. Cannarsa and G. Da Prato, A semigroup approach to Kolmogoroff equations in Hilbert spaces, Appl. Math. Lett. 4 (1991), 49-52.
  • [3] P. Cannarsa and G. Da Prato, On functional analysis approach to parabolic equations in infinite dimensions, J. Funct. Anal. 118 (1993), 22-42.
  • [4] Yu. Daleckij, Differential equations with functional derivatives and stochastic equations for generalized random processes, Dokl. Akad. Nauk SSSR 166 (1966), 1035-1038 (in Russian).
  • [5] G. Da Prato, Parabolic Equations in Hilbert Spaces, Scuola Normale Superiore Pisa, Lecture Notes, May 1996.
  • [6] G. Da Prato, Some results on parabolic evolution equations with infinitely many variables, J. Differential Equations 68 (1987), 281-297.
  • [7] G. Da Prato, Stochastic Evolution Equations by Semigroups Methods, Centre De Recerca Matematica, Barcelona, Quaderns 11 (1998).
  • [8] G. Da Prato, B. Gołdys and J. Zabczyk, Ornstein-Uhlenbeck semigroups in open sets of Hilbert spaces, C. R. Acad. Sci. Paris Sér. I 325 (1997), 433-438.
  • [9] G. Da Prato and J. Zabczyk, Smoothing properties of transition semigroups in Hilbert Spaces, Stochastics Stochastics Rep. 35 (1991), 63-77.
  • [10] G. Da Prato and J. Zabczyk, Stochastic Equations in Infinite Dimensions, Encyclopedia Math. Appl. 44, Cambridge Univ. Press, 1992.
  • [11] E. B. Davies, One-Parameter Semigroups, Academic Press, 1980.
  • [12] E. B. Dynkin, Markov Processes, Springer, 1965.
  • [13] A. Friedman, Stochastic Differential Equations and Applications, Academic Press, New York, 1975.
  • [14] L. Gross, Potential theory on Hilbert space, J. Funct. Anal. 1 (1967), 123-181.
  • [15] H. H. Kuo, Gaussian Measures in Banach Spaces, Lecture Notes in Math. 463, Springer, Berlin, 1975.
  • [16] H. H. Kuo and M. A. Piech, Stochastic integrals and parabolic equations in abstract Wiener space, Bull. Amer. Math. Soc. 79 (1973), 478-482.
  • [17] A. Lunardi, Schauder theorems for linear elliptic and parabolic problems with unbounded coefficients in $ℝ^n$, Studia Math. 128 (1998), 171-198.
  • [18] M. A. Piech, A fundamental solution of the parabolic equation on Hilbert space, J. Funct. Anal. 3 (1969), 85-114.
  • [19] E. Priola, Maximal regularity results for a homogeneous Dirichlet problem in a general half space of a Hilbert space, preprint, Scuola Normale Superiore di Pisa.
  • [20] B. Simon, The $P(ϕ)_2$ Euclidean (Quantum) Field Theory, Princeton Univ. Press, 1974.
  • [21] D. W. Stroock, Probability Theory, an Analytic View, Cambridge Univ. Press, 1993.
  • [22] J. Zabczyk, Infinite dimensional diffusions in modeling and analysis, Jahresber. Deutsch. Math.-Verein. 101 (1999), 47-59.
  • [23] J. Zabczyk, Parabolic Equations on Hilbert Spaces, Lecture Notes in Math. 1715, Springer, 1999.
  • [24] J. Zabczyk, Stopping problems on Polish spaces, Ann. Univ. Mariae Curie- Skłodowska 51 (1997), 181-199
Typ dokumentu
Bibliografia
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bwmeta1.element.bwnjournal-article-smv141i2p109bwm
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