Dirichlet problem for parabolic equations on Hilbert spaces

• Autorzy: Anna Talarczyk
• 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.

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Studia Mathematica
• Rocznik: 2000
• Tom: 141
• Numer: 2
• Strony: 109-142

Daty

wydano

2000

otrzymano

1999-08-19

poprawiono

2000-02-02

Twórcy

autor

Anna Talarczyk

• 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