Article
Full entry |
PDF
(0.3 MB)
Feedback
PDF
(0.3 MB)
Feedback
Keywords:
Hamilton-Jacobi equation; stochastic semilinear equation; invariant measure; Log-Sobolev inequality; hypercontractivity
Hamilton-Jacobi equation; stochastic semilinear equation; invariant measure; Log-Sobolev inequality; hypercontractivity
Summary:
We show that solutions to some Hamilton-Jacobi Equations associated to the problem of optimal control of stochastic semilinear equations enjoy the hypercontractivity property.
References:
[1] D. Bakry: L’hypercontractivité et son utilisation en théorie des semigroupes. Lectures on Probability Theory (Saint-Flour, 1992). Lecture Notes in Math. Vol. 1581, Springer, Berlin, 1994, pp. 1–114. MR 1307413 | Zbl 0856.47026
[2] P. Cannarsa and G. Da Prato: Some results on nonlinear optimal control problems and Hamilton-Jacobi equations in infinite dimensions. J. Funct. Anal. 90 (1990), 27–47. DOI 10.1016/0022-1236(90)90079-Z | MR 1047576
[3] A. Chojnowska-Michalik: Transition semigroups for stochastic semilinear equations on Hilbert spaces. Dissertationes Math. (Rozprawy Mat.) 396 (2001), 1–59. DOI 10.4064/dm396-0-1 | MR 1841090 | Zbl 0991.60049
[4] A. Chojnowska-Michalik and B. Goldys: Existence, uniqueness and invariant measures for stochastic semilinear equations on Hilbert spaces. Probab. Theory Related Fields 102 (1995), 331–356. DOI 10.1007/BF01192465 | MR 1339737
[5] A. Chojnowska-Michalik and B. Goldys: Nonsymmetric Ornstein-Uhlenbeck generators. Infinite Dimensional Stochastic Analysis (Amsterdam, 1999), R. Neth. Acad. Arts Sci., Amsterdam, 2000, pp. 99–116. MR 1831413
[6] G. Da Prato, A. Debussche and B. Goldys: Invariant measures of non-symmetric dissipative stochastic systems. (to appear).
[7] G. Da Prato and J. Zabczyk: Stochastic Equations in Infinite Dimensions. Cambridge University Press, Cambridge, 1992. MR 1207136
[8] G. Da Prato and J. Zabczyk: Ergodicity for Infinite Dimensional Systems. Cambridge University Press, Cambridge, 1996. MR 1417491
[9] B. Goldys and F. Gozzi: Second order parabolic HJ Equations in Hilbert Spaces: $L^2$ Approach. Submitted.
[10] B. Goldys and B. Maslowski: Ergodic control of semilinear stochastic equations and the Hamilton-Jacobi equation. J. Math. Anal. Appl. 234 (1999), 592–631. DOI 10.1006/jmaa.1999.6387 | MR 1689410
[11] F. Gozzi: Regularity of solutions of a second order Hamilton-Jacobi equation and application to a control problem. Comm. Partial Differential Equations 20 (1995), 775–826. DOI 10.1080/03605309508821115 | MR 1326907 | Zbl 0842.49021
[12] L. Gross: Logarithmic Sobolev inequalities. Amer. J. Math. 97 (1975), 1061–1083. DOI 10.2307/2373688 | MR 0420249 | Zbl 0359.46038
[13] R. Phelps: Gaussian null sets and differentiability of Lipschitz maps on Banach Spaces. Pacific J. Math. 77 (1978), 523–531. DOI 10.2140/pjm.1978.77.523 | MR 0510938
Search
Partner of
