Geodesic
Hypercontractivity of solutions to Hamilton-Jacobi equations
Czechoslovak Mathematical Journal, Tome 51 (2001) no. 4, pp. 733-743 Cet article a éte moissonné depuis la source Czech Digital Mathematics Library

Voir la notice de l'article

We show that solutions to some Hamilton-Jacobi Equations associated to the problem of optimal control of stochastic semilinear equations enjoy the hypercontractivity property.
We show that solutions to some Hamilton-Jacobi Equations associated to the problem of optimal control of stochastic semilinear equations enjoy the hypercontractivity property.
Classification : 49L20, 60H15, 93E20
Keywords: Hamilton-Jacobi equation; stochastic semilinear equation; invariant measure; Log-Sobolev inequality; hypercontractivity 
@article{CMJ_2001_51_4_a5,
     author = {Goldys, Beniamin},
     title = {Hypercontractivity of solutions to {Hamilton-Jacobi} equations},
     journal = {Czechoslovak Mathematical Journal},
     pages = {733--743},
     year = {2001},
     volume = {51},
     number = {4},
     mrnumber = {1864039},
     zbl = {1001.60066},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/CMJ_2001_51_4_a5/}
}
TY  - JOUR
AU  - Goldys, Beniamin
TI  - Hypercontractivity of solutions to Hamilton-Jacobi equations
JO  - Czechoslovak Mathematical Journal
PY  - 2001
SP  - 733
EP  - 743
VL  - 51
IS  - 4
UR  - http://geodesic.mathdoc.fr/item/CMJ_2001_51_4_a5/
LA  - en
ID  - CMJ_2001_51_4_a5
ER  - 
%0 Journal Article
%A Goldys, Beniamin
%T Hypercontractivity of solutions to Hamilton-Jacobi equations
%J Czechoslovak Mathematical Journal
%D 2001
%P 733-743
%V 51
%N 4
%U http://geodesic.mathdoc.fr/item/CMJ_2001_51_4_a5/
%G en
%F CMJ_2001_51_4_a5
Goldys, Beniamin. Hypercontractivity of solutions to Hamilton-Jacobi equations. Czechoslovak Mathematical Journal, Tome 51 (2001) no. 4, pp. 733-743. http://geodesic.mathdoc.fr/item/CMJ_2001_51_4_a5/

[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 | Zbl

[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 | MR

[3] A. Chojnowska-Michalik: Transition semigroups for stochastic semilinear equations on Hilbert spaces. Dissertationes Math. (Rozprawy Mat.) 396 (2001), 1–59. | DOI | MR | Zbl

[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 | MR

[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

[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

[8] G. Da Prato and J. Zabczyk: Ergodicity for Infinite Dimensional Systems. Cambridge University Press, Cambridge, 1996. | MR

[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 | MR

[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 | MR | Zbl

[12] L. Gross: Logarithmic Sobolev inequalities. Amer. J.  Math. 97 (1975), 1061–1083. | DOI | MR | Zbl

[13] R. Phelps: Gaussian null sets and differentiability of Lipschitz maps on Banach Spaces. Pacific J.  Math. 77 (1978), 523–531. | DOI | MR