| Title:
|
Hypercontractivity of solutions to Hamilton-Jacobi equations (English) |
| Author:
|
Goldys, Beniamin |
| Language:
|
English |
| Journal:
|
Czechoslovak Mathematical Journal |
| ISSN:
|
0011-4642 (print) |
| ISSN:
|
1572-9141 (online) |
| Volume:
|
51 |
| Issue:
|
4 |
| Year:
|
2001 |
| Pages:
|
733-743 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| 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. (English) |
| Keyword:
|
Hamilton-Jacobi equation |
| Keyword:
|
stochastic semilinear equation |
| Keyword:
|
invariant measure |
| Keyword:
|
Log-Sobolev inequality |
| Keyword:
|
hypercontractivity |
| MSC:
|
49L20 |
| MSC:
|
60H15 |
| MSC:
|
93E20 |
| idZBL:
|
Zbl 1001.60066 |
| idMR:
|
MR1864039 |
| . |
| Date available:
|
2009-09-24T10:46:51Z |
| Last updated:
|
2020-07-03 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/127683 |
| . |
| Reference:
|
[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. Zbl 0856.47026, MR 1307413 |
| Reference:
|
[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. MR 1047576, 10.1016/0022-1236(90)90079-Z |
| Reference:
|
[3] A. Chojnowska-Michalik: Transition semigroups for stochastic semilinear equations on Hilbert spaces.Dissertationes Math. (Rozprawy Mat.) 396 (2001), 1–59. Zbl 0991.60049, MR 1841090, 10.4064/dm396-0-1 |
| Reference:
|
[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. MR 1339737, 10.1007/BF01192465 |
| Reference:
|
[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 |
| Reference:
|
[6] G. Da Prato, A. Debussche and B. Goldys: Invariant measures of non-symmetric dissipative stochastic systems.(to appear). |
| Reference:
|
[7] G. Da Prato and J. Zabczyk: Stochastic Equations in Infinite Dimensions.Cambridge University Press, Cambridge, 1992. MR 1207136 |
| Reference:
|
[8] G. Da Prato and J. Zabczyk: Ergodicity for Infinite Dimensional Systems.Cambridge University Press, Cambridge, 1996. MR 1417491 |
| Reference:
|
[9] B. Goldys and F. Gozzi: Second order parabolic HJ Equations in Hilbert Spaces: $L^2$ Approach.Submitted. |
| Reference:
|
[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. MR 1689410, 10.1006/jmaa.1999.6387 |
| Reference:
|
[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. Zbl 0842.49021, MR 1326907, 10.1080/03605309508821115 |
| Reference:
|
[12] L. Gross: Logarithmic Sobolev inequalities.Amer. J. Math. 97 (1975), 1061–1083. Zbl 0359.46038, MR 0420249, 10.2307/2373688 |
| Reference:
|
[13] R. Phelps: Gaussian null sets and differentiability of Lipschitz maps on Banach Spaces.Pacific J. Math. 77 (1978), 523–531. MR 0510938, 10.2140/pjm.1978.77.523 |
| . |
