Differentiability of the transition semigroup of the stochastic Burgers equation, and application to the corresponding Hamilton-Jacobi equation

Giuseppe Da Prato; Arnaud Debussche

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni (1998)

  • Volume: 9, Issue: 4, page 267-277
  • ISSN: 1120-6330

Abstract

top
We consider a stochastic Burgers equation. We show that the gradient of the corresponding transition semigroup P t ⁢ φ does exist for any bounded φ ; and can be estimated by a suitable exponential weight. An application to some Hamilton-Jacobi equation arising in Stochastic Control is given.

How to cite

top

Da Prato, Giuseppe, and Debussche, Arnaud. "Differentiability of the transition semigroup of the stochastic Burgers equation, and application to the corresponding Hamilton-Jacobi equation." Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni 9.4 (1998): 267-277. <http://eudml.org/doc/252308>.

@article{DaPrato1998,
abstract = {We consider a stochastic Burgers equation. We show that the gradient of the corresponding transition semigroup \( P\_\{t\} \varphi \) does exist for any bounded \( \varphi \); and can be estimated by a suitable exponential weight. An application to some Hamilton-Jacobi equation arising in Stochastic Control is given.},
author = {Da Prato, Giuseppe, Debussche, Arnaud},
journal = {Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni},
keywords = {Stochastic control problem; Burgers equation; Hamilton-Jacobi equation; transition semigroup; Burger's equation; fixed point theorem; optimal control},
language = {eng},
month = {12},
number = {4},
pages = {267-277},
publisher = {Accademia Nazionale dei Lincei},
title = {Differentiability of the transition semigroup of the stochastic Burgers equation, and application to the corresponding Hamilton-Jacobi equation},
url = {http://eudml.org/doc/252308},
volume = {9},
year = {1998},
}

TY - JOUR
AU - Da Prato, Giuseppe
AU - Debussche, Arnaud
TI - Differentiability of the transition semigroup of the stochastic Burgers equation, and application to the corresponding Hamilton-Jacobi equation
JO - Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni
DA - 1998/12//
PB - Accademia Nazionale dei Lincei
VL - 9
IS - 4
SP - 267
EP - 277
AB - We consider a stochastic Burgers equation. We show that the gradient of the corresponding transition semigroup \( P_{t} \varphi \) does exist for any bounded \( \varphi \); and can be estimated by a suitable exponential weight. An application to some Hamilton-Jacobi equation arising in Stochastic Control is given.
LA - eng
KW - Stochastic control problem; Burgers equation; Hamilton-Jacobi equation; transition semigroup; Burger's equation; fixed point theorem; optimal control
UR - http://eudml.org/doc/252308
ER -

References

top
  1. Bismut, J. M., Large deviations and the Malliavin Calculus. Birkhäuser, 1984. Zbl0537.35003MR755001
  2. Cannarsa, P. - Da Prato, G., Some results on nonlinear optimal control problems and Hamilton-Jacobi equations in infinite dimensions. J. Funct. Anal., 90, 1990, 27-47. Zbl0717.49022MR1047576DOI10.1016/0022-1236(90)90079-Z
  3. Cerrai, S., Optimal control problems for reaction-diffusion equations by a Dynamic Programming approach. Scuola Normale Superiore, Pisa1998, preprint. 
  4. Da Prato, G. - Debussche, A., Control of the stochastic Burgers model of turbulence. Siam Journal on Control and Optimization, to appear. Zbl1111.49302MR1691934DOI10.1137/S0363012996311307
  5. Da Prato, G. - Debussche, A. - Temam, R., Stochastic Burgers equation. NoDEA, 1994, 389-402. Zbl0824.35112MR1300149DOI10.1007/BF01194987
  6. Da Prato, G. - Zabczyk, J., Ergodicity for Infinite Dimensional Systems. London Mathematical Society Lecture Notes, 229, 1996. Zbl0849.60052MR1417491DOI10.1017/CBO9780511662829
  7. Elworthy, K. D., Stochastic flows on Riemannian manifolds. In: M. A. Pinsky - V. Vihstutz (eds.), Diffusion processes and related problems in analysis. Birkhäuser, 1992, vol. II, 33-72. Zbl0758.58035MR1187985
  8. Fleming, W. H. - Rishel, R. W., Deterministic and Stochastic Optimal Control. Springer-Verlag, 1975. Zbl0323.49001MR454768
  9. Gozzi, F., Regularity of solutions of a second order Hamilton-Jacobi equation and application to a control problem. Commun. in partial differential equations, 20, 1995, 775-826. Zbl0842.49021MR1326907DOI10.1080/03605309508821115

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.