Geodesic
Differentiability of the transition semigroup of the stochastic Burgers equation, and application to the corresponding Hamilton-Jacobi equation
Atti della Accademia nazionale dei Lincei. Rendiconti Lincei. Matematica e applicazioni, Série 9, Tome 9 (1998) no. 4, pp. 267-277.

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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.
Si considera un’equazione di Burgers stocastica. Si prova che il gradiente del semigruppo di transizione corrispondente \( P_{t} \varphi \) esiste per ogni \( \varphi \) limitata e che può essere stimato con un opportuno peso esponenziale. Viene data un’applicazione ad una equazione di Hamilton-Jacobi che interviene in un problema di controllo stocastico. 
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Da Prato, Giuseppe; 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. Rendiconti Lincei. Matematica e applicazioni, Série 9, Tome 9 (1998) no. 4, pp. 267-277. http://geodesic.mathdoc.fr/item/RLIN_1998_9_9_4_a3/

[1] J. M. Bismut, Large deviations and the Malliavin Calculus. Birkhäuser, 1984. | MR | Zbl

[2] P. Cannarsa - 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 | Zbl

[3] S. Cerrai, Optimal control problems for reaction-diffusion equations by a Dynamic Programming approach. Scuola Normale Superiore, Pisa 1998, preprint.

[4] G. Da Prato - A. Debussche, Control of the stochastic Burgers model of turbulence. Siam Journal on Control and Optimization, to appear. | DOI | MR | Zbl

[5] G. Da Prato - A. Debussche - R. Temam, Stochastic Burgers equation. NoDEA, 1994, 389-402. | DOI | MR | Zbl

[6] G. Da Prato - J. Zabczyk, Ergodicity for Infinite Dimensional Systems. London Mathematical Society Lecture Notes, 229, 1996. | DOI | MR | Zbl

[7] K. D. Elworthy, 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. | MR | Zbl

[8] W. H. Fleming - R. W. Rishel, Deterministic and Stochastic Optimal Control. Springer-Verlag, 1975. | MR | Zbl

[9] F. Gozzi, 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. | DOI | MR | Zbl