Stochastic homogenization of Hamilton-Jacobi-Bellman equations. (English) Zbl 1111.60055

Summary: We study the homogenization of some Hamilton-Jacobi-Bellman equations with a vanishing second-order term in a stationary ergodic random medium under the hyperbolic scaling of time and space. Imposing certain convexity, growth, and regularity assumptions on the Hamiltonian, we show the locally uniform convergence of solutions of such equations to the solution of a deterministic “effective” first-order Hamilton-Jacobi equation. The effective Hamiltonian is obtained from the original stochastic Hamiltonian by a minimax formula. Our homogenization results have a large-deviations interpretation for a diffusion in a random environment.


60H35 Computational methods for stochastic equations (aspects of stochastic analysis)
60H15 Stochastic partial differential equations (aspects of stochastic analysis)
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[1] ; ; Asymptotic analysis for periodic structures. Studies in Mathematics and Its Applications, 5. North-Holland, Amsterdam-New York, 1978.
[2] ; Homogenization of multiple integrals. Oxford Lecture Series in Mathematics and Its Applications, 12. The Clarendon Press, New York, 1998. · Zbl 0911.49010
[3] Caffarelli, Comm Pure Appl Math 58 pp 319– (2005)
[4] Capuzzo-Dolcetta, Indiana Univ Math J 50 pp 1113– (2001)
[5] ; An introduction to homogenization. Oxford Lecture Series in Mathematics and Its Applications, 17. The Clarendon Press, New York, 1999. · Zbl 0939.35001
[6] Concordel, Indiana Univ Math J 45 pp 1095– (1996)
[7] An introduction to {\(\Gamma\)}-convergence. Progress in Nonlinear Differential Equations and Their Applications, 8. Birkhäuser-Boston, Boston, 1993.
[8] Evans, Proc Roy Soc Edinburgh Sect A 120 pp 245– (1992) · Zbl 0796.35011
[9] ; Controlled Markov processes and viscosity solutions. Applications of Mathematics (New York), 25. Springer, New York, 1993. · Zbl 0773.60070
[10] ; ; Homogenization of differential operators and integral functionals. Translated from the Russian by G. A. Yosifian. Springer, Berlin, 1994.
[11] ; ; Homogenization of Hamilton-Jacobi equations. Unpublished manuscript, 1986.
[12] Lions, Comm Pure Appl Math 56 pp 1501– (2003)
[13] ; Homogenization of ”viscous” Hamilton-Jacobi equations in stationary ergodic media. Preprint, 2004.
[14] Stochastic integrals. Probability and Mathematical Statistics, 5. Academic Press, New York, 1969.
[15] G-convergence and homogenization of nonlinear partial differential operators. Mathematics and Its Applications, 422. Kluwer, Dordrecht, 1997.
[16] ; Diffusions with random coefficients. North-Holland, Amsterdam, 1982.
[17] Rezakhanlou, Arch Ration Mech Anal 151 pp 277– (2000)
[18] Sion, Pacific J Math 8 pp 171– (1958) · Zbl 0081.11502
[19] Souganidis, Asymptot Anal 20 pp 1– (1999)
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