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S. R. S. Varadhan :
“Homogenization of random Hamilton–Jacobi–Bellman equations ,”
pp. 397–403
in
Probability, geometry and integrable systems .
Edited by M. Pinsky and B. Birnir .
Mathematical Sciencies Research Institute Publications 55 .
Cambridge University Press ,
2008 .
MR
2407606
Zbl
1160.35334
incollection

Abstract
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We consider nonlinear parabolic equations of Hamilton–Jacobi–Bellman type. The Lagrangian is assumed to be convex, but with a spatial dependence which is stationary and random. Rescaling in space and time produces a similar equation with a rapidly varying spatial dependence and a small viscosity term. Motivated by corresponding results for the linear elliptic equation with small viscosity, we seek to find the limiting behaviour of the solution of the Cauchy (final value) problem in terms of a homogenized problem, described by a convex function of the gradient of the solution. The main idea is to use the principle of dynamic programming to write a variational formula for the solution in terms of solutions of linear problems. We then show that asymptotically it is enough to restrict the optimization to a subclass, one for which the asymptotic behavior can be fully analyzed. The paper outlines these steps and refers to the recently published work of Kosygina, Rezakhanlou and the author for full details.

@incollection {key2407606m,
AUTHOR = {Varadhan, S. R. Srinivasa},
TITLE = {Homogenization of random {H}amilton--{J}acobi--{B}ellman
equations},
BOOKTITLE = {Probability, geometry and integrable
systems},
EDITOR = {Pinsky, Mark and Birnir, Bj\"orn},
SERIES = {Mathematical Sciencies Research Institute
Publications},
NUMBER = {55},
PUBLISHER = {Cambridge University Press},
YEAR = {2008},
PAGES = {397--403},
NOTE = {MR:2407606. Zbl:1160.35334.},
ISBN = {9780521895279},
}
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