Celebratio Mathematica

M. D. Donsker and S. R. S. Varadhan: “Large deviations for Markov processes and the asymptotic evaluation of certain Markov process expectations for large times,” pp. 82–88 in Probabilistic methods in differential equations (Victoria, BC, August 19–20, 1974). Edited by M. A. Pinsky. Lecture Notes in Mathematics 451. Springer (Berlin), 1975. MR 0410942 incollection

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M. D. Donsker and S. R. S. Varadhan: “On the number of distinct sites visited by a random walk,” pp. 57–62 in Stochastic analysis (Evanston, IL, April 10–14, 1978). Edited by A. Friedman and M. A. Pinsky. Academic Press (New York), 1978. MR 517233 Zbl 0442.60068 incollection

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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

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.

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S. R. Srinivasa Varadhan

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