Abstract
In this expository paper we give an overview of the statistical properties of Hamilton-Jacobi Equations and Scalar Conservation Laws. The first part (Sects. 2–4) is devoted to the recent proof of Menon-Srinivasan Conjecture. This conjecture provides a Smoluchowski-type kinetic equation for the evolution of a Markovian solution of a scalar conservation law with convex flux. In the second part of the paper (Sects. 5 and 6) we discuss the question of homogenization for Hamilton-Jacobi PDEs and Hamiltonian ODEs with deterministic and stochastic Hamiltonian functions.
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Acknowledgments
These notes are based on the mini course that was given by the author at Institut Henri Poincaré, Centre Emile Borel during the trimester Stochastic Dynamics Out of Equilibrium. The author thanks this institution for hospitality and support. He is also very grateful to the organizers of the program for invitation, and an excellent research environment. Special thanks to two anonymous referees for their careful reading of these notes and their many insightful comments and suggestions.
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Rezakhanlou, F. (2019). Stochastic Solutions to Hamilton-Jacobi Equations. In: Giacomin, G., Olla, S., Saada, E., Spohn, H., Stoltz, G. (eds) Stochastic Dynamics Out of Equilibrium. IHPStochDyn 2017. Springer Proceedings in Mathematics & Statistics, vol 282. Springer, Cham. https://doi.org/10.1007/978-3-030-15096-9_5
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