Stochastic Solutions to Hamilton-Jacobi Equations

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.