Title: Stochastic homogenization of viscous Hamilton-Jacobi equations with non-convex Hamiltonians in one space dimension

Abstract: Recently constructed counter-examples showed that homogenization of viscous and non-viscous Hamilton-Jacobi equations in stationary ergodic random media can fail in dimensions 2 or higher if the momentum part of the Hamiltonian has a strict saddle point (while the Hamiltonian is uniformly super-linear as |p| goes to infinity). It is expected that in one space dimension the non-convexity of the Hamiltonian should not be an obstacle to homogenization. I shall present a recent result  (joint work with Andrea Davini, Sapienza Università di Roma, and Atilla Yilmaz, Temple University) which shows homogenization for general one-dimensional viscous Hamilton-Jacobi equations with Hamiltonians of the form H(p,x,ω)=G(p)+V(x,ω) under the scaled hill-and-valley condition on random environment and pose several open questions.