Stochastic homogenization of a class of quasiconvex viscous Hamilton-Jacobi equations in one space dimension.

We prove homogenization for a class of viscous Hamilton-Jacobi equations in the stationary & ergodic setting in one space dimension. Our assumptions include most notably the following: the Hamiltonian is of the form G (p) + β V (x , ω) , the function G is coercive and strictly quasiconvex, min ⁡ G = 0 , β > 0 , the random potential V takes values in [ 0 , 1 ] with full support and it satisfies a hill condition that involves the diffusion coefficient. Our approach is based on showing that, for every direction outside of a bounded interval (θ 1 (β) , θ 2 (β)) , there is a unique sublinear corrector with certain properties. We obtain a formula for the effective Hamiltonian and deduce that it is coercive, identically equal to β on (θ 1 (β) , θ 2 (β)) , and strictly monotone elsewhere. [ABSTRACT FROM AUTHOR]