Homogenization of nonconvex viscous Hamilton–Jacobi equations in stationary ergodic media in one dimension
Elena Kosygina, Atilla Yilmaz · Nonlinearity · 2025
Abstract We establish homogenization for nondegenerate viscous Hamilton–Jacobi equations in one space dimension when the diffusion coefficient a ( x , ω ) > 0 and the Hamiltonian H ( p , x , ω ) are general stationary ergodic processes in x . Our result is valid under mild regularity assumptions on a and H plus standard coercivity and growth assumptions (in p ) on the latter. In particular, we impose neither any additional condition on the law of the media nor any shape restriction on the graph of p ↦ H ( p , x , ω ) . Our approach consists of two main steps: (i) constructing a suitable candidate H ― for the effective Hamiltonian; (ii) proving homogenization. In the first step, we work with the set E of all points at which H ― is naturally determined by correctors with stationary derivatives. We prove that E is a closed subset of R that is unbounded from above and below, and, if E ≠ R , then H ― can be extended continuously to R by setting it to be constant on each connected component of E c . In the second step, we use a key bridging lemma, comparison arguments and several general results to verify that homogenization holds with this H ― as the effective Hamiltonian.