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In this note we revisit the homogenization theory of Hamilton-Jacobi and “viscous”Hamilton-Jacobi partial differential equations with convex nonlinearities in stationary ergodic environments. We present a new simple proof for the homogenization in probability. The argument uses some a priori bounds (uniform modulus of continuity) on the solution and the convexity and coercivity (growth) of the nonlinearity. It does not rely, however, on the control interpretation formula of the solution as was the case with all previously known proofs. We also introduce a new formula for the effective Hamiltonian for Hamilton-Jacobi and “viscous” Hamilton-Jacobi equations.

Stochastic homogenization of Hamilon-Jacobi and "viscous"-Hamilton-Jacobi equations with convex nonlinearities -- Revisited