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

In this note we revisit the homogenization theory of Hamilton-Jacobi and "viscous"- Hamilton-Jacobi partial differential equations with convex nonlinearities in stationary ergodic envi- ronments. 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 coer- civity (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.

Keywords

Stochastic homogenization, Hamilton-Jacobi equations, viscosity solutions

2010 Mathematics Subject Classification

35B27, 35D40

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Published 1 January 2010