Communications in Mathematical Sciences

Volume 8 (2010)

Number 2

Special Issue on the Occasion of Andrew Majda’s Sixtieth Birthday: Part II

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

Pages: 627 – 637



Pierre-Louis Lions

Panagiotis E. Souganidis


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.


Stochastic homogenization; Hamilton-Jacobi equations; viscosity solutions

2010 Mathematics Subject Classification

35B27, 35D40

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