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Stochastic homogenization of Hamilton-Jacobi and "Viscous"-Hamilton-Jacobi equations with convex

Souganidis, Panagiotis E.; Lions, Pierre-Louis (2010), Stochastic homogenization of Hamilton-Jacobi and "Viscous"-Hamilton-Jacobi equations with convex, Communications in Mathematical Sciences, 8, 2, p. 627-637

Type
Article accepté pour publication ou publié
Date
2010
Journal name
Communications in Mathematical Sciences
Volume
8
Number
2
Publisher
International Press
Pages
627-637
Metadata
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Author(s)
Souganidis, Panagiotis E.

Lions, Pierre-Louis
Abstract (EN)
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.
Subjects / Keywords
viscosity solutions; Hamilton-Jacobi equations; Stochastic homogenization