## Homogenization and enhancement of the $$G$$-equation in random environments.(English)Zbl 1284.60126

The homogenization limit when the perturbation parameter tends to zero can be thought of as a small parameter perturbation technique. Here one applies this method to a $$G$$-equation submitted to a general stationary ergodic environment. The framework of the study is the so-called viscosity solution. The averaging properties of this $$G$$-equation cannot be studied by using subadditive theorem which is the standard approach to the homogenization of Hamilton-Jacobi equation in random media. The main contribution of the paper is to propose a new approach to circumvent these problems, which reduces to a controllability estimate and the construction of a random sequence which defines a long-time asymptotic limit.

### MSC:

 60H15 Stochastic partial differential equations (aspects of stochastic analysis) 35R60 PDEs with randomness, stochastic partial differential equations 60K37 Processes in random environments
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