### Abstract

We consider the homogenization of Hamilton-Jacobi equations and degenerate Bellman equations in stationary, ergodic, unbounded environments. We prove that, as the microscopic scale tends to zero, the equation averages to a deterministic Hamilton-Jacobi equation and study some properties of the effective Hamiltonian. We discover a connection between the effective Hamiltonian and an eikonal-type equation in exterior domains. In particular, we obtain a new formula for the effective Hamiltonian. To prove the results we introduce a new strategy to obtain almost sure homogenization, completing a program proposed by Lions and Souganidis that previously yielded homogenization in probability. The class of problems we study is strongly motivated by Sznitman's study of the quenched large deviations of Brownian motion interacting with a Poissonian potential, but applies to a general class of problems which are not amenable to probabilistic tools.

Original language | English (US) |
---|---|

Pages (from-to) | 460-504 |

Number of pages | 45 |

Journal | Journal des Mathematiques Pures et Appliquees |

Volume | 97 |

Issue number | 5 |

DOIs | |

State | Published - May 2012 |

### Fingerprint

### Keywords

- Poissonian potential
- Stochastic homogenization
- Viscous Hamilton-Jacobi equation

### ASJC Scopus subject areas

- Applied Mathematics
- Mathematics(all)

### Cite this

*Journal des Mathematiques Pures et Appliquees*,

*97*(5), 460-504. https://doi.org/10.1016/j.matpur.2011.09.009

**Stochastic homogenization of Hamilton-Jacobi and degenerate Bellman equations in unbounded environments.** / Armstrong, Scott; Souganidis, Panagiotis E.

Research output: Contribution to journal › Article

*Journal des Mathematiques Pures et Appliquees*, vol. 97, no. 5, pp. 460-504. https://doi.org/10.1016/j.matpur.2011.09.009

}

TY - JOUR

T1 - Stochastic homogenization of Hamilton-Jacobi and degenerate Bellman equations in unbounded environments

AU - Armstrong, Scott

AU - Souganidis, Panagiotis E.

PY - 2012/5

Y1 - 2012/5

N2 - We consider the homogenization of Hamilton-Jacobi equations and degenerate Bellman equations in stationary, ergodic, unbounded environments. We prove that, as the microscopic scale tends to zero, the equation averages to a deterministic Hamilton-Jacobi equation and study some properties of the effective Hamiltonian. We discover a connection between the effective Hamiltonian and an eikonal-type equation in exterior domains. In particular, we obtain a new formula for the effective Hamiltonian. To prove the results we introduce a new strategy to obtain almost sure homogenization, completing a program proposed by Lions and Souganidis that previously yielded homogenization in probability. The class of problems we study is strongly motivated by Sznitman's study of the quenched large deviations of Brownian motion interacting with a Poissonian potential, but applies to a general class of problems which are not amenable to probabilistic tools.

AB - We consider the homogenization of Hamilton-Jacobi equations and degenerate Bellman equations in stationary, ergodic, unbounded environments. We prove that, as the microscopic scale tends to zero, the equation averages to a deterministic Hamilton-Jacobi equation and study some properties of the effective Hamiltonian. We discover a connection between the effective Hamiltonian and an eikonal-type equation in exterior domains. In particular, we obtain a new formula for the effective Hamiltonian. To prove the results we introduce a new strategy to obtain almost sure homogenization, completing a program proposed by Lions and Souganidis that previously yielded homogenization in probability. The class of problems we study is strongly motivated by Sznitman's study of the quenched large deviations of Brownian motion interacting with a Poissonian potential, but applies to a general class of problems which are not amenable to probabilistic tools.

KW - Poissonian potential

KW - Stochastic homogenization

KW - Viscous Hamilton-Jacobi equation

UR - http://www.scopus.com/inward/record.url?scp=84859722346&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84859722346&partnerID=8YFLogxK

U2 - 10.1016/j.matpur.2011.09.009

DO - 10.1016/j.matpur.2011.09.009

M3 - Article

VL - 97

SP - 460

EP - 504

JO - Journal des Mathematiques Pures et Appliquees

JF - Journal des Mathematiques Pures et Appliquees

SN - 0021-7824

IS - 5

ER -