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

Scott Armstrong, Panagiotis E. Souganidis

Research output: Contribution to journalArticle

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 languageEnglish (US)
Pages (from-to)460-504
Number of pages45
JournalJournal des Mathematiques Pures et Appliquees
Volume97
Issue number5
DOIs
StatePublished - May 2012

Fingerprint

Stochastic Homogenization
Hamiltonians
Bellman Equation
Degenerate Equations
Hamilton-Jacobi
Homogenization
Hamilton-Jacobi Equation
Brownian movement
Exterior Domain
Large Deviations
Brownian motion
Tend
Zero
Class

Keywords

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

ASJC Scopus subject areas

  • Applied Mathematics
  • Mathematics(all)

Cite this

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

In: Journal des Mathematiques Pures et Appliquees, Vol. 97, No. 5, 05.2012, p. 460-504.

Research output: Contribution to journalArticle

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