### Abstract

We consider a family {u_{ε} (t, x, ω)}, ε > 0, of solutions to the equation ∂u_{ε}/∂t + εΔu_{ε}/2 + H(t/ε, x/ε, ∇u _{ε}, ω) = 0 with the terminal data u_{ε}(T, x, ω) = U(x). Assuming that the dependence of the Hamiltonian H(t, x, p, ω) on time and space is realized through shifts in a stationary ergodic random medium, and that H is convex in p and satisfies certain growth and regularity conditions, we show the almost sure locally uniform convergence, in time and space, of u_{ε}(t, x, ω) as ε → 0 to the solution u(t, x) of a deterministic averaged equation ∂u/∂t + H̄(∇u) = 0, u(T, x) = U(x). The "effective" Hamiltonian H̄ is given by a variational formula.

Original language | English (US) |
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Pages (from-to) | 816-847 |

Number of pages | 32 |

Journal | Communications on Pure and Applied Mathematics |

Volume | 61 |

Issue number | 6 |

DOIs | |

State | Published - Jun 1 2008 |

### ASJC Scopus subject areas

- Mathematics(all)
- Applied Mathematics

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## Cite this

Kosygina, E., & Varadhan, S. R. S. (2008). Homogenization of Hamilton-Jacobi-Bellman equations with respect to time-space shifts in a stationary ergodic medium.

*Communications on Pure and Applied Mathematics*,*61*(6), 816-847. https://doi.org/10.1002/cpa.20220