Homogenization of Hamilton-Jacobi-Bellman equations with respect to time-space shifts in a stationary ergodic medium

Elena Kosygina, S. R S Varadhan

Research output: Contribution to journalArticle

Abstract

We consider a family {uε (t, x, ω)}, ε > 0, of solutions to the equation ∂uε/∂t + εΔuε/2 + H(t/ε, x/ε, ∇u ε, ω) = 0 with the terminal data uε(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 uε(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 languageEnglish (US)
Pages (from-to)816-847
Number of pages32
JournalCommunications on Pure and Applied Mathematics
Volume61
Issue number6
DOIs
StatePublished - Jun 1 2008

ASJC Scopus subject areas

  • Mathematics(all)
  • Applied Mathematics

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