TY - JOUR

T1 - Homogenization of Hamilton-Jacobi-Bellman equations with respect to time-space shifts in a stationary ergodic medium

AU - Kosygina, Elena

AU - Varadhan, S. R S

PY - 2008/6

Y1 - 2008/6

N2 - 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.

AB - 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.

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U2 - 10.1002/cpa.20220

DO - 10.1002/cpa.20220

M3 - Article

AN - SCOPUS:52349108062

SN - 0010-3640

VL - 61

SP - 816

EP - 847

JO - Communications on Pure and Applied Mathematics

JF - Communications on Pure and Applied Mathematics

IS - 6

ER -