Premium
Homogenization of Hamilton‐Jacobi‐Bellman equations with respect to time‐space shifts in a stationary ergodic medium
Author(s) -
Kosygina Elena,
Varadhan S. R. S.
Publication year - 2008
Publication title -
communications on pure and applied mathematics
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 3.12
H-Index - 115
eISSN - 1097-0312
pISSN - 0010-3640
DOI - 10.1002/cpa.20220
Subject(s) - ergodic theory , mathematics , homogenization (climate) , hamiltonian (control theory) , regular polygon , hamilton–jacobi equation , combinatorics , mathematical analysis , mathematical physics , hamiltonian system , pure mathematics , geometry , mathematical optimization , biodiversity , ecology , biology
We consider a family { u ϵ ( t, x , ω)}, ϵ < 0, of solutions to the equation ∂ u ϵ /∂ t + ϵΔ u 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. © 2007 Wiley Periodicals, Inc.