Stochastic homogenization of Hamilton-Jacobi-Bellman equations

Abstract

We study the homogenization of some Hamilton-Jacobi-Bellman equations with a vanishing second-order term in a stationary ergodic random medium under the hyperbolic scaling of time and space. Imposing certain convexity, growth, and regularity assumptions on the Hamiltonian, we show the locally uniform convergence of solutions of such equations to the solution of a deterministic "effective" first-order Hamilton-Jacobi equation. The effective Hamiltonian is obtained from the original stochastic Hamiltonian by a minimax formula. Our homogenization results have a large-deviations interpretation for a diffusion in a random environment.

Stochastic Homogenization of Nonconvex Hamilton‐Jacobi Equations: A Counterexample

“…They have proved independently that when H is convex with respect to p, then u ǫ converges P-almost surely to the unique solution of a system of the form ∂ t u(x, t) +H(Du(x, t)) = 0 in R n × (0, +∞) u(x, 0) = 0 in R n whereH is the effective Hamiltonian. This result has been extended to various frameworks, still under the assumption that the Hamiltonian is convex in p (see [19,15,18,17,22,4,6]). Quantitative results about the speed of convergence have been obtained in [3,20,2].…”

Variational Formula for the Time Constant of First‐Passage Percolation

“…This problem is considered in Kosygina et al [23], Caffarelli et al [8], Lions and Souganidis [27], and Lions and Souganidis [28]. Caffarelli et al [8] and Kosygina et al [23] require uniform ellipticity of the matrix A; i.e., they assume ∃ λ 1 , λ 2 > 0 such that for all x and ω,…”

Stochastic homogenization of viscous superquadratic Hamilton–Jacobi equations in dynamic random environment

“…Note that if ψ ∈ S is also differentiable with respect to (x, t), then the stationarity of increments is equivalent to ψ t and Dψ being stationary, and the sublinearity is equivalent to E[ψ t ] = 0 and E[Dψ] = 0. Another formula for effective Hamiltonian was introduced in [18] for time homogeneous random environment, and then generalized in [19] to space-time random environment, both under the assumption that the diffusion term is given by the identity matrix. We recall how this formula was obtained, and write it in the form that it should take when the diffusion matrix is more general.…”

“…We note that for any v ∈ R n , the pair (b, Φ), where b j = v j + D i A ij and Φ ≡ 1, satisfies the equation above and, hence, E is non-empty. Following [18,19], the effective Hamiltonian, for each p ∈ R n , should be given by…”