Stochastic homogenization of Hamilton-Jacobi and degenerate Bellman equations in unbounded environments

By S. Armstrong, Panagiotis E. Souganidis. 2011

72 citations recorded by the source corpus.

Abstract We consider the homogenization of Hamilton–Jacobi equations and degenerate Bellman equations in stationary, ergodic, unbounded environments. We prove that, as the microscopic scale tends to zero, the equation averages to a deterministic Hamilton–Jacobi equation and study some properties of the effective Hamiltonian. We discover a connection between the effective Hamiltonian and an eikonal-type equation in exterior domains. In particular, we obtain a new formula for the effective Hamiltonian. To prove the results we introduce a new strategy to obtain almost sure homogenization, completing a program proposed by Lions and Souganidis that previously yielded homogenization in probability. The class of problems we study is strongly motivated by Sznitmanʼs study of the quenched large deviations of Brownian motion interacting with a Poissonian potential, but applies to a general class of problems which are not amenable to probabilistic tools.

Subfields: Applied Mathematics, Computational Theory and Mathematics, Modeling and Simulation

View Paper: https://www.semanticscholar.org/paper/f8383e324a1423a9a6f98b7c97d70e3dd7e996eb

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In Applied Mathematics

In Computational Theory and Mathematics

In Modeling and Simulation

Citations in the corpus

Stationary cocycles and Busemann functions for the corner growth model (2015, background): “ ·(x−y)−Bξ(ˆω,x,y). Then for bP-almost every ωˆ gpl(h) = max i=1,2 (5.10) {ω0 +h·ei +Fh(ˆω,0,ei)}= t(h). (ii) ξ(h) = ξ(h) = ξ∈D: (5.10) holds for Fh deﬁned as in (5.9). (iii) ξ(h) = ξ(h) = ξ∈D: Let θ∈[0,1] be such that h−t(h)(e1 +e2) = θh−(ξ) +(1 −θ)h+(ξ) CORNER GROWTH MODEL 17 and deﬁne Fξ±(ˆω,x,y)”
Stochastic Homogenization of Level-Set Convex Hamilton–Jacobi Equations (2012, background): “This is easy, and since the argument is nearly the same as in [2, 3, 9], we omit the proof.”
Stochastic Homogenization of Nonconvex Hamilton‐Jacobi Equations: A Counterexample (2015, background): “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]).”
Viscosity solutions of general viscous Hamilton–Jacobi equations (2013, background): “See also Barles and Da Lio [6] as well as [11, 15, 2, 5], wherein special cases of the results presented in this section can also be found.”