Stochastic Homogenization of the Hamilton-Jacobi Equation on Manifolds

TL;DR

This work proves a stochastic homogenization result for the Hamilton–Jacobi equation on a Riemannian manifold under a stationary ergodic abelian-group action. By scaling the metric and Hamiltonian, it shows that viscosity solutions converge a.s. and locally uniformly to a deterministic limit on the asymptotic cone $oldsymbol R^b$, governed by an effective Hamiltonian $ar{H}$ that is the convex conjugate of an effective Lagrangian $ar{L}$. The analysis leverages action-minimizing methods inspired by Aubry–Mather theory, Kingman’s subadditive ergodic theorem, and a notion of convergence from the rescaled manifold to the cone. The construction extends the Mather β-function to the stochastic setting and yields a stationary-ergodic stable-norm framework for families of Riemannian metrics, linking large-scale geometry to homogenization. The results unify periodic, quasi-periodic, and random environments under a geometric-analytic homogenization paradigm on manifolds.

Abstract

This article establishes a stochastic homogenization result for the first order Hamilton-Jacobi equation on a Riemannian manifold $M$, in the context of a stationary ergodic random environment. The setting involves a finitely generated abelian group $ \mathtt{G}$ of rank $b$ acting on $M$ by isometries in a free, totally discontinuous, and co-compact manner, and a family of Hamiltonians $H: T^*M \times Ω\to \mathbb{R}$, parametrized over a probability space $(Ω, \mathbb{P})$, which are stationary with respect to a $\mathbb{P}$-ergodic action of $\mathtt{G}$ on $Ω$. Under standard assumptions, including strict convexity and coercivity in the momentum variable, we prove that as the scaling parameter $\varepsilon$ goes to $0$, the viscosity solutions to the rescaled equation converge almost surely and locally uniformly to the solution to a deterministic homogenized Hamilton-Jacobi equation posed on $\mathbb{R}^b$, which corresponds to the asymptotic cone of $\mathtt{G}$. In particular, this approach sheds light on the relation between the limit problem, the limit space, and the complexity of the acting group. The classical periodic case corresponds to a randomness set $Ω$ that reduces to a singleton; other interesting examples of this setting are also described. We remark that the effective Hamiltonian $\overline{H}$ is obtained as the convex conjugate of an effective Lagrangian $\overline{L}$, which generalizes Mather's $β$-function to the stochastic setting; this represents a first step towards the development of a stationary-ergodic version of Aubry-Mather theory. As a geometric application, we introduce a notion of stable-like norm for stationary ergodic families of Riemannian metrics on $M$, which generalizes the classical Federer-Gromov's stable norm for closed manifolds.

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