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Stochastic homogenization of levelset convex HamiltonJacobi equations
 INT. MATH. RES. NOT
, 2012
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Error estimates and convergence rates for the stochastic homogenization of HamiltonJacobi equations
 J. AMER. MATH. SOC
, 2013
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Stochastic homogenization of nonconvex HamiltonJacobi equations in one space dimension
, 2014
"... We prove stochastic homogenization for a general class of coercive, nonconvex HamiltonJacobi equations in one space dimension. Some properties of the effective Hamiltonian arising in the nonconvex case are also discussed. ..."
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We prove stochastic homogenization for a general class of coercive, nonconvex HamiltonJacobi equations in one space dimension. Some properties of the effective Hamiltonian arising in the nonconvex case are also discussed.
EXISTENCE AND REGULARITY OF STRICT CRITICAL SUBSOLUTIONS IN THE STATIONARY ERGODIC SETTING
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A METRIC METHOD FOR THE ANALYSIS OF STATIONARY ERGODIC HAMILTONJACOBI EQUATIONS
"... The scope of this contribution is to explain how the socalled metric method, which has revealed to be a powerful tool for the analysis of deterministic HamiltonJacobi equations, see [4], can be used in the stationary ergodic setting. The material is taken from [1], [2], [3], and to these papers we ..."
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The scope of this contribution is to explain how the socalled metric method, which has revealed to be a powerful tool for the analysis of deterministic HamiltonJacobi equations, see [4], can be used in the stationary ergodic setting. The material is taken from [1], [2], [3], and to these papers we refer for a more formal and complete treatment of the subject. Other papers of interest are [6] and [7]. We focus on two basic issues, namely the role of random closed stationary sets and the asymptotic analysis of the intrinsic distances leading to the notion of stable norm. These items are of crucial relevance. In a sense the stationary ergodic structure of the Hamiltonian induces a stochastic geometry in the state variable space $\mathbb{R}^{N} $, where the fiindamental entities are indeed the closed random stationary sets which, somehow, play the same role as the points in the deterministic case, see [5] for a general treatment of of random sets theory. Secondly, the ergodicity can be viewed as an extremely weak form of compactness, mostly thanks to some powerful asymptotic results, like Birkhoff and Kingman subadditive theorem, and especially the latter is a fundamental tool for proving the existence of asymptotic norms. In Section 2 we start by recalling the basic points of the metric method in the deterministic case, then in Section 3 we discuss the notion(s) of critical value. 2. DETERMINISTIC CASE The basic idea of the metric methodology is very simple: we consider an Hamiltonian $H $ : $\mathbb{R}^{N}\cross \mathbb{R}^{N}arrow \mathbb{R} $ and we assume three conditions, which will be kept throughout the paper, on it: $H $ is continuous in both arguments; (1) $H $ is convex in the momentum variable; (2) $\lim_{parrow+\infty}H(x,p)=+\infty $ uniformly in $x $. (3) Then, given an associate HamiltonJacobi equations in $\mathbb{R}^{N} $ of the form