Stochastic homogenization for variational solutions of Hamilton–Jacobi equations

Viterbo, Claude

doi:10.2140/apde.2025.18.805

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Stochastic homogenization for variational solutions of Hamilton–Jacobi equations

Claude Viterbo

Vol. 18 (2025), No. 4, 805–856

DOI: 10.2140/apde.2025.18.805

Abstract

Let $(Ω, μ)$ be a probability space endowed with an ergodic action $τ$ of $(ℝ^{n}, +)$ . Let $H (x, p; ω) = H_{ω} (x, p)$ be a smooth Hamiltonian on $T^{*} ℝ^{n}$ parametrized by $ω \in Ω$ and such that $H (a + x, p; τ_{a} ω) = H (x, p; ω)$ . We consider for an initial condition $f \in C^{0} (ℝ^{n}, ℝ)$ the family of variational solutions of the stochastic Hamilton–Jacobi equations

{\begin{matrix} \frac{\partial u^{𝜀}}{\partial t} (t, x; ω) + H (\frac{x}{𝜀}, \frac{\partial u^{𝜀}}{\partial x} (t, x; ω)) = 0, \\ u^{𝜀} (0, x; ω) = f (x) . \end{matrix}

Under some coercivity assumptions on $p$ — but without any convexity assumption — we prove that for a.e. $ω \in Ω$ we have $C^{0} - \lim u^{𝜀} (t, x; ω) = v (t, x)$ , where $v$ is the variational solution of the homogenized equation

{\begin{matrix} \frac{\partial v}{\partial t} (t, x) + \bar{H} (x, \frac{\partial v}{\partial x} (t, x)) = 0, \\ v (0, x) = f (x) . \end{matrix}

Keywords

stochastic PDE, Hamilton–Jacobi equations, variational solutions

Mathematical Subject Classification

Primary: 35F21, 35R60, 37J39, 57R17

Milestones

Received: 28 July 2021

Revised: 27 December 2023

Accepted: 18 February 2024

Published: 27 March 2025

Authors

Claude Viterbo
Laboratoire Mathématique d’Orsay, UMR 8628 Université de Paris-Saclay, CNRS Orsay France

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Vol. 18, No. 4, 2025