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Stochastic homogenization for variational solutions of Hamilton-Jacobi equations

Claude Viterbo

TL;DR

This work develops a stochastic homogenization framework for variational solutions of Hamilton–Jacobi equations with random Hamiltonians $H(x,p;oldsymbol{ ext ω})$ under an ergodic ${oldsymbol{ ext R}}^n$-action. Using generating functions, spectral invariants, and a $oldsymbol{ ext γ}_c$-topology on non-compact cotangent bundles, the authors establish almost-sure convergence of $u^oldsymbol{ ext ε}$ to a homogenized equation with effective Hamiltonian $oldsymbol{ar H}$, without requiring convexity in $p$. The proof combines non-compact, almost periodic, and coercive regimes through regularization to finite-dimensional tori, ergodic arguments, and a detailed comparison of Hamiltonian flows with their variational solutions. The results unify stochastic and variational approaches, extend viscosity-based homogenization in the convex case, and apply to both non-compact and discrete settings, with extensions to time-dependent and partially homogenized scenarios. Altogether, the paper provides a rigorous pathway from random Hamiltonians to deterministic effective dynamics in a broad, non-convex, variational context.

Abstract

Let $(Ω, μ)$ be a probability space endowed with an ergodic action, $τ$ of $( {\mathbb R} ^n, +)$. Let $H(x,p; ω)=H_ω(x,p)$ be a smooth Hamiltonian on $T^* {\mathbb R} ^n$ parametrized by $ω\in Ω$ and such that $ H(a+x,p;τ_aω)=H(x,p;ω)$. We consider for an initial condition $f\in C^0 ( {\mathbb R}^n)$, the family of variational solutions of the stochastic Hamilton-Jacobi equations $$\left\{ \begin{aligned} \frac{\partial u^{ \varepsilon }}{\partial t}(t,x;ω)+H\left (\frac{x}{ \varepsilon } , \frac{\partial u^\varepsilon }{\partial x}(t,x;ω);ω\right )=0 &\\ u^\varepsilon (0,x;ω)=f(x)& \end{aligned} \right .$$ Under some coercivity assumptions on $p$ -- but without any convexity assumption -- we prove that for a.e. $ω\in Ω$ we have $C^0-\lim u^{\varepsilon}(t,x;ω)=v(t,x)$ where $v$ is the variational solution of the homogenized equation $$\left\{ \begin{aligned} \frac{\partial v}{\partial t}(x)+{\overline H}\left (\frac{\partial v }{\partial x}(x) \right )=0 &\\ v (0,x)=f(x)& \end{aligned} \right.$$

Stochastic homogenization for variational solutions of Hamilton-Jacobi equations

TL;DR

This work develops a stochastic homogenization framework for variational solutions of Hamilton–Jacobi equations with random Hamiltonians under an ergodic -action. Using generating functions, spectral invariants, and a -topology on non-compact cotangent bundles, the authors establish almost-sure convergence of to a homogenized equation with effective Hamiltonian , without requiring convexity in . The proof combines non-compact, almost periodic, and coercive regimes through regularization to finite-dimensional tori, ergodic arguments, and a detailed comparison of Hamiltonian flows with their variational solutions. The results unify stochastic and variational approaches, extend viscosity-based homogenization in the convex case, and apply to both non-compact and discrete settings, with extensions to time-dependent and partially homogenized scenarios. Altogether, the paper provides a rigorous pathway from random Hamiltonians to deterministic effective dynamics in a broad, non-convex, variational context.

Abstract

Let be a probability space endowed with an ergodic action, of . Let be a smooth Hamiltonian on parametrized by and such that . We consider for an initial condition , the family of variational solutions of the stochastic Hamilton-Jacobi equations Under some coercivity assumptions on -- but without any convexity assumption -- we prove that for a.e. we have where is the variational solution of the homogenized equation

Paper Structure

This paper contains 15 sections, 50 theorems, 183 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Notations and abbreviations
  3. Acknowledgments and general remarks.
  4. Non-compact supported Hamiltonians
  5. Spectral invariants in cotangent bundles of non-compact manifolds
  6. The case of Lagrangians
  7. The case of Hamiltonians in $T^*{\mathbb R}^{n}$
  8. Compactness and ergodicity
  9. Some results on compact abelian metric groups
  10. Regularization of the Hamiltonians in $\widehat{\mathfrak {Ham}}_{fc}$
  11. Homogenization in the almost periodic case
  12. Proof of the Main Theorem
  13. The coercive case
  14. The discrete case (Proof of Corollary \ref{['Cor-1.6']})
  15. Extending the Main Theorem

Key Result

Theorem 1.1

Let $H(x,p;\omega)$ be a stochastic Hamiltonian on $T^*{\mathbb R}^n\times \Omega$, where $(\Omega,\mu)$ is a probability space endowed with an action $\tau$ of ${\mathbb R}^n$. We assume the following conditions are satisfied : Then if $\varphi_{\varepsilon ,\omega}$ is the flow of $H_{ \varepsilon ,\omega}(x,p)=H( \frac{x}{ \varepsilon },p)$ there is a function $\overline H$ in $C^0( {\mathbb R

Figures (1)

  • Figure 2: $\widetilde{\Gamma}_m$ in red, $\widetilde{\Gamma}_{m+1}$ in blue and $D_m$ in green on the left and $\Gamma_{m+1}$ on the right

Theorems & Definitions (121)

  • Theorem 1.1: Main theorem
  • Remark 1.2
  • Corollary 1.3: Main corollary
  • Remark 1.5
  • Corollary 1.6
  • Corollary 1.7
  • Definition 4.1
  • Lemma 4.2
  • proof
  • Proposition 4.3
  • ...and 111 more