Stochastic homogenization for variational solutions of Hamilton-Jacobi equations

Abstract

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

56 pages, 3 figures

References in corpus (1)

• Stochastic homogenization of nonconvex Hamilton-Jacobi equations: a counterexample