Sharp global and almost everywhere convergence rates for periodic homogenization of viscous quadratic Hamilton-Jacobi equations

Abstract

We study the periodic homogenization of the viscous Hamilton--Jacobi equation \[ u_t^\varepsilon + \frac{1}{2}|Du^\varepsilon|^2 + V\!\left(\frac{x}{\varepsilon}\right) = \frac{\varepsilon}{2}\Delta u^\varepsilon \qquad \text{in } \mathbb{R}^n \times (0,\infty), \] with initial datum $g \in W^{1,\infty}(\mathbb{R}^n)$, where $V$ is Lipschitz continuous and $\mathbb{Z}^n$-periodic. We prove the sharp global estimate \[ |u^\varepsilon(x,t)-u(x,t)| \leq \varepsilon\!\left(C+\frac{n}{2}\log\!\left(\frac{\max\{t,\varepsilon\}}{\varepsilon}\right)\right) \qquad \text{for all } (x,t)\in \mathbb{R}^n \times [0,\infty), \] where $\varepsilon \in (0,1]$, $u$ solves the limiting (homogenized) equation and $C>0$ is a constant depending only on $\|Dg\|_{L^\infty(\mathbb{R}^n)}$, $\|DV\|_{L^\infty(\mathbb{R}^n)}$, and $n$. We further show that if $g$ is locally semiconcave, then \[|u^\varepsilon(x,t)-u(x,t)| \leq C_{x,t}\varepsilon \qquad \text{for a.e. } (x,t)\in \mathbb{R}^n \times (0,\infty),\] where $C_{x,t}$ depends on $(x,t)$, $\|Dg\|_{L^\infty(\mathbb{R}^n)}$, and $\|DV\|_{L^\infty(\mathbb{R}^n)}$. More precisely, the above improved rate holds at every point $(x,t)$ where $u(\cdot,t)$ is twice differentiable at $x$. In particular, this occurs for a.e. $x\in \mathbb{R}^n$, since $u(\cdot,t)$ is locally semiconcave. We conclude by raising the open problem of whether the same $O(\varepsilon |\log \varepsilon|)$ rate remains valid for general strictly convex Hamiltonians or general periodic diffusions.