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

We study the periodic homogenization of the viscous Hamilton–Jacobi equation $$u_t^{\epsilon }+\frac{1}{2}\mathrm{\mid }D{u}^{\epsilon }{\mathrm{\mid }}^2+V\text{\hspace{0.17em}⁣}\left(\frac{x}{\epsilon }\right)=\frac{\epsilon }{2}\mathrm{\Delta }{u}^{\epsilon }\text{in }{\mathbb{R}}^n\times (0,\mathrm{\infty }),$$ with initial datum $g\in W^{1,\mathrm{\infty }}({\mathbb{R}}^n)$, where $V$ is Lipschitz continuous and ${\mathbb{Z}}^n$-periodic. We prove the sharp global estimate $$\mathrm{\mid }{u}^{\epsilon }(x,t)-u(x,t)\mathrm{\mid }\le \epsilon \text{\hspace{0.17em}⁣}(C+\frac{n}{2}\log \text{\hspace{0.17em}⁣}\left(\frac{\max \{t,\epsilon \}}{\epsilon }\right))\text{for all }(x,t)\in {\mathbb{R}}^n\times [0,\mathrm{\infty }),$$ where $\epsilon \in (0,1]$, $u$ solves the limiting (homogenized) equation and $C>0$ is a constant depending only on $\mathrm{\parallel }Dg{\mathrm{\parallel }}_{L^{\mathrm{\infty }}({\mathbb{R}}^n)}$, $\mathrm{\parallel }DV{\mathrm{\parallel }}_{L^{\mathrm{\infty }}({\mathbb{R}}^n)}$, and $n$. We further show that if $g$ is locally semiconcave, then $$\mathrm{\mid }{u}^{\epsilon }(x,t)-u(x,t)\mathrm{\mid }\le C_{x,t}\epsilon \text{for a.e. }(x,t)\in {\mathbb{R}}^n\times (0,\mathrm{\infty }),$$ where $C_{x,t}$ depends on $(x,t)$, $\mathrm{\parallel }Dg{\mathrm{\parallel }}_{L^{\mathrm{\infty }}({\mathbb{R}}^n)}$, and $\mathrm{\parallel }DV{\mathrm{\parallel }}_{L^{\mathrm{\infty }}({\mathbb{R}}^n)}$. 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 $O(\epsilon \mathrm{\mid }\log \epsilon \mathrm{\mid })$ rate remains valid for general strictly convex Hamiltonians or general periodic diffusions.