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We study the rate of convergence of $u^\epsilon$, as $\epsilon \to 0+$, to$u$ in periodic homogenization of Hamilton-Jacobi equations. Here, $u^\epsilon$and $u$ are viscosity solutions to the oscillatory Hamilton-Jacobi equation andits effective equation \begin{equation*} {\rm (C)_\epsilon} \qquad\begin{cases} u_t^\epsilon+H\left(\frac{x}{\epsilon},Du^\epsilon\right)=0\qquad &\text{in} \ \mathbb{R}^n \times (0,\infty), u^\epsilon(x,0)=g(x) \qquad&\text{on} \ \mathbb{R}^n, \end{cases} \end{equation*} and \begin{equation*}{\rm (C)} \qquad \begin{cases} u_t+\overline{H}\left(Du\right)=0 \qquad&\text{in} \ \mathbb{R}^n \times (0,\infty), u(x,0)=g(x) \qquad &\text{on} \\mathbb{R}^n, \end{cases} \end{equation*} respectively. We assume that theHamiltonian $H=H(y,p)$ is coercive and convex in the $p$ variable and is$\mathbb{Z}^n$-periodic in the $y$ variable, and the initial data $g$ isbounded and Lipschitz continuous.

Rate of convergence in periodic homogenization of Hamilton-Jacobi equations: the convex setting