Note on quantitative homogenization results for parabolic systems in ${\mathbb{R}}^d$

In $L_2({\mathbb{R}}^d;{\mathbb{C}}^n)$, we consider a semigroup $e^{-t{A}_{\epsilon }}$, $t⩾0$, generated by a matrix elliptic second order differential operator $A_{\epsilon }⩾0$. Coefficients of $A_{\epsilon }$ are periodic, depend on $\mathbf{x}\mathrm{/}\epsilon$ and oscillate rapidly as $\epsilon \to 0$. Approximations for $e^{-t{A}_{\epsilon }}$ were obtained by T. A. Suslina (2004, 2010) via the spectral method and by V. V. Zhikov and S. E. Pastukhova (2006) via the shift method. In the present note, we give another short proof based on the contour integral representation for the semigroup and approximations for the resolvent with two-parametric error estimates obtained by T. A. Suslina (2015).