Gaussian bounds and asymptotic expansions of Green function in parabolic homogenization (Q6098491)

This paper investigates the asymptotic behavior of the Green function for a family of second-order parabolic operators \(\partial_t+\mathcal{L}_\varepsilon\) with rapidly oscillating and time-dependent periodic coefficients, where \(\mathcal{L}_\varepsilon=-\operatorname{div}(A(x/\varepsilon, t/\varepsilon^2)\nabla)\), and the periodic coefficient tensor \(A(y, s)=(a_{ij}^{\alpha\beta}(y, s))\) is uniformly bounded and elliptic. The asymptotic behavior, as \(\varepsilon\rightarrow 0\), of \(G_\varepsilon(x, t;y, s)\) and \(\nabla_x G_\varepsilon(x, t; y, s)\) as well as \(\nabla_x \nabla_y G_\varepsilon(x, t; y, s)\) are obtained under some smoothness assumptions on \(\Omega\) and the parabolic Hölder continuity condition on \(A\).