Long-time behaviour of heat flow: Global estimates and exact asymptotics

We obtain global upper and lower bounds on the heat kernel of an elliptic second-order differential operator, which become sharp in certain long-time and large-space asymptotics. We prove a generalization of ARONSON'S Gaussian bounds which identifies correctly an effective drift for heat flow. In the case of periodic coefficients we give variational characterizations of the effective conductivity, which is then made to appear in heat kernel bounds. These results are for heat kernels with measurable coefficients. For differentiable coefficients we prove tighter estimates, in which the rate of homogenization is known to be optimal.