  Questions concerning quantitative and asymptotic properties of the elliptic
measure corresponding to a uniformly elliptic divergence form operator have
been the focus of recent studies. In this setting we show that the elliptic
measure of an operator with coefficients satisfying a vanishing Carleson
condition in the upper half space is an asymptotically optimal $A_\infty$
weight. In particular, for such operators the logarithm of the elliptic kernel
is in the space of (locally) vanishing mean oscillation.
  To achieve this, we prove local, quantitative estimates on a quantity
(introduced by Fefferman, Kenig and Pipher) that controls the $A_\infty$
constant. Our work uses recent results obtained by David, Li and Mayboroda.
These quantitative estimates may offer a new framework to approach similar
problems.
