A stability result for parabolic measures of operators with singular drifts

We study the operator $${\mathrm{\partial }}_t-\text{div}A\mathrm{\nabla }+B\cdot \mathrm{\nabla }$$ in parabolic upper-half-space, where $A$ is an elliptic matrix satisfying an oscillation condition and $B$ is a singular drift with a Carleson control. Our main result establishes quantitative $A_{\mathrm{\infty }}$-estimates for the parabolic measure in terms of oscillation of $A$ and smallness of $B$. The proof relies on new estimates for parabolic Green functions that quantify their deviations from linear functions of the normal variable and on a novel, quantitative Carleson measure criterion for anisotropic $A_{\mathrm{\infty }}$-weights.