Caloric measure in parabolic flat domains

Abstract

In this paper we define parabolic chord arc domains and show that in a parabolic chord arc domain with vanishing constant, the logarithm of the density of caloric measure with respect to a certain projective Lebesgue measure is of vanishing mean oscillation. We also obtain a partial converse to this result. Our work generalizes to the heat equation recent work of Kenig and Toro for the Laplacian.