Deficit functions and the log Sobolev inequality

Abstract

There is a long history of parabolic monotonicity formulas that developed independently from several different fields and a much more recent elliptic theory. The elliptic theory can be localized and there are additional monotone quantities. There is also a surprising link: Taking a high-dimensional limit of the right elliptic monotonicity can give a parabolic one as a limit. Poincar\'e was the first to observe such a connection. We introduce two deficit functions, one elliptic and one parabolic, then show that the parabolic deficit is pointwise the limit of the elliptic and, that the elliptic satisfies an equation that converges to the equation for the parabolic. These pointwise quantities and their equations recover the monotonicities and leads to an elliptic proof of the log Sobolev inequality as well as new concentration of measure phenomena.