A chain rule for a class of evolutive nonlocal hypoelliptic equations

Abstract

We prove a chain rule of local type for a class of fractional hypoelliptic equations of Kolmogorov-Fokker-Planck type. We introduce a semigroup based notion of nonlocal \emph{carré du champ} which works successfully in situations in which the infinitesimal generator of the semigroup itself does not necessarily possess a gradient. Our results extend and sharpen the original 2004 chain rule due to A. Córdoba and D. Córdoba.