A chain rule for a class of evolutive nonlocal hypoelliptic equations

TL;DR

This paper establishes a new chain rule for a class of fractional hypoelliptic equations, introducing a semigroup-based nonlocal arre9 du champ, extending previous results and applicable even when the generator lacks a gradient.

Contribution

It introduces a novel semigroup-based nonlocal arre9 du champ and extends the 2004 chain rule to broader hypoelliptic equations.

Findings

Developed a local-type chain rule for fractional hypoelliptic equations.

Introduced a semigroup-based nonlocal arre9 du champ applicable without a gradient.

Extended and sharpened the original 2004 chain rule by Cf3rdoba and Cf3rdoba.

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{carr\'e 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. C\'ordoba and D. C\'ordoba.