Regularity results for linear parabolic equations on Carnot tori via mollifier kernel construction

TL;DR

This paper establishes existence, uniqueness, and regularity of solutions to linear parabolic equations on Carnot tori, utilizing mollifier kernels and singular integral operators, with applications to dual Fokker-Planck equations.

Contribution

It introduces mollifier constructions adapted to Carnot groups and tori, advancing regularity theory for parabolic equations on sub-Riemannian manifolds.

Findings

Proved existence and regularity of solutions on Carnot tori.

Constructed mollifiers adapted to H"{o}rmander vector fields.

Applied results to dual Fokker-Planck equations.

Abstract

This paper first proves the existence, uniqueness and regularity of the solution to a class of linear backward parabolic equations on Carnot tori, namely the periodic linear parabolic equation on Carnot groups. Such groups are non-commutative and typical examples of sub-Riemannian manifolds. Moreover, we apply the results for this equation to its dual equation (i.e., the Fokker-Planck-Kolmogorov equation in the general form), and derive the existence, uniqueness and regularity of its weak solution. To obtain the regularity results for solutions to the linear parabolic equation and its dual equation, firstly, we construct several families of mollifiers adapted respectively to the H\"{o}rmander vector fields generating Carnot groups, Carnot tori and dual spaces of non-isotropic H\"{o}lder spaces; secondly, we use the theory of singular integral operators to establish stronger a priori regularity for the solutions.