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

Yiming Jiang, Yawei Wei, Yiyun Yang

TL;DR

The paper addresses linear backward parabolic equations on Carnot tori and their dual forward FPK equations in a sub-Riemannian setting. It develops mollifier constructions adapted to Hörmander vector fields, Carnot tori, and dual Hölder spaces, and leverages singular integral theory to obtain Schauder estimates and Hölder regularity. The main contributions are well-posedness and regularity results for the backward equation, corresponding regularity for the dual FPK equation, and a generalized framework for analyzing mean field type master equations on Carnot tori. The methods provide a robust toolkit for degenerate parabolic diffusion on non-commutative, anisotropic spaces with periodic geometry, enabling applications to mean field games on Carnot manifolds.

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 Hörmander vector fields generating Carnot groups, Carnot tori and dual spaces of non-isotropic Hölder spaces; secondly, we use the theory of singular integral operators to establish stronger a priori regularity for the solutions.

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

TL;DR

The paper addresses linear backward parabolic equations on Carnot tori and their dual forward FPK equations in a sub-Riemannian setting. It develops mollifier constructions adapted to Hörmander vector fields, Carnot tori, and dual Hölder spaces, and leverages singular integral theory to obtain Schauder estimates and Hölder regularity. The main contributions are well-posedness and regularity results for the backward equation, corresponding regularity for the dual FPK equation, and a generalized framework for analyzing mean field type master equations on Carnot tori. The methods provide a robust toolkit for degenerate parabolic diffusion on non-commutative, anisotropic spaces with periodic geometry, enabling applications to mean field games on Carnot manifolds.

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 Hörmander vector fields generating Carnot groups, Carnot tori and dual spaces of non-isotropic Hölder spaces; secondly, we use the theory of singular integral operators to establish stronger a priori regularity for the solutions.
Paper Structure (8 sections, 20 theorems, 236 equations)

This paper contains 8 sections, 20 theorems, 236 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Brief overview of Carnot groups and Carnot tori
  4. Theory of singular integral operators and continuity on Hölder spaces
  5. A priori Schauder estimates for the Cauchy problem on Carnot groups with rough coefficients
  6. Construction of several types of Mollifiers
  7. Results for the linear degenerate parabolic equation
  8. Results for the degenerate FPK equation

Key Result

Proposition 1.1

For each $\varepsilon>0$, set where $\varphi \in C_0^{\infty}(\mathbb{R})$ is a standard mollified function such that $\int_{\mathbb{R}}\varphi(t)dt=1$. For any function $f:\mathbb{R}\times\mathbb{R}^{n}\to\mathbb{R}$ satisfying for some constants $M>0$ and $\mu \in [0,(2c)^{-1})$ with $c$ given in Gamma_0 Lie derivatives GE, define the function where $\phi_{\varepsilon}$ is the function define

Theorems & Definitions (51)

  • Proposition 1.1: Mollifiers adapted to $\mathcal{X}$
  • Remark 1.1
  • Proposition 1.2: Mollifiers adapted to Carnot tori
  • Proposition 1.3: Mollifiers adapted to dual spaces
  • Corollary 1.1
  • Theorem 1.1
  • Theorem 1.2
  • Definition 1.1
  • Theorem 1.3
  • Definition 2.1: Homogeneous group, see 07BLU22BB
  • ...and 41 more