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Fractional heat content asymptotics for Carnot groups

Rohan Sarkar

TL;DR

This work extends small-time fractional heat-content asymptotics to Carnot groups by linking the deficit |Ω|−Q^{(α)}_Ω(t) to the horizontal perimeter |∂Ω|_H via a universal rate μ_α(t). The authors represent the fractional heat content through subordination of horizontal Brownian motion and develop a Taylor-based Carnot-group framework plus precise exit-time estimates to obtain both lower and upper bounds. For 1≤α≤2, they prove lim_{t→0} (|Ω|−Q^{(α)}_Ω(t))/μ_α(t) = |∂Ω|_H, with μ_α(t) matching the Euclidean rate; the lower bound holds for any finite horizontal perimeter, while the upper bound is established for C^2 domains with no characteristic points. The results highlight a universal, geometry-driven mechanism in sub-Riemannian settings and demonstrate how probabilistic constructions combined with Carnot-group Taylor expansions can yield sharp heat-content asymptotics on non-Euclidean spaces.

Abstract

We propose a novel approach for studying small-time asymptotics of the fractional heat content of $C^2$ non-characteristic domains in Carnot groups. Denoting the sub-Laplacian operator by $\mathcal{L}$, the fractional heat content of a bounded domain $Ω$ is defined as $Q^{(α)}_Ω(t)=\int_Ω u_α(t,x)dx$, where $u_α$ is the solution to the heat equation corresponding to the fractional sub-Laplacian $\mathcal{L}_α:=\mathcal{L}^{α/2}$ with Dirichlet boundary condition on $Ω$. We prove that for $1\leα\le 2$, there exists explict rate function $μ_α: (0,\infty)\to (0,\infty)$ such that \[ \lim_{t\to 0}\frac{|Ω|-Q^{(α)}_Ω(t)}{μ_α(t)}=|\partialΩ|_H, \] where $|\partialΩ|_H$ is the horizontal perimeter of $Ω$. Moreover, the rate function $μ_α$ coincides with the same for the Euclidean case.

Fractional heat content asymptotics for Carnot groups

TL;DR

This work extends small-time fractional heat-content asymptotics to Carnot groups by linking the deficit |Ω|−Q^{(α)}_Ω(t) to the horizontal perimeter |∂Ω|_H via a universal rate μ_α(t). The authors represent the fractional heat content through subordination of horizontal Brownian motion and develop a Taylor-based Carnot-group framework plus precise exit-time estimates to obtain both lower and upper bounds. For 1≤α≤2, they prove lim_{t→0} (|Ω|−Q^{(α)}_Ω(t))/μ_α(t) = |∂Ω|_H, with μ_α(t) matching the Euclidean rate; the lower bound holds for any finite horizontal perimeter, while the upper bound is established for C^2 domains with no characteristic points. The results highlight a universal, geometry-driven mechanism in sub-Riemannian settings and demonstrate how probabilistic constructions combined with Carnot-group Taylor expansions can yield sharp heat-content asymptotics on non-Euclidean spaces.

Abstract

We propose a novel approach for studying small-time asymptotics of the fractional heat content of non-characteristic domains in Carnot groups. Denoting the sub-Laplacian operator by , the fractional heat content of a bounded domain is defined as , where is the solution to the heat equation corresponding to the fractional sub-Laplacian with Dirichlet boundary condition on . We prove that for , there exists explict rate function such that where is the horizontal perimeter of . Moreover, the rate function coincides with the same for the Euclidean case.
Paper Structure (12 sections, 20 theorems, 126 equations)

This paper contains 12 sections, 20 theorems, 126 equations.

Table of Contents

  1. Introduction
  2. Preliminaries on Carnot group
  3. Global coordinates of $\mathbb{G}$
  4. Homogeneous norms and distances
  5. Horizontal perimeter
  6. Taylor formula
  7. Horizontal Brownian motion and subordination
  8. Small time estimates of the exit probability
  9. Asymptotic of the minimum functional
  10. Proof of the lower bound
  11. Proof of Theorem \ref{['thm:main']}
  12. Some estimates related to $\alpha/2$-stable subordinators on $\mathbb{R}$

Key Result

Theorem 1.2

Let $\Omega$ be a bounded, open subset of $\mathbb{G}$ with $C^{2}$ boundary having no characteristic points. Then for all $1\leqslant \alpha\leqslant 2$, where $\mu_\alpha(t)$ is defined in eq:mu_alpha.

Theorems & Definitions (41)

  • Definition 1.1: Characteristic points
  • Theorem 1.2
  • Lemma 2.1
  • Remark 2.2
  • Lemma 2.3
  • Lemma 2.4
  • Theorem 2.5
  • Lemma 2.6
  • proof
  • Theorem 3.1
  • ...and 31 more