Fractional heat content asymptotics for Carnot groups

Technical Summary: Fractional Heat Content Asymptotics for Carnot Groups

Problem Statement
This paper investigates the small-time asymptotics of the fractional heat content for bounded domains within Carnot groups. Let $G$ be a Carnot group equipped with a sub-Laplacian operator $L$. For a bounded domain $\Omega \subset G$ and a parameter $\alpha \in [1, 2]$, the fractional heat content is defined as:
$$Q^{(\alpha)}_{\Omega}(t) = \int_{\Omega} u_{\alpha}(x, t) \, dx$$
where $u_{\alpha}$ is the solution to the parabolic equation $(\partial_t + L^{\alpha})u_{\alpha} = 0$ with Dirichlet boundary conditions ($u_{\alpha} = 0$ on $\partial\Omega$) and initial condition $u_{\alpha}(0, x) = 1$. The central problem is to determine the rate at which the heat content decays from the initial volume $|\Omega|$ as $t \to 0$, specifically identifying the limit:
$$\lim_{t \to 0} \frac{|\Omega| - Q^{(\alpha)}_{\Omega}(t)}{\mu_{\alpha}(t)}$$
where $\mu_{\alpha}(t)$ is a specific rate function derived from the Euclidean case. While such asymptotics are well-established for the classical diffusion case ($\alpha=2$) on Riemannian and sub-Riemannian manifolds, and for fractional cases in Euclidean space, this work addresses the gap for non-local fractional sub-Laplacians on sub-Riemannian manifolds (specifically Carnot groups).

Methodology
The authors propose a novel approach combining probabilistic representations with analytic estimates on Carnot groups, distinct from the geometric heat kernel expansion techniques used in prior works for the local case ($\alpha=2$).

1. Probabilistic Representation: Utilizing the subordination argument for fractional powers of Markov generators, the heat content is represented via the subordinated horizontal Brownian motion $B^{\alpha}(t) = B(S^{\alpha}_t)$, where $B$ is the horizontal Brownian motion and $S^{\alpha}$ is an $\alpha/2$-stable subordinator. The heat content is expressed as:
   $$Q^{(\alpha)}_{\Omega}(t) = \int_{\Omega} P_x(\tau^{(\alpha)}_{\Omega} > t) \, dx$$
   where $\tau^{(\alpha)}_{\Omega}$ is the first exit time of $B^{\alpha}$ from $\Omega$.

2. Functional Approach: Instead of analyzing the domain directly, the authors study a functional version of the heat content for non-negative measurable functions $f$:
   $$Q^{(\alpha)}_f(t) = \int_G E_x \left[ \inf_{0 \le s \le t} f(B^{\alpha}(s)) \right] dx$$
   This allows the use of Taylor expansions on the group structure.

3. Analytic Tools:

   • Taylor Formula: The proof relies on a Taylor expansion for $C^{1,1}$ functions on Carnot groups (due to Bonfiglioli), which approximates the function value along the path of the process.
   • Exit Probability Estimates: A critical component is establishing sharp upper bounds for the probability that the horizontal Brownian motion exits a ball of radius $R$ within time $t$. Theorem 3.1 provides these bounds, showing that the exit probability decays exponentially with terms involving the homogeneous dimension and the stratification of the group.
   • Geometric Measure Theory: The horizontal perimeter $|\partial\Omega|_H$ is utilized, defined via the horizontal variation of the indicator function. The authors employ the coarea formula and properties of the spherical Hausdorff measure with respect to homogeneous metrics.

Key Contributions and Results
The main result (Theorem 1.2) establishes that for a bounded open subset $\Omega$ of a Carnot group $G$ with a $C^2$ boundary containing no characteristic points, and for $1 \le \alpha \le 2$:
$$\lim_{t \to 0} \frac{|\Omega| - Q^{(\alpha)}_{\Omega}(t)}{\mu_{\alpha}(t)} = |\partial\Omega|_H$$
Here, $|\partial\Omega|_H$ denotes the horizontal perimeter of $\Omega$, and $\mu_{\alpha}(t)$ is the rate function defined in the Euclidean context (involving $t^{1/\alpha}$ for $\alpha > 1$, $t \log(1/t)$ for $\alpha=1$, and $t$ for $\alpha < 1$).

Specific contributions include:

• Extension to Non-Local Operators: The paper provides the first asymptotic expansion for fractional heat content involving non-local operators (fractional sub-Laplacians) on sub-Riemannian manifolds.
• Universality of the Rate Function: The result confirms that the rate function $\mu_{\alpha}(t)$, previously known for Euclidean spaces, remains valid for Carnot groups when $1 \le \alpha \le 2$.
• Lower Bound for General Sets: Proposition 5.2 proves that the lower bound of the asymptotic limit holds for any bounded open set with finite horizontal perimeter, without requiring $C^2$ regularity or the absence of characteristic points.
• Simplified Argument: The authors replace the complex exit distribution arguments used in previous Heisenberg group studies with a simplified argument based on Taylor formulas and refined exit probability estimates.

Significance and Claims
The paper claims that its primary significance lies in bridging the gap between fractional heat content asymptotics in Euclidean spaces and sub-Riemannian geometry. The authors emphasize that their method is robust and relies on estimates of small exit times and Taylor expansions, suggesting the approach could be applicable to proving heat content asymptotics for non-local operators on more general spaces.

The authors explicitly state that the assumption of a non-characteristic boundary is crucial for their method, as it ensures the signed distance function is $C^2$ and Lipschitz continuous, which is necessary for the Taylor expansion and the upper bound proof. They acknowledge that while they believe the result should hold for any sub-Riemannian manifold, a proof for the general case (beyond Carnot groups) is currently unknown to them. The work does not propose new applications or experimental setups but focuses strictly on the theoretical derivation of these asymptotic limits.