We study the small-time asymptotics of the heat content of smooth non-characteristic domains of a general rank-varying sub-Riemannian structure, equipped with an arbitrary smooth measure. By adapting to the sub-Riemannian case a technique due to Savo, we establish the existence of the full asymptotic series:$Q_{\mathrm{\Omega }}(t)=\sum _{k=0}^{\infty }{a}_k{t}^{k/2},\text{as }t\to 0.$ We compute explicitly the coefficients up to order $k=5$, in terms of sub-Riemannian invariants of the domain. Furthermore, we prove that every coefficient can be obtained as the limit of the corresponding one for a suitable Riemannian extension.

As a particular case we recover, using non-probabilistic techniques, the order 2 formula recently obtained by Tyson and Wang in the Heisenberg group [48]. A consequence of our fifth-order analysis is the evidence for new phenomena in presence of characteristic points. In particular, we prove that the higher order coefficients in the asymptotics can blow-up in their presence. A key tool for this last result is an exact formula for the distance from a specific surface with an isolated characteristic point in the Heisenberg group, which is of independent interest.

我们研究了一般随机变亚黎曼结构的光滑非特征域的热含量的小时间渐近性，该光滑非特征域配备了任意光滑量度。通过将萨沃因技术应用于次黎曼情形，我们建立了完整渐近级数的存在：${问}_{\mathrm{\Omega }}（Ť）=\sum _{ķ=0}^{\infty }{\mathrm{一种}}_{ķ}{Ť}^{ķ/\mathrm{2个}}，\text{作为 }Ť\to 0。$ 我们显式计算系数，直到阶数 $ķ=5$，以该域的次黎曼不变量表示。此外，我们证明，对于合适的黎曼扩展，可以获取每个系数作为相应系数的极限。

作为一个特例，我们使用非概率技术恢复了海森堡小组中泰森和王最近获得的2阶公式[48]。我们的五阶分析的结果是存在特征点时出现新现象的证据。特别是，我们证明了渐近线中的高阶系数在它们出现时会爆炸。最后一个结果的关键工具是一个精确的公式，用于计算与海森堡组中具有孤立特征点的特定表面之间的距离，这是一个独立的问题。