论文标题
统一的，局部的渐近物，用于亚riemannian热核，它们的对数衍生物和相关的扩散桥
Uniform, localized asymptotics for sub-Riemannian heat kernels, their logarithmic derivatives, and associated diffusion bridges
论文作者
论文摘要
我们表明，可以定位的伊曼尼亚热核，其衍生物及其对数衍生物的小渐近渐进式，即使在不完全的歧管下，在无限度的距离上基本最佳条件下也可以研究它们。继续远离异常最小化器，我们表明，渐近学与在两个相关点之间（在切割基因座上是不平凡的）之间最小化的大地测量学结构密切相关。在各种情况下，这给出了紧凑型的均匀热核边界，并允许热核及其衍生物的完全扩展。 该方法自然地扩展到热核的对数衍生物，在大多数情况下，我们再次在紧凑型上获得统一的界限，并且在任何特定对点上都具有更精确的扩展。特别是，我们确定了给出相应扩散桥的大数量定律的度量，对数衍生物的主要术语是由相对于该度量的几何自然随机变量的累积给出的。结果是，非急性切割基因座的特征是热内核的对数巢穴的行为。
We show that the small-time asymptotics of the sub-Riemannian heat kernel, its derivatives, and its logarithmic derivatives can be localized, allowing them to be studied even on incomplete manifolds, under essentially optimal conditions on the distance to infinity. Continuing, away from abnormal minimizers, we show that the asymptotics are closely connected to the structure of the minimizing geodesics between the two relevant points (which is non-trivial on the cut locus). This gives uniform heat kernel bounds on compacts, and also allows a complete expansion of the heat kernel, and its derivatives, in a wide variety of cases. The method extends naturally to logarithmic derivatives of the heat kernel, where we again get uniform bounds on compacts and a more precise expansion for any particular pair of points, in most cases. In particular, we determine the measure giving the law of large numbers for the corresponding diffusion bridge, and the leading terms of the logarithmic derivatives are given by the cumulants of geometrically natural random variables with respect to this measure. One consequence is that the non-abnormal cut locus is characterized by the behavior of the log-Hessian of the heat kernel.
