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Heat kernel asymptotics on the free step-two Carnot group with $3$ generators
Hong-Quan Li, Sheng-Chen Mao, Ye Zhang · 2023-12-23 · via math.PR updates on arXiv.org

In this work, we establish the uniform heat kernel asymptotics as well as sharp bounds for its derivatives on the free step-two Carnot group with $3$ generators. As a by-product, on this highly non-trivial toy model, we completely solve the Gaveau-Brockett problem, in other words, we obtain the expression of the squared Carnot-Carathéodory distance, as explicitly as one can possibly hope for. Furthermore, the precise estimates of the heat kernel, and its small-time asymptotic behaviors are deduced.

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