2023/12/23 by Li, Hong-Quan, Mao, Sheng-Chen, Zhang, Ye
#35B40 #35B45 #35H10 #35K08 #43A80 #43A85 #58J35 #58J37 #Analysis of PDEs (math.AP) #FOS: Mathematics #Probability (math.PR)
paper · doi:10.48550/arxiv.2312.15168
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.