COUPLING IN THE HEISENBERG GROUP AND ITS APPLICATIONS TO GRADIENT ESTIMATES

• 成果类型: Article
• 署名作者: Banerjee, Sayan; Gordina, Maria; Mariano, Phanuel
• 署名单位: University of North Carolina; University of North Carolina Chapel Hill; University of Connecticut
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• DOI: 10.1214/17-AOP1247
• 发表日期: 2018
• 页码: 3275-3312
• 关键词: sub-riemannian manifolds; heat kernel; differential-equations; inequalities; bounds

摘要

We construct a non-Markovian coupling for hypoelliptic diffusions which are Brownian motions in the three-dimensional Heisenberg group. We then derive properties of this coupling such as estimates on the coupling rate, and upper and lower bounds on the total variation distance between the laws of the Brownian motions. Finally, we use these properties to prove gradient estimates for harmonic functions for the hypoelliptic Laplacian which is the generator of Brownian motion in the Heisenberg group.