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November 2018 Coupling in the Heisenberg group and its applications to gradient estimates
Sayan Banerjee, Maria Gordina, Phanuel Mariano
Ann. Probab. 46(6): 3275-3312 (November 2018). DOI: 10.1214/17-AOP1247


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.


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Sayan Banerjee. Maria Gordina. Phanuel Mariano. "Coupling in the Heisenberg group and its applications to gradient estimates." Ann. Probab. 46 (6) 3275 - 3312, November 2018.


Received: 1 October 2016; Revised: 1 November 2017; Published: November 2018
First available in Project Euclid: 25 September 2018

zbMATH: 06975487
MathSciNet: MR3857856
Digital Object Identifier: 10.1214/17-AOP1247

Primary: 60D05
Secondary: 60H30

Keywords: Brownian motion , coupling , Gradient estimate , Heisenberg group , Karhunen–Loeve expansion , non-Markovian coupling , sub-Riemannian manifold , total variation distance

Rights: Copyright © 2018 Institute of Mathematical Statistics


Vol.46 • No. 6 • November 2018
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