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Coupling in the Heisenberg group and its applications to gradient estimates. (English) Zbl 1441.60013

A coupling of two probability measures \(\mu_1\) and \(\mu_2\), defined on respective measure spaces \((\Omega_1, \mathcal{A}_1)\) and \((\Omega_2, \mathcal{A}_2)\), is a measure \(\mu\) on the product space \((\Omega_1\times \Omega_2, \mathcal{A}_1 \times\mathcal{A}_2)\) with marginals \(\mu_1\) and \(\mu_2\). In this paper, coupling of two Markov processes having the same generator, but starting from different points joining together (coupling) at some random time is considered, and how these can be used to obtain total variation bounds and prove gradient estimates for harmonic functions on \(\mathbf{H}^3\). The paper is organized as follows. Section 2 gives basics on sub-Riemannian manifolds and the Heisenberg group \(\mathbf{H}^3\) including Brownian motion on \(\mathbf{H}^3\). In Section 3, the non-Markovian coupling of Brownian motions in \(\mathbf{H}^3\) is constructed and describes its properties. In Section 4, the authors prove the gradient estimates for harmonic functions for the hypoelliptic Laplacian which is the generator of Brownian motion in the Heisenberg group.

MSC:

60D05 Geometric probability and stochastic geometry
60H30 Applications of stochastic analysis (to PDEs, etc.)
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