Coupling in the Heisenberg group and its applications to gradient estimates

Abstract

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.

Gradient bounds for Kolmogorov type diffusions

“…As pointed out in [46], Kuwada's duality between L 1 -gradient bounds and L ∞ -Wasserstein control shows that for each t > 0, and p, p ∈ G, there exists a coupling B almost surely for some constant K 1 that does not depend on p, p, t. We remark that in [21], the authors show that any coupling that satisfies (4.25) on G must be non-Markovian. This further highlights the need for more non-Markovian coupling techniques as in [9,10].…”

Couplings in $L^{p}$ distance of two Brownian motions and their Lévy area

“…If we moreover assume that the full process (X (1) t ,X (2) t ) t≥0 is Markovian, we say that the co-adapted coupling is Markovian.…”

“…In the general context of co-adapted coupling, we were not able to obtain any results for p ∈ [1, 2): we ignore whether there exist co-adapted couplings satisfying (6) or (7), or not, for p ∈ [1,2). One difficulty in this study is to obtain estimates for the expectation of nonnegative (nonconvex) functionals of martingales as typically x → |x| 1/2 , see Remark 3.3.…”

“…We mention the interesting recent work by S. Banerjee, M. Gordina and P. Mariano [2] where the authors also use non co-adapted couplings to study the decay in total variation for the laws of Heisenberg Brownian motions and obtain gradient estimates for harmonic functions. This work and our work seems to deliver a common message namely that co-adapted couplings are not the unique relevant couplings, what concerns obtaining gradient estimates.…”

On the Cheng-Yau gradient estimate for Carnot groups and sub-Riemannian manifolds

“…The Cheng-Yau estimate was proved in [2,Corollary 4.6] for the simplest noncommutative Carnot group, the Heisenberg group, using probabilistic (coupling) techniques. The purpose of this note is to show that this estimate can be proven on two classes of sub-Riemannian manifolds, namely, sub-Riemannian manifolds satisfying a generalized curvature-dimension inequality and Carnot groups, by relying on results from [4,6,11].…”