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

Abstract

We study co-adapted couplings of (canonical hypoelliptic) diffusions on the (subRiemannian) Heisenberg group, that we call (Heisenberg) Brownian motions and are the joint laws of a planar Brownian motion with its Lévy area. We show that contrary to the situation observed on Riemannian manifolds of non-negative Ricci curvature, for any co-adapted coupling, two Heisenberg Brownian motions starting at two given points can not stay at bounded distance for all time t ≥ 0. Actually, we prove the stronger result that they can not stay bounded in L p for p ≥ 2.We also prove two positive results. We first study the coupling by reflection and show that it stays bounded in L p for 0 ≤ p < 1. Secondly, we construct an explicit static (and in particular non co-adapted) coupling between the laws of two Brownian motions, which provides L 1 -Wasserstein control uniformly in time.Finally, we explain how the results generalise to the Heisenberg groups of higher dimension.

Gradient bounds for Kolmogorov type diffusions

“…In particular, the L 1 -gradient bounds for the heat semigroup have been proven first in [47] and also in [7]. 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].…”

“…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 of Brownian Motions on $$SU(2,\mathbb {C})$$

“…Then, a strategy for coupling such processes consists in coupling the driving noises and see the effects on the "driven processes". It gives numerous examples for couplings, successful or not [22,10,16,12,28,27,26,3,4,17,3,5]. This strategy can be used to study the subelliptic Brownian motion B t = (X t , z t ) for three model spaces of subRiemannian manifolds:…”

“…This method has been studied these last decades in the case of Riemannian manifolds (see for example [21,27]). The case of subRiemannian manifolds is a current topic of interest and have been investigated on the Heisenberg group in [9,8,21,20,19,4,11,5,25], on SU(2) [12,13,25] and on SL(2, R) [13,25]. On the spaces considered in this work, the subRiemannian Brownian motion can be written under the form (X t , z t ) t where (X t ) t is a Brownian motion on an Riemannian base and z t can be interpreted as an area swept by (X s ) s≤t .…”