Couplings of Brownian Motions of Deterministic Distance in Model Spaces of Constant Curvature

Abstract

We consider the model space M n K of constant curvature K and dimension n ≥ 1 (Euclidean space for K = 0, sphere for K > 0 and hyperbolic space for K < 0), and we show that given a functionfor every t ≥ 0 if and only if ρ is continuous and satisfies for almost every t ≥ 0 the differential inequality −(n − 1). In other words, we characterize all co-adapted couplings of Brownian motions on the model space M n K for which the distance between the processes is deterministic. In addition, the construction of the coupling is explicit for every choice of ρ satisfying the above hypotheses.2010 Mathematics Subject Classification. Primary 60J65. Secondary 60G99, 58J65.

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

“…Aside from providing a better understanding of the geometry of the state space, the coupling method for Brownian motions is a great tool for many analysis results involving the harmonic functions and the heat semi-group such as Harnack, Poincaré, Sobolev or Wasserstein inequalities (see [23,30,15,14] for some examples). 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].…”

“…If we choose m := t 2βn with t such that t 2βn ≥ 1 2 x−x 2 2 , we can use Lemma 3.4 together with (27) to obtain the expected result. This ends the proof of Theorem 1.1.…”