Abstract

We exhibit some explicit co-adapted couplings for n-dimensional Brownian motion and all its Lévy stochastic areas. In the two-dimensional case we show how to derive exact asymptotics for the coupling time under various mixed coupling strategies, using Dufresne's formula for the distribution of exponential functionals of Brownian motion. This yields quantitative asymptotics for the distributions of random times required for certain simultaneous couplings of stochastic area and Brownian motion. The approach also applies to higher dimensions, but will then lead to upper and lower bounds rather than exact asymptotics.

Keywords Brownian motion, co-adapted coupling, coupling time distribution, Dufresne formula, exponential functional of Brownian motion, Kolmogorov diffusion, Lévy stochastic area, maximal coupling, Morse– Thue sequence, non-co-adapted coupling, reflection coupling, rotation coupling, stochastic differential, synchronous coupling

AMS subject classification (MSC2010) 60J65, 60H10

Introduction

It is a pleasure to present this paper as a homage to my DPhil supervisor John Kingman, in grateful acknowledgement of the formative period which I spent as his research student at Oxford, which launched me into a deeply satisfying exploration of the world of mathematical research. It seems fitting in this paper to present an overview of a particular aspect of probabilistic coupling theory which has fascinated me for a considerable time; given that one can couple two copies of a random process, when can one in addition couple other associated functionals of the processes? How far can one go?