Consider a Brownian motion on the circumference of the unit circle, which jumps to the opposite point of the circumference at incident times of an independent Poisson process of rate $\lambda$. We examine the problem of coupling two copies of this ‘jumpy Brownian motion’ started from different locations, so as to optimise certain functions of the coupling time. We describe two intuitive co-adapted couplings (‘Mirror’ and ‘Synchronous’) which differ only when the two processes are directly opposite one another, and show that the question of which strategy is best depends upon the jump rate $\lambda$ in a non-trivial way. We also provide an explicit description of a (non-co-adapted) maximal coupling for any jump rate in the case that the two jumpy Brownian motions begin at antipodal points of the circle.

This is a case study concerning the rate at which probabilistic coupling occurs for nilpotent diffusions. We focus on the simplest case of Kolmogorov diffusion (Brownian motion together with its time integral or, more generally, together with a finite number of iterated time integrals). We show that in this case there can be no Markovian maximal coupling. Indeed, there can be no efficient Markovian coupling strategy (efficient for all pairs of distinct starting values), where the notion of efficiency extends the terminology of Burdzy and Kendall (2000). Finally, at least in the classical case of a single time integral, it is not possible to choose a Markovian coupling that is optimal in the sense of simultaneously minimizing the probability of failing to couple by time t for all positive t. In recompense for all these negative results, we exhibit a simple efficient non-Markovian coupling strategy.