Optimal Coupling of Jumpy Brownian Motion on the Circle (2203.14791v2)

Abstract: 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. More precisely, we use the theory of stochastic control to show that there exists a critical value $\lambda^\star = 0.083\dots$ such that the Mirror coupling minimises the mean coupling time within the class of all co-adapted couplings when $\lambda<\lambda^\star$, but for $\lambda\ge \lambda^\star$ the Synchronous coupling uniquely maximises the Laplace transform $\mathbb{E}[e^{-\gamma T}]$ of all coupling times $T$ within this class. 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.