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

We consider the model space (mathbb {M}^{n}_{K}) of constant curvature K and dimension (nge 1) (Euclidean space for (K=0), sphere for (K>0) and hyperbolic space for (K<0)), and we show that given a function (rho :[0,infty )rightarrow [0, infty )) with (rho (0)=mathrm {dist}(x,y)) there exists a coadapted coupling (X(t), Y(t)) of Brownian motions on (mathbb {M}^{n}_{K}) starting at (x, y) such that (rho (t)=mathrm {dist}(X(t),Y(t))) for every (tge 0) if and only if (rho ) is continuous and satisfies for almost every (tge 0) the differential inequality

$$begin{aligned} -(n-1)sqrt{K}tan left( tfrac{sqrt{K}rho (t)}{2}right) le rho '(t)le -(n-1)sqrt{K}tan left( tfrac{sqrt{K}rho (t)}{2}right) +tfrac{2(n-1)sqrt{K}}{sin (sqrt{K}rho (t))}. end{aligned}$$

In other words, we characterize all coadapted couplings of Brownian motions on the model space (mathbb {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 (rho ) satisfying the above hypotheses.