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

We consider the model space M K n 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 function ρ : [ 0 , ∞ ) → [ 0 , ∞ ) with ρ ( 0 ) = dist ( x , y ) there exists a coadapted coupling ( X ( t ), Y ( t )) of Brownian motions on M K n starting at ( x , y ) such that ρ ( t ) = dist ( X ( t ) , Y ( t ) ) for every t ≥ 0 if and only if ρ is continuous and satisfies for almost every t ≥ 0 the differential inequality - ( n - 1 ) K tan K ρ ( t ) 2 ≤ ρ ′ ( t ) ≤ - ( n - 1 ) K tan K ρ ( t ) 2 + 2 ( n - 1 ) K sin ( K ρ ( t ) ) . In other words, we characterize all coadapted couplings of Brownian motions on the model space M K n 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.