Finitary isomorphisms of Brownian motions

Ornstein and Shields (Advances in Math. 10 (1973) 143–146) proved that Brownian motion reflected on a bounded region is an infinite entropy Bernoulli flow, and, thus, Ornstein theory yielded the existence of a measure-preserving isomorphism between any two such Brownian motions. For fixed h>0, we construct by elementary methods, isomorphisms with almost surely finite coding windows between Brownian motions reflected on the intervals [0,qh] for all positive rationals q.