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 $q$.
"Finitary isomorphisms of Brownian motions." Ann. Probab. 48 (4) 1966 - 1979, July 2020. https://doi.org/10.1214/19-AOP1412