Finitary isomorphisms of Brownian motions

TL;DR

This paper constructs explicit finitary isomorphisms between reflected Brownian motions on various intervals, demonstrating finite coding windows and deepening understanding of their measure-preserving properties.

Contribution

It provides elementary constructions of finitary isomorphisms between reflected Brownian motions on different intervals, extending classical results.

Findings

Finitary isomorphisms with finite coding windows are constructed.

The methods apply to Brownian motions reflected on intervals with rational endpoints.

The work offers elementary proofs of measure-preserving isomorphisms.

Abstract

Ornstein and Shields (Advances in Math., 10:143-146, 1973) 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.