# 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.

## Key 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.

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Source: https://tomesphere.com/paper/1905.07867