# Randomized Computation of Continuous Data: Is Brownian Motion Computable?

## TL;DR

This paper investigates the computability of continuous data, specifically Brownian motion, within the framework of Computable Analysis, establishing conditions under which it is computable based on moduli of continuity and extending previous results on randomized computation.

## Contribution

It confirms the sufficiency of using the Cantor space of infinite fair coin flips for randomized computation and characterizes the computability of Brownian motion via computable moduli of continuity and their distributions.

## Key findings

- Randomized computation can be confined to Cantor space without loss of generality.
- Brownian motion is computable iff its moduli of continuity have a computable distribution.
- Extends prior work on biased coin sequences to the case of continuous stochastic processes.

## Abstract

We consider randomized computation of continuous data in the sense of Computable Analysis. Our first contribution formally confirms that it is no loss of generality to take as sample space the Cantor space of infinite FAIR coin flips. This extends [Schr\"oder&Simpson'05] and [Hoyrup&Rojas'09] considering sequences of suitably and adaptively BIASED coins. Our second contribution is concerned with 1D Brownian Motion (aka Wiener Process), a probability distribution on the space of continuous functions f:[0,1]->R with f(0)=0 whose computability has been conjectured [Davie&Fouch\'e'13; arXiv:1409.4667,S6]. We establish that this (higher-type) random variable is computable iff some/every computable family of moduli of continuity (as ordinary random variables) has a computable probability distribution with respect to the Wiener Measure.