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arXiv · 1906.06684

Randomized Computation of Continuous Data: Is Brownian Motion Computable?

Willem Fouché·Hyunwoo Lee·Donghyun Lim·Sewon Park·Matthias Schröder·Martin Ziegler
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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 [Schröder&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&Fouché'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.

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Willem Fouché, Hyunwoo Lee, Donghyun Lim, Sewon Park, Matthias Schröder, Martin Ziegler. 2019-06-16. Randomized Computation of Continuous Data: Is Brownian Motion Computable?. https://arxiv.org/abs/1906.06684

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