On zeros of Martin-L\"of random Brownian motion

TL;DR

This paper explores the properties of Martin-L"of random Brownian motion paths, establishing their zero sets' dimensions, and introduces a new proof of the computability of the Dirichlet problem solution.

Contribution

It provides new insights into the zero sets of Martin-L"of random Brownian paths and proves the computability of the Dirichlet problem solution in the plane.

Findings

Zero sets of Martin-L"of random Brownian paths have effective dimension at least 1/2.

Every real with effective dimension greater than 1/2 is a zero of some Martin-L"of random Brownian path.

The Dirichlet problem solution in the plane is computable.

Abstract

We investigate the sample path properties of Martin-L\"of random Brownian motion. We show (1) that many classical results which are known to hold almost surely hold for every Martin-L\"of random Brownian path, (2) that the effective dimension of zeroes of a Martin-L\"of random Brownian path must be at least 1/2, and conversely that every real with effective dimension greater than 1/2 must be a zero of some Martin-L\"of random Brownian path, and (3) we will demonstrate a new proof that the solution to the Dirichlet problem in the plane is computable.