On zeros of Martin-Löf random Brownian motion

Abstract

We investigate the sample path properties of Martin-Löf random Brownian motion.
We show (1) that many classical results which are known to hold almost surely hold
for every Martin-Löf random Brownian path, (2) that the effective dimension of
zeroes of a Martin-Löf 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-Löf random Brownian path, and (3) we will demonstrate a new proof that
the solution to the Dirichlet problem in the plane is computable.