### 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.

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.

#### Full Text:

9. [PDF]DOI: https://doi.org/10.4115/jla.2014.6.9

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Journal of Logic and Analysis ISSN: 1759-9008