Martin-Löf Randomness and Brownian Motion
We investigate the Martin-L�of random sample paths of Brownian motion, applying techniques from algorithmic randomness to Brownian motion, an active area of research in probability theory.
In Chapter 2, we investigate many classical results about one dimensional Brownian motion in the context of Martin-L�of randomness. We show that many results which are known to hold almost surely for the Brownian motion process - including results concerning the modulus of continuity, points of increase, time inversion, and law of large numbers - hold for every Martin-L�of random sample path. We also show that scaling invariance and the strong Markov property hold for every Martin-L�of random path, with suitable effectivization.
In Chapter 3, we investigate the zero set of one-dimensional Brownian motion. We prove that the set of zeroes is characterized by having high effective dimension. We also demonstrate that, although the zeroes are highly noncomputable in the sense of effective dimension, many of them are layerwise computable from a Brownian path.
In Chapter 4, we give a new proof that the solution to the Dirichlet problem in the plane is computable. It is a well-known result of Kakutani that the solution to the Dirichlet problem can be found using expected hitting times of Brownian paths to the boundary. We show that the hitting times of Martin-L�of random Brownian paths on a computable boundary are layerwise computable in the path, and thus the expected value of the hitting times is computable, and so the solution to the Dirichlet problem is computable.
In Chapter 5, we further investigate planar Martin-L�of random Brownian motion. We demonstrate that a Martin-L�of random planar Brownian path only hits points such that the path is not random relative to those points (except the origin), which implies that a Martin-L�of random planar path has area zero and that every point except the origin is hit by only measure 0 many paths. We also show that every Martin-L�of random planar Brownian path has points of uncountable multiplicity.