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

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

Algorithmic randomness and layerwise computability

International audienceIn this article we present the framework of layerwise computability. We explain the origin of this notion, its main features and properties, and we illustrate it with several concrete examples: decomposition of measures, random closed sets, Brownian motion

Computable Measure Theory and Algorithmic Randomness

International audienceWe provide a survey of recent results in computable measure and probability theory, from both the perspectives of computable analysis and algorithmic randomness, and discuss the relations between them

Randomness and the ergodic decomposition

International audienceThe interaction between algorithmic randomness and ergodic theory is a rich field of investigation. In this paper we study the particular case of the ergodic decomposition. We give several positive partial answers, leaving the general problem open. We shortly illustrate how the effectivity of the ergodic decomposition allows one to easily extend results from the ergodic case to the non-ergodic one (namely Poincaré recurrence theorem). We also show that in some cases the ergodic measures can be computed from the typical realizations of the process