6 research outputs found
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
Local time of Martin-Lof Brownian motion
In this paper we study the local times of Brownian motion from the point of
view of algorithmic randomness. We introduce the notion of effective local time
and show that any path which is Martin-L\"of random with respect to the Wiener
measure has continuous effective local times at every computable point. Finally
we obtain a new simple representation of classical Brownian local times,
computationally expressed
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