On zeros of Martin-Löf random Brownian motion

  title={On zeros of Martin-L{\"o}f random Brownian motion},
  author={Laurent Bienvenu and Kelty Allen and Theodore A. Slaman},
  journal={J. Log. Anal.},
We investigate the sample path properties of Martin-Lof random Brownian motion. We show (1) that many classical results which are known to hold almost surely hold for every Martin-Lof random Brownian path, (2) that the effective dimension of zeroes of a Martin-Lof 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-Lof random Brownian path, and (3) we will demonstrate a new proof that the solution to… 
Schnorr randomness for noncomputable measures
A number of theorems are proved demonstrating that this is the correct definition of Schnorr randomness which enjoys many of the same properties as Martin-Lof randomness for noncomputable measures, Nonetheless, a number of proofs significantly differ from the Martin- Lof case, requiring new ideas from computable analysis.
On the close interaction between algorithmic randomness and constructive/computable measure theory
A number of recent results are surveyed showing that classical almost everywhere convergence theorems can be used to characterize many of the common randomness notions including Schnorr randomness, computablerandomness, and Martin-L\"of randomness.
Algorithmic Randomness and Fourier Analysis
It is shown that the Schnorr random points are precisely those that satisfy Carleson’s Theorem for every f ∈ Lp[−π, π] given natural computability conditions on f and p.
Energy randomness
This paper shows that X ∈ 2ω is s-energy random if and only if∑n∈ω2sn−KM(X↾n)<∞, providing a characterization of energy randomness via a priori complexity KM.
I T ] 3 0 O ct 2 01 9 Algorithmic Randomness in Continuous-Time Markov Chains ∗
In this paper we develop the elements of the theory of algorithmic randomness in continuous-time Markov chains (CTMCs). Our main contribution is a rigorous, useful notion of what it means for an
We study the relative computational power of structures related to the ordered field of reals, specifically using the notion of generic Muchnik reducibility. We show that any expansion of the reals
16w5072: Algorithmic Randomness Interacts with Analysis and Ergodic Theory
This workshop brought together four logical approaches to understanding the algorithmic and computational content of mathematical theorems. In Computable or Effective Analysis, one aims to understand
When does randomness come from randomness?
  • Jason M. Rute
  • Computer Science, Mathematics
    Theor. Comput. Sci.
  • 2016
It is implied that the set of Martin-L\"of randoms is the largest subset of $2^{\mathbb{N}}$ satisfying this property and also satisfying randomness preservation.
Algorithmic Randomness in Continuous-Time Markov Chains
This paper develops the elements of the theory of algorithmic randomness in continuous-time Markov chains (CTMCs) and defines the randomness of trajectories in terms of a new kind of martingale (algorithmic betting strategy), and proves equivalent characterizations in Terms of constructive measure theory and Kolmogorov complexity.


Effective dimension of points visited by Brownian motion
It is shown that Khintchine's law of the iterated logarithm holds at almost all points; and there exist points having effective dimension <1.
The Descriptive Complexity of Brownian Motion
Abstract A continuous function x on the unit interval is a generic Brownian motion when every probabilistic event which holds almost surely with respect to the Wiener measure is reflected in x ,
Kolmogorov complexity and strong approximation of Brownian motion
Brownian motion and scaled and interpolated simple random walk can be jointly embedded in a probability space in such a way that almost surely the $n$-step walk is within a uniform distance
Applications of Effective Probability Theory to Martin-Löf Randomness
The study of the framework of layerwise computability is pursued and a general version of Birkhoff's ergodic theorem for random points, where the transformation and the observable are supposed to be effectively measurable instead of computable is proved.
Two . dimensional Brownian Motion and Harmonic Functions
1. The purpose of this paper is to investigate the properties of two-dimensional Brownian motions’ and to apply the results thus obtained to the theory of harmonic functions in the Gaussian plane.
An Application of Martin-Löf Randomness to Effective Probability Theory
A framework for computable analysis of measure, probability and integration theories, which lies on Martin-Lof randomness and the existence of a universal randomness test is provided and characterizations of effective notions of measurability and integrability are proved in terms of layerwise computability.
Some theorems concerning 2-dimensional Brownian motion
This paper consists of three separate parts(1) which are related mainly in that they treat different stochastic processes which arise in the study of plane brownian motion. §1 is concerned with the
On the computability of a construction of Brownian motion†
  • G. Davie, W. Fouché
  • Computer Science, Mathematics
    Mathematical Structures in Computer Science
  • 2013
A construction due to Fouché in which a Brownian motion is constructed from an algorithmically random infinite binary sequence is examined, showing that although the construction is provably not computable in the sense of computable analysis, a lower bound for the rate of convergence is computable, making the construction layerwise computable.
Some Random Series of Functions
1. A few tools from probability theory 2. Random series in a Banach space 3. Random series in a Hilbert space 4. Random Taylor series 5. Random Fourier series 6. A bound for random trigonometric
Ergodic-Type Characterizations of Algorithmic Randomness
This work shows that a theorem of Kucera that states that given a Martin-Lof random infinite binary sequence ω and an effectively open set A of measure less than 1, some tail of ω is not in A can be seen as an effective version of Birkhoff's ergodic theorem.