Willem L. Fouché

Dynamics of a generic Brownian motion: Recursive aspects

We study the local fluctuations of Brownian motions which are represented by infinite binary strings which are random in the sense of Kolmogorov-Chaitin. We show how the dynamical properties of such a Brownian motion at a point depend on its recursive properties.

Symmetry and the Ramsey degree of posets

In this paper we introduce a measure of the extent to which a given finite poset deviates from being a Ramsey object in the class of finite posets. We show how this measure depends on the symmetry properties of a poset.

Symmetries and Ramsey properties of trees

In this paper we show the extent to which a finite tree of fixed height is a Ramsey object m the class of trees of the same height can be measured by its symmetry group. @ I999 Elsevier Science B.V. All rights reserved

Weihrauch-completeness for layerwise computability

We introduce the notion of being Weihrauch-complete for layerwise computability and provide several natural examples related to complex oscillations, the law of the iterated logarithm and Birkhoffs theorem. We also consider hitting time operators, which share the Weihrauch degree of the former examples but fail to be layerwise computable.

On the computability of a construction of Brownian motion

We examine a construction due to Fouche in which a Brownian motion is constructed from an algorithmically random infinite binary sequence. We show that although the construction is provably not computable in the sense of computable analysis, a lower bound for the rate of convergence is computable in any upper bound for the compressibilty of the sequence, making the construction layerwise computable.

Fractals Generated by Algorithmically Random Brownian Motion

A continuous function x on the unit interval is an algorithmically random Brownian motion when every probabilistic event A which holds almost surely with respect to the Wiener measure, is reflected in x , provided A has a suitably effective description. In this paper we study the zero sets and global maxima from the left as well as the images of compact sets of reals of Hausdorff dimension zero under such a Brownian motion. In this way we shall be able to find arithmetical definitions of perfect sets of reals whose elements are linearly independent over the field of recursive real numbers.

Fourier spectra of measures associated with algorithmically random brownian motion

In this paper we study the behaviour at infinity of the Fourier transform of Radon measures supported by the images of fractal sets under an algorithmically random Brownian motion. We show that, under some computability conditions on these sets, the Fourier transform of the associated measures have, relative to the Hausdorff dimensions of these sets, optimal asymptotic decay at infinity. The argument relies heavily on a direct characterisation, due to Asarin and Pokrovskii, of algorithmically random Brownian motion in terms of the prefix free Kolmogorov complexity of finite binary sequences. The study also necessitates a closer look at the potential theory over fractals from a computable point of view.

Symmetry and the Ramsey Degrees of Finite Relational Structures

In this paper, we introduce a measure of the extent to which a finite combinatorial structure is a Ramsey object in the class of objects with a similar structure. We show for classes of finite relational structures, including graphs, binary posets, and bipartite graphs, how this measure depends on the symmetries of the structure.

Kolmogorov Complexity and Symmetric Relational Structures

We study partitions of Fraiss6 limits of classes of finite relational structures where the partitions are encoded by infinite binary strings which are random in the sense of Kolmogorov-Chaitin. ?

Martin-Löf Randomness, Invariant Measures and Countable Homogeneous Structures

We use ideas from topological dynamics (amenability), combinatorics (structural Ramsey theory) and model theory (Fraïssé limits) to study closed amenable subgroups G of the symmetric group S∞ of a countable set, where S∞ has the topology of pointwise convergence. We construct G-invariant measures on the universal minimal flows associated with these groups G in, moreover, an algorithmic manner. This leads to an identification of the generic elements, in the sense of being Martin-Löf random, of these flows with respect to the constructed invariant measures. Along these lines we study the random elements of S∞, which are permutations that transform recursively presented universal structures into such structures which are Martin-Löf random.