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 Birkhoff's theorem. We also consider hitting time operators, which share the Weihrauch degree of the former examples but fail to be layerwise computable.

In this paper, we continue the study of the geometry of Brownian motions which are encoded by Kolmogorov–Chaitin random reals (complex oscillations). We unfold Kolmogorov–Chaitin complexity in the context of Brownian motion and specifically to phenomena emerging from the random geometric patterns generated by a Brownian motion.

We examine a construction due to Fouché 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.

Abstract We discuss ways in which a typical one-dimensional Brownian motion can be approximated by oscillations which are encoded by finite binary strings of high descriptive complexity. We study the recursive properties of Brownian motions that can be thus obtained.

Abstract We study partitions of Fraïssé 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.