On zeros of Martin-Löf random Brownian motion

Diophantine properties of brownian motion: recursive aspects

We use recent results on the Fourier analysis of the zero sets of Brownian motion to explore the diophantine properties of an algorithmically random Brownian motion ( also known as a complex oscillation). We discuss the construction and definability of perfect sets which are linearly independent over the rationals directly from Martin-Löf random reals. Finally we explore the recent work of Tsir...

On the average number of sharp crossings of certain Gaussian random polynomials

Effective dimension of points visited by Brownian motion

We consider the individual points on a Martin-Löf random path of Brownian motion. We show (1) that Khintchine’s law of the iterated logarithm holds at almost all points; and (2) there exist points (besides the trivial example of the origin) having effective dimension < 1. The proof of (1) amounts essentially to showing for almost all times t, the path f is random relative to t and the dimension...

Expected Number of Local Maxima of Some Gaussian Random Polnomials

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 O(n−1/2 log n) of the Brownian path, for all but finitely many positive integers n. Almost surely, this n-step walk will be incompressible in the sense of Kolmogorov complexity, and all Martin-Löf random path...