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 paths of Brownian motion have such an incompressible close approximant. This strengthens a result of Asarin, who obtained instead the bound O(n−1/6 log n). The result cannot be improved to o(n−1/2 √ log n).