  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-L\"of random} paths of Brownian motion have such an incompressible
close approximant. This strengthens a result of Asarin, who obtained the bound
$O(n^{-1/6} \log n)$. The result cannot be improved to $o(n^{-1/2}{\sqrt{\log
n}})$.
