I was just looking through a book which proves many interesting and rather difficult results on Brownian motion (<a href="http://people.bath.ac.uk/maspm/book.pdf">pdf link</a>, <a href="http://people.bath.ac.uk/maspm">website link</a>), and it seems that the Kolmogorov zero-one law applies to most of these.

Using Fourier transforms, a standard Brownian motion X<sub>t</sub> on the range 0&le;t&le;1 can be decomposed as
$$
X_t = At + \sum_{n=1}^\infty\frac{1}{\sqrt{2}\pi n}\left(B_n(\cos 2\pi nt - 1)+C_n\sin 2\pi nt\right)
$$
where A, B<sub>n</sub>, C<sub>n</sub> are independent normals with mean 0 and variance 1.
It follows that any property of the Brownian motion which is unchanged under addition of a linear combination of sines, cosines and linear terms is a tail event and, by Kolmogorov's zero-one law, has probability zero or one. Eg, Brownian motion is known to be nowhere differentiable (with probability 1).