Let $B(t)$ be a Brownian motion. The ordering of $(0, B(1), ..., B(n-1)) $ is a random permutation in $S_n$. This is not uniform for $n>2$ since the probabilities of the identity permutation $[123...n]$ and of $[n(n-1)...1]$ are $1/2^{n-1}$, not $1/n!$. (For $n=3$, the other $4$ permutations all have probability $1/8$.)

If $B(t)$ is conditioned to return to the origin at time $n$, this gives a different distribution on $S_n$. For $n=3$ it is uniform, and I had hoped it would be uniform for all $n$, but this is not the case. For $n=4$, permutations appear to fall into $4$ classes of size $4$ or $8$, with constant probabilities on these classes.

## What are these distributions on $S_n$? In particular, which permutations are the least/most likely and what are their probabilities?

One approach for the unpinned Brownian motion is to consider $(B(1)-B(0),...,B(n)-B(n-1))$, a spherically symmetric Gaussian. The permutation is a function of the direction from the origin, and this direction is uniformly distributed on the sphere. The probability of a permutation corresponds to the ratio between the volume of a spherical simplex and the volume of the whole sphere. An analogous method works for pinned Brownian motion. However, I don't see a good way to compute those volumes.