TITLE: Brownian motion and spheres
QUESTION [17 upvotes]: Consider a Brownian motion on $[0;1]$. A (finite) discrete approximation of this Brownian motion consists of $N$ iid Gaussian random variables $\Delta W_i$ of variance $\frac{1}{N}$:
$$ W\left(\frac{k}{N}\right) = \sum_{i=1}^k \Delta W_i. $$
The vector $V_N = (\Delta W_1, \ldots, \Delta W_N) \in \mathbb{R}^N$ has a norm approximately equal to $1$ since the random variable $\|V_N\|^2$ has a variance equal to $\frac{C}{N}$ and $\frac{C}{N} \to 0$ (basic concentration of measure results can make this statement more precise). This is why (?) one can approximately say that in order to sample a Brownian path, it suffices to sample a point uniformly on the unit sphere of $\mathbb{R}^N$.
Question: letting $N$ go to $\infty$, how can one formalize (if possible and/or correct) the idea that a Brownian path on $[0;1]$ is like a point uniformly chosen on the unit sphere of an infinite dimensional Banach space ?

REPLY [3 votes]: H.P. McKean wrote a fun paper on this topic. He argues that one can think of Wiener measure as uniform measure on an infinite-dimensional sphere with radius "$\sqrt{\infty}$".
See http://www.jstor.org/pss/2959482
for a copy of the paper.