Is it reasonable to view the Brownian bridge as a kind of Brownian motion indexed by points on the circle? The Brownian bridge has some strange connections with the Riemann zeta function (see Williams' article http://www.statslab.cam.ac.uk/~grg/books/hammfest/22-dw.ps, for example).

I'm looking for a heuristic explanation of why this might be the case. If one could interpret the Brownian bridge as described above, then the heuristic would be that Brownian motion is naturally associated with heat flow, which goes hand in hand with theta functions, which goes some way toward explaining the appearance of the zeta function.

I don't know where to begin reading about Brownian motion indexed by anything other than $\mathbb{R}^{+}$. The standard stochastic analysis texts don't really address the idea.

Many thanks.