Interpretation of the distribution of brownian motion

[coolnessitself]

Hi all,

I feel like there's a missing link in my understanding of brownian motion. I'm comfortable with the "method of http://fraden.brandeis.edu/courses/phys39/simulations/Uhlenbeck%20Brownian%20Motion%20Rev%20Mod%20Phys%201945.pdf" where the signal is written as a Fourier series, and with fokker-planck equations and diffusion. I'm somewhat comfortable with an introductory theory of stochastic processes.

What bothers me is that I can't explain to myself what the distribution of sample paths means. For example, a statistician might want to do inference on an unknown scalar field. They place a gaussian process prior on the field, and from that can get a pretty good fit. I think of a GP as a distribution over functions.
So brownian motion is a GP, with some added conditions. But if it's a GP, I don't understand how W(t), dW(t), or \int W(t) create a distribution over functions. I can see how there's some probability that the sample path will be in a particular interval (y,y+dy), but that's not quite the same to me.

Help me out?

 

[marcusl]

The first part of this article
http://bayes.wustl.edu/etj/articles/cmystery.pdf
has a brilliant description of inference applied to a related problem, that of diffusion. His approach to thinking about paths may be helpful.