Interpretation of the distribution of brownian motion

AI Thread Summary
The discussion centers on the complexities of understanding the distribution of sample paths in Brownian motion, particularly in relation to Gaussian processes (GP). The author is comfortable with mathematical concepts like Fourier series and Fokker-Planck equations but struggles to grasp how Brownian motion, as a GP, translates into a distribution over functions. They recognize that while there is a probability associated with sample paths falling within specific intervals, this does not fully clarify the underlying distribution. The author references an article that offers insights into inference related to diffusion, suggesting that it may provide a clearer perspective on the topic. Overall, the discussion highlights the need for a deeper understanding of the relationship between Brownian motion and its probabilistic interpretations.
coolnessitself
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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?
 
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