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?
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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