Interpretation of the distribution of brownian motion

Click For Summary
SUMMARY

This discussion centers on the interpretation of Brownian motion as a Gaussian process (GP) and the challenges in understanding the distribution of sample paths. The participant is familiar with Fourier series and Fokker-Planck equations but struggles with the implications of Brownian motion being a GP. They seek clarity on how the stochastic elements W(t), dW(t), and the integral of W(t) relate to the distribution over functions, particularly in the context of statistical inference on unknown scalar fields.

PREREQUISITES
  • Understanding of Gaussian processes and their properties
  • Familiarity with Fourier series in the context of signal representation
  • Knowledge of Fokker-Planck equations and diffusion processes
  • Basic concepts of stochastic processes and their applications
NEXT STEPS
  • Study the mathematical foundations of Gaussian processes in detail
  • Explore the implications of Brownian motion in stochastic calculus
  • Investigate the role of sample paths in statistical inference
  • Review the article on diffusion and its relation to Brownian motion for deeper insights
USEFUL FOR

Researchers, statisticians, and students in physics or applied mathematics who are looking to deepen their understanding of Brownian motion and its applications in statistical inference and stochastic processes.

coolnessitself
Messages
29
Reaction score
0
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?
 
Last edited:
Physics news on Phys.org

Similar threads

Is (Wt)2 a Brownian Motion? Analyzing the Distribution of Vt+s - Vs
  • · Replies 6 ·
Replies
6
Views
3K
I need to prove this (seemingly simple) property of Brownian motion
  • · Replies 1 ·
Replies
1
Views
2K
Undergrad Interpretation of QM in the Heisenberg Picture
  • · Replies 34 ·
2
Replies
34
Views
6K