Study of the path probability of a Brownian motion

TL;DR

This paper investigates the probability distribution of paths taken by Brownian particles under Gaussian noise, demonstrating that path probability decreases exponentially with the action, through numerical experiments.

Contribution

It provides numerical evidence for the existence of path probability in stochastic systems and quantifies its exponential decay relative to the action.

Findings

Path probability decreases exponentially with action.

Numerical experiments confirm the existence of path probability.

Analysis applies to Brownian particles under conservative forces.

Abstract

This work is a numerical experiment of stochastic motion of conservative Hamiltonian system or weakly damped Brownian particles. The objective is to prove the existence of path probability and to compute its values. By observing a large number of particles moving from one point to another under Gaussian noise and conservative forces, it is determined that the path probability decreases exponentially with action along the paths.