Stochastic representation of deterministic interactions and Brownian motion

Abstract

Exact generalized stochastic representation of deterministic interaction between two dynamical (quantum or classical) systems is derived which helps when considering one of them to replace another by equivalent commutative ($c$-number valued) random sources. The method is applied to classical Brownian motion of a particle in a gas, and statistics of this motion is reduced to statistics of the gas response to perturbations.