On Brownian Motion Equations

A study is made of the relationships between the different descriptions of Brownian motion expressed as an integro-differential equation of Boltzmann type, as a Langevin equation and a partial differential equation corresponding to it, and as Fokker–Plank–Kolmogorov equations. Special attention is devoted to the relationships between the last two descriptions. Let the mass of a particle be $m$, the temperature of the medium $T$, the viscosity of the medium $A$, and let the intensity of the power field in $x$ be $F(x)$. Then the Brownian motion equation in the phase space $(x,y=\dot x)$ has the form (3.4). In the appendix to this paper the existence of the Green function for equation (3.4) is proved. An asymptotic series is obtained as the solution to the Cauchy problem for equation (3.4) for $\varepsilon={m/{A\ll 1}}$. In particular it is proved that the zero term of this asymptotic series for $t\gg\varepsilon$ is the solution to the Cauchy problem for equation (4.13) under suitable initial conditions.