A connection between the Boltzmann equation and the Ornstein-Uhlenbeck process (Q797426)

In the present paper one considers a hard sphere suffering elastic collisions with the spherical molecules of a gas in thermal equilibrium uninfluenced by the movement of a Brownian particle (the considered hard sphere). It is shown that the evolution of the velocity of the Brownian particle, as described by the Ornstein-Uhlenbeck process is a limit case of the description given by the linear Boltzmann equation. It was shown [\textit{G. Wilemski}, J. Stat. Phys. 24, 153 ff. (1976)] that if the mass of the Brownian particle is large compared with the mass of gas molecule and if its velocity never departs from the equipartition value, one can give the linear Boltzmann equation a form which looks nearly like the diffusion equation of the Ornstein-Uhlenbeck velocity process. In the present paper one demonstrates that firstly the linear Boltzmann equation with hard spherical molecules has a unique solution, and subsequently that if the mass m of the gas molecules decreases while their number \(N_ m\) increases in such a way that \(N_ m\sqrt{m}\) remains constant, this solution converges in \(L^ 1(R^ 3)\) uniformly on bounded time intervals to the solution of the Ornstein-Uhlenbeck diffusion equation.