Chapman-Enskog-Hilbert expansion for the Ornstein-Uhlenbeck process and the approximation of Brownian motion

Abstract

Let ( x ( t ) , υ ( t ) ) (x(t),\upsilon (t)) denote the joint Ornstein-Uhlenbeck position-velocity process. Special solutions of the backward equation of this process are studied by a technique used in statistical mechanics. This leads to a new proof of the fact that, as ε ↓ 0 , ε x ( t / ε 2 ) \varepsilon \downarrow 0,\varepsilon x(t/{\varepsilon ^2}) tends weakly to Brownian motion. The same problem is then considered for υ ( t ) \upsilon (t) belonging to a large class of diffusion processes.