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Let (x(t) , v(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 2 of the fact that, as e 4-0, ex(t/e ) tends weakly to Brownian motion. The same problem is then considered for u(f) belonging to a large class of diffusion processes.doi:10.1090/s0002-9947-1974-0353469-4 fatcat:jlzy6newcvezzbmduxp6z2a77q