In this paper we shall show that every linear regular diffusion is a pathwise limit of processes which are rather simple in nature. These simple processes, which may be considered to be the basic building blocks of a general diffusion, are called stretched Brownian motions. They behave like Brownian motion except that the variance is a constant which depends on the region of the state space in which the particle is located, and there are a finite number of such regions. The precise description of stretched Brownian motion, its characteristics, and the convergence theorem are presented in Section 1.
Processes called stretched random walks are defined in Section 2. These processes have the characteristics of a symmetric random walk with variance CJ~ when they are located in (xi, xi+ ,), where -co < x1 < ..a < xk < co, and when they reach xi they have probability pi of moving to the right of xi and probability qi = 1 -pi of moving to the left of xi. It will be shown that every stretched Brownian motion is a weak limit of stretched random walks, and this in turn will imply the existence of a sequence of stretched random walks that converge weakly to a general given diffusion. The last topic deals with the time change T (called natural time), relative to Brownian motion, associated with a given scale function S. A particle undergoing the diffusion X = S-'BT then moves as closely as possible to a particle undergoing motion which imitates Rrownian motion in the sense that its traversal time is quadratic.
The convergence theorem in Section 1 motivated a more general limit theorem (which we present in 121) involving processes obtained from Brownian motion undergoing time changes which do not necessarily arise * Supported in part by an NRC research grant while visiting the University of British Columbia.