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Let X(t) (05 t < 00) be the Brownian motion process. Concerning the uniform continuity of X(t), there exists P. Levy’s result. Before stating his result, let us define the concept of upper class and lower class with regard to the uniform continuity of X(t) (0 2 ts 1). If there exists a positive number E such that 1 t’--t[ 5 E implies the relation (1) lf(~‘)-f(t)IIg(It’--tl) I where g(t) is a non-negative, continuous, non-decreasing function defined in some finite interval (0, T) and vanishing with t, then we say that f(t) satisfies Lipschitz’s condition relative to g(t). Putting p(t) = +(+)2/” if X(t) (05 t S 1) satisfies Lipschitz’s condition relative to y(t) with probability 1 we say that e(t) belongs to the upper class. If X(t) (0 5 t 5 1) does not satisfy Lipschitz’s condition relative to y(t) with probability 1 we say that q(t) belongs to the lower class. P. Levy [l] proved that the function

On the Lipschitz's condition for Brownian motion.