The Brownian Movement and Stochastic Equations

The irregular movements of small particles immersed in a liquid, caused by the impacts of the molecules of the liquid, were described by Brown in 1828. 1 Since 1905 the Brownian movement has been treated statistically, on the basis of the fundamental work of Einstein and Smoluchowski. Let x (t) be the x-coordinate of a particle at time t. Einstein and Smoluchowski treated x (t) as a chance variable. They found the distribution of x (t) x (O) to be Gaussian, with mean 0 and variance a I t l, where a is a positive constant which can be calculated from the physical characteristics of the moving particles and the given liquid. More exactly, such a family of chance variables x (t) is now described as the family of chance variables determining a temporally homogeneous differential stochastic process: the distribution of x (s + t) x (t) is Gaussian, with mean 0, variance a I t, and if t1 < < tn.