The Brownian Movement and Stochastic Equations

Abstract

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.