Homogeneous stochastic processes*

Abstract

The form of a stationary translation-invariant Markov process on the real line has been known for some time, and these processes have been variously characterized as infinitely divisible or infinitely decomposable. The purpose of this paper is to study a natural generalization of these processes on a homogeneous space (X, G). Aside from the lack of structure inherent in the very generality of the spaces (X, G), the basic obstacles to be surmounted stem from the presence of non trivial compact subgroups in G and the non commutativity of G, which precludes the use of an extended Fourier analysis of characteristic functions, a tool which played a dominant role in the classical studies. Even in the general situation there is a striking similarity between homogeneous processes and their counterparts on the real line. © 1959 by Pacific Journal of Mathematics.