Some analytical properties of continuous stationary Markov transition functions

Abstract

A systematic treatment of Markov processes with Euclidean state-spaces has recently been presented by Doob [1], the restriction on the nature of the state-space being associated with the very illuminating probabilistic method which he uses throughout. At about the same time a new step was taken by Kolmogorov [4] who established for countable state-spaces the existence and finiteness of the derivative of the transition-function p i j ( t ) {p_{ij}}(t) at t = 0 + t = 0 + when i ≠ j i \ne j . In this paper some of Doob’s and Kolmogorov’s results are combined and shown to be valid (when suitably formulated) for an arbitrary state-space. For the sake of a generality which proves useful in the discussion of existence theorems the transition-function P t ( x , A ) {P_t}(x,\;A) is not assumed to be “honest"; i.e., if X X is the state-space then it is supposed that P t ( x , X ) ≦ 1 {P_t}(x,\;X) \leqq 1 .