Some Analytical Properties of Continuous Stationary Markov Transition Functions
David George Kendall · Transactions of the American Mathematical Society · 1955
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 ${p_{ij}}(t)$ at $t = 0 +$ when $i e j$. 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 ${P_t}(x,\;A)$ is not assumed to be “honest"; i.e., if $X$ is the state-space then it is supposed that ${P_t}(x,\;X) \leqq 1$.