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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Some analytical properties of continuous stationary Markov transition functions

Author: David G. Kendall
Journal: Trans. Amer. Math. Soc. 78 (1955), 529-540
MSC: Primary 60.0X
MathSciNet review: 0067401
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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_{ij}}(t)$ at $ t = 0 + $ when $ 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)$ 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$.

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Article copyright: © Copyright 1955 American Mathematical Society