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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

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

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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Some analytical properties of continuous stationary Markov transition functions
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by David G. Kendall PDF
Trans. Amer. Math. Soc. 78 (1955), 529-540 Request permission

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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Additional Information
  • © Copyright 1955 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 78 (1955), 529-540
  • MSC: Primary 60.0X
  • DOI: https://doi.org/10.1090/S0002-9947-1955-0067401-2
  • MathSciNet review: 0067401