Some analytical properties of continuous stationary Markov transition functions

Kendall, David G.

doi:10.1090/S0002-9947-1955-0067401-2

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 ${P_t}(x,\;X) \leqq 1$.

References

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