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
- J. L. Doob, Stochastic processes, John Wiley & Sons, Inc., New York; Chapman & Hall, Ltd., London, 1953. MR 0058896
- Willy Feller, On the integro-differential equations of purely discontinuous Markoff processes, Trans. Amer. Math. Soc. 48 (1940), 488–515. MR 2697, DOI 10.1090/S0002-9947-1940-0002697-3
- Einar Hille, Functional Analysis and Semi-Groups, American Mathematical Society Colloquium Publications, Vol. 31, American Mathematical Society, New York, 1948. MR 0025077
- A. N. Kolmogorov, On the differentiability of the transition probabilities in stationary Markov processes with a denumberable number of states, Moskov. Gos. Univ. Učenye Zapiski Matematika 148(4) (1951), 53–59 (Russian). MR 0050210
- Paul Lévy, Systèmes markoviens et stationnaires. Cas dénombrable, Ann. Sci. École Norm. Sup. (3) 68 (1951), 327–381 (French). MR 0047961
- Paul Lévy, Complément à l’étude des processus de Markoff, Ann. Sci. Ecole Norm. Sup. (3) 69 (1952), 203–212 (French). MR 0052718
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