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Transactions of the American Mathematical Society

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Minimal excessive measures and functions


Author: E. B. Dynkin
Journal: Trans. Amer. Math. Soc. 258 (1980), 217-244
MSC: Primary 60J50; Secondary 28D99, 47D07
DOI: https://doi.org/10.1090/S0002-9947-1980-0554330-5
MathSciNet review: 554330
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Abstract: Let H be a class of measures or functions. An element h of H is minimal if the relation $ h\, = \,{h_1}\, + \,{h_2}$, $ {h_1}$, $ {h_2} \in H$ implies that $ {h_1}$, $ {h_2}$ are proportional to h. We give a limit procedure for computing minimal excessive measures for an arbitrary Markov semigroup $ {T_t}$ in a standard Borel space E. Analogous results for excessive functions are obtained assuming that an excessive measure $ \gamma $ on E exists such that $ {T_t}f\, = 0$ if $ f\, = \,0$ $ \gamma $-a.e. In the Appendix, we prove that each excessive element can be decomposed into minimal elements and that such a decomposition is unique.


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Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1980-0554330-5
Keywords: Martin boundary, excessive measures and functions, entrance and exit laws, decomposition into minimal (extreme) elements
Article copyright: © Copyright 1980 American Mathematical Society

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