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April, 1975 A Characterization of the Kernel $\lim{_\lambda \downarrow 0}V_\lambda$ for Sub-Markovian Resolvents $(V_\lambda)^1$
J. C. Taylor
Ann. Probab. 3(2): 355-357 (April, 1975). DOI: 10.1214/aop/1176996407

Abstract

Let $(V_\lambda)$ be a sub-Markovian resolvent of kernels $V_\lambda$ on a measurable space $(E, \mathscr{E})$. Assume that $V = \lim_{\lambda \downarrow 0}V_\lambda$ is a proper kernel. The proper kernels $V$ on $(E, \mathscr{E})$ that are of the form $V = \lim_{\lambda \downarrow 0}V_\lambda, (V_\lambda)$ a sub-Markovian resolvent of kernels on $(E, \mathscr{E})$, are proved to be precisely those proper kernels $V$ which satisfy the complete maximum principle and for which the following condition holds: there exists an increasing sequence $(A_n) \subset \mathscr{E}$ with $\mathbf{\bigcup}_n A_n = E$ such that (i) $V1_{A_n} < \infty$ for all $n$; and (ii) if $f \in \mathscr{E}^+$ and $Vf < \infty$ then $\inf_nR_{\mathscr{C} A_n} Vf < \infty$, where $R_Bu = \inf \{v \text{supermedian} \mid u \geqq v \text{on} B\}$.

Citation

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J. C. Taylor. "A Characterization of the Kernel $\lim{_\lambda \downarrow 0}V_\lambda$ for Sub-Markovian Resolvents $(V_\lambda)^1$." Ann. Probab. 3 (2) 355 - 357, April, 1975. https://doi.org/10.1214/aop/1176996407

Information

Published: April, 1975
First available in Project Euclid: 19 April 2007

zbMATH: 0303.60068
MathSciNet: MR373025
Digital Object Identifier: 10.1214/aop/1176996407

Subjects:
Primary: 60J35
Secondary: 47D05

Keywords: maximum principle , potentials , Ray processes , resolvent , sub-Markovian resolvent

Rights: Copyright © 1975 Institute of Mathematical Statistics

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Vol.3 • No. 2 • April, 1975
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