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On a convergence of positive continuous additive functionals in terms of the smooth measures

Part of: Limit theorems Other generalizations

Published online by Cambridge University Press:  16 May 2025

Yasuhito Nishimori ,
Matsuyo Tomisaki ,
Kaneharu Tsuchida  and
Toshihiro Uemura
Yasuhito Nishimori
Affiliation:
Faculty of Education, University of Yamanashi, Yamanashi 400-8510, Japan e-mail: y.nishimori@yamanashi.ac.jp
Matsuyo Tomisaki
Affiliation:
Department of Mathematics, Nara Women’s University, Nara, 630-8506, Japan e-mail: tomisaki@cc.nara-wu.ac.jp
Kaneharu Tsuchida*
Affiliation:
Department of Mathematics, National Defense Academy, Yokosuka, 239-8686, Japan
Toshihiro Uemura
Affiliation:
Department of Mathematics, Faculty of Engineering Science, Kansai University, Osaka 564-8680, Japan e-mail: t-uemura@kansai-u.ac.jp
*
e-mail: tsuchida@nda.ac.jp
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Abstract

A compactness of the Revuz map is established in the sense that the locally uniform convergence of a sequence of positive continuous additive functionals (PCAFs) is derived in terms of their smooth measures. To this end, we first introduce a metric on the space of measures of finite energy integrals and show some structures of the metric. Then, we show the compactness and give some examples of PCAFs that the convergence holds in terms of the associated smooth measures.

Keywords

Positive continuous additive functional (PCAF)smooth measureDirichlet form

MSC classification

Primary: 31C25: Dirichlet spaces
Secondary: 60F99: None of the above, but in this section
Type
Article
Information
Canadian Journal of Mathematics , First View , pp. 1 - 30
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Copyright
© The Author(s), 2025. Published by Cambridge University Press on behalf of Canadian Mathematical Society

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Footnotes

This work was supported by JSPS KAKENHI Grant Numbers JP19H00643 (T.U.), JP22K03340 (M.T., T.U.), JP22K03427 (Y.N.).

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