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Quasi-homeomorphisms of Dirichlet forms

Published online by Cambridge University Press:  22 January 2016

Zhen-Qing Chen ,
Zhi-Ming Ma  and
Michael Röckner
Zhen-Qing Chen
Affiliation:
Department of Mathematics, University of California at, San Diege, La folla, CA 92093, USA
Zhi-Ming Ma
Affiliation:
Institute of Applied Mathematics, Academia Sinica, 100090 Beijing, China
Michael Röckner
Affiliation:
Faculty of Mathematics, University of Bielefeld, D-33501 Bielefeld, Germany
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Extending fundamental work of M. Fukushima, M. L. Silverstein, S. Carrillo Menendez, and Y. Le Jan (cf. [F71a, 80], [Si74], [Ca-Me75], [Le77]) it was recently discovered that there is a one-to-one correspondence between (equivalent classes of) all pairs of sectorial right processes and quasi-regular Dirichlet forms (see [AM91], [AM92], [AMR90], [AMR92a], [AMR92b], [MR92]). Based on the potential theory for quasi-regular Dirichlet forms, it was shown that any quasi-regular Dirichlet form on a general state space can be considered as a regular Dirichlet form on a locally compact separable metric space by “local compactification”. There are several ways to implement this local compactification. One relies on h-transformation which was mentioned in [AMR90, Remark 1.4]. A direct way using a modified Ray-Knight compactification was announced on the “5th French-German meeting: Bielefeld Encounters in Mathematics and Physics IX. Dynamics in Complex and Irregular Systems”, Bielefeld, December 16 to 21, 1991, and the “Third European Symposium on Analysis and Probability”, Paris, January 3-10, 1992, and appeared in [MR92, Chap. VI] and [AMR92b] (see also the proof of Theorem 3.7 below). One can also do this by Gelfand-transform. This way was found by the first named author independetly and announced in the “12th Seminar on Stochastic Processes”, Seattle, March 26-28, 1992. It will be discussed in Section 4.

Type
Research Article
Information
Nagoya Mathematical Journal , Volume 136 , December 1994 , pp. 1 - 15
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 1994

References

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