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April 2003 New perspectives on Ray's theorem for the local times of diffusions
M. B. Marcus, J. Rosen
Ann. Probab. 31(2): 882-913 (April 2003). DOI: 10.1214/aop/1048516539


A new global isomorphism theorem is obtained that expresses the local times of transient regular diffusions under $P^{x,y}$, in terms of related Gaussian processes. This theorem immediately gives an explicit description of the local times of diffusions in terms of $0$th order squared Bessel processes similar to that of Eisenbaum and Ray's classical description in terms of certain randomized fourth order squared Bessel processes. The proofs given are very simple. They depend on a new version of Kac's lemma for $h$-transformed Markov processes and employ little more than standard linear algebra. The global isomorphism theorem leads to an elementary proof of the Markov property of the local times of diffusions and to other recent results about the local times of general strongly symmetric Markov processes. The new version of Kac's lemma gives simple, short proofs of Dynkin's isomorphism theorem and an unconditioned isomorphism theorem due to Eisenbaum.


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M. B. Marcus. J. Rosen. "New perspectives on Ray's theorem for the local times of diffusions." Ann. Probab. 31 (2) 882 - 913, April 2003.


Published: April 2003
First available in Project Euclid: 24 March 2003

zbMATH: 1038.60075
MathSciNet: MR1964952
Digital Object Identifier: 10.1214/aop/1048516539

Primary: 60G15 , 60J55
Secondary: 60G17

Keywords: Diffusions , Gaussian processes , Kac's lemma , Ray's theorem , symmetric Markov processes

Rights: Copyright © 2003 Institute of Mathematical Statistics


Vol.31 • No. 2 • April 2003
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