## The Annals of Probability

### New perspectives on Ray's theorem for the local times of diffusions

#### Abstract

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.

#### Article information

Source
Ann. Probab., Volume 31, Number 2 (2003), 882-913.

Dates
First available in Project Euclid: 24 March 2003

https://projecteuclid.org/euclid.aop/1048516539

Digital Object Identifier
doi:10.1214/aop/1048516539

Mathematical Reviews number (MathSciNet)
MR1964952

Zentralblatt MATH identifier
1038.60075

Subjects
Primary: 60J55: Local time and additive functionals 60G15: Gaussian processes
Secondary: 60G17: Sample path properties

#### Citation

Marcus, M. B.; Rosen, J. New perspectives on Ray's theorem for the local times of diffusions. Ann. Probab. 31 (2003), no. 2, 882--913. doi:10.1214/aop/1048516539. https://projecteuclid.org/euclid.aop/1048516539

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• NEW YORK, NEW YORK 10031 E-MAIL: mbmarcus@earthlink.net DEPARTMENT OF MATHEMATICS
• COLLEGE OF STATEN ISLAND, CUNY
• STATEN ISLAND, NEW YORK 10314 E-MAIL: jrosen3@earthlink.net