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



The Annals of Probability

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

M. B. Marcus and J. Rosen

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

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.

Primary Subjects: 60J55, 60G15
Secondary Subjects: 60G17
Keywords: Diffusions; Ray's theorem; Kac's lemma; symmetric Markov processes; Gaussian processes

Full-text: Open access

Links and Identifiers

Permanent link to this document: http://projecteuclid.org/euclid.aop/1048516539
Digital Object Identifier: doi:10.1214/aop/1048516539
Mathematical Reviews number (MathSciNet): MR1964952
Zentralblatt MATH identifier: 01933034

References

[1] BIANE, P. and YOR, M. (1988). Sur la loi des temps locaux browniens en un temps exponential. Seminaire de Probabilités XXII. Lecture Notes in Math. 1321 454-466. Springer, Berlin.
[2] EISENBAUM, N. (1994). Dy nkin's isomorphism theorem and the Ray-Knight theorems. Probab. Theory Related Fields 99 321-335.
Mathematical Reviews (MathSciNet): MR95d:60128
[3] EISENBAUM, N. (1995). Une version sans conditionnement du theoreme d'isomorphisme de Dy nkin. Seminaire de Probabilités XXIX. Lecture Notes in Math. 1613 266-289. Springer, Berlin.
[4] EISENBAUM, N., KASPI, H., MARCUS, M. B., ROSEN, J. and SHI, Z. (2000). A Ray-Knight theorem for sy mmetric Markov processes. Ann. Probab. 28 1781-1796.
Mathematical Reviews (MathSciNet): MR2002j:60138
Zentralblatt MATH: 01905963
[5] FITZSIMMONS, P. and PITMAN, J. (1999). Kac's moment formula and the Fey nman-Kac formula for additive functionals of a Markov process. Stochastic Process. Appl. 79 117- 134.
Mathematical Reviews (MathSciNet): MR2000a:60136
[6] MARCUS, M. B. and ROSEN, J. (1992). Sample path properties of the local times of strongly sy mmetric Markov processes via Gaussian processes. Ann. Probab. 20 1603-1684.
[7] MARCUS, M. B. and ROSEN, J. (2001). Gaussian processes and the local times of sy mmetric Levy processes. In Lévy Processes-Theory and Applications (O. Barnsdorff-Nielsen, T. Mikosch and S. Resnick, eds.) 67-88. Birkhäuser, Boston.
[8] RAY, D. (1963). Sojourn times of a diffusion processs. Illinois J. Math. 7 615-630.
[9] REVUZ, D. and YOR, M. (1999). Continuous Martingales and Brownian Motion, 3rd ed. Springer, Berlin,
Mathematical Reviews (MathSciNet): MR2000h:60050
[10] ROGERS, L. C. G. and WILLIAMS, D. (1987). Diffusions, Markov Processes and Martingales 1, 2nd ed. Wiley, New York.
[11] ROGERS, L. C. G. and WILLIAMS, D. (1994). Diffusions, Markov Processes and Martingales 2. Wiley, New York.
[12] SHARPE, M. (1988). General Theory of Markov Processes. Academic Press, New York.
Mathematical Reviews (MathSciNet): MR89m:60169
[13] SHEPPARD, P. (1985). On the Ray-Knight property of local times. J. London Math. Soc. (2) 31 377-384.
Mathematical Reviews (MathSciNet): MR87h:60140
[14] WALSH, J. B. (1978). Excursions and local times. Temps Locaux. Astérisque 52/53 159-192.
[15] WILLIAMS, D. (1974). Path decomposition and continuity of local time for one-dimensional diffusion, I. Proc. London Math. Soc. (3) 28 738-768.
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

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