Electronic Communications in Probability

Squared Bessel processes of positive and negative dimension embedded in Brownian local times

Abstract

The Ray–Knight theorems show that the local time processes of various path fragments derived from a one-dimensional Brownian motion $B$ are squared Bessel processes of dimensions $0$, $2$, and $4$. It is also known that for various singular perturbations $X= |B| + \mu \ell$ of a reflecting Brownian motion $|B|$ by a multiple $\mu$ of its local time process $\ell$ at $0$, corresponding local time processes of $X$ are squared Bessel with other real dimension parameters, both positive and negative. Here, we embed squared Bessel processes of all real dimensions directly in the local time process of $B$. This is done by decomposing the path of $B$ into its excursions above and below a family of continuous random levels determined by the Harrison–Shepp construction of skew Brownian motion as the strong solution of an SDE driven by $B$. This embedding connects to Brownian local times a framework of point processes of squared Bessel excursions of negative dimension and associated stable processes, recently introduced by Forman, Pal, Rizzolo and Winkel to set up interval partition evolutions that arise in their approach to the Aldous diffusion on a space of continuum trees.

Article information

Source
Electron. Commun. Probab., Volume 23 (2018), paper no. 74, 13 pp.

Dates
Accepted: 1 October 2018
First available in Project Euclid: 17 October 2018

https://projecteuclid.org/euclid.ecp/1539763347

Digital Object Identifier
doi:10.1214/18-ECP174

Mathematical Reviews number (MathSciNet)
MR3866047

Zentralblatt MATH identifier
06964417

Citation

Pitman, Jim; Winkel, Matthias. Squared Bessel processes of positive and negative dimension embedded in Brownian local times. Electron. Commun. Probab. 23 (2018), paper no. 74, 13 pp. doi:10.1214/18-ECP174. https://projecteuclid.org/euclid.ecp/1539763347

References

• [1] K. Alexander. Excursions and local limit theorems for Bessel-like random walks. Electron. J. Probab., 16:1–44, 2011.
• [2] J. Bertoin and I. Kortchemski. Self-similar scaling limits of Markov chains on the positive integers. Ann. Appl. Probab., 26(4):2556–2595, 08 2016.
• [3] K. Burdzy and Z.-Q. Chen. Local time flow related to skew Brownian motion. Ann. Probab., 29(4):1693–1715, 2001.
• [4] K. Burdzy and H. Kaspi. Lenses in skew Brownian flow. Ann. Probab., 32(4):3085–3115, 2004.
• [5] P. Carmona, F. Petit, and M. Yor. Some extensions of the arc sine law as partial consequences of the scaling property of Brownian motion. Probab. Theory Related Fields, 100(1):1–29, 1994.
• [6] S. N. Evans. Right inverses of Lévy processes and stationary stopped local times. Probab. Theory Related Fields, 118(1):37–48, 2000.
• [7] N. Forman, G. Brito, Y. Chou, A. Forney, and C. Li. WXML final report: Chinese restaurant process. 2017.
• [8] N. Forman, S. Pal, D. Rizzolo, and M. Winkel. Diffusions on a space of interval partitions with Poisson–Dirichlet stationary distributions. arXiv:1609.06706v2, 2017.
• [9] N. Forman, S. Pal, D. Rizzolo, and M. Winkel. Interval partition evolutions with emigration related to the Aldous diffusion. arXiv:1804.01205 [math.PR], 2018.
• [10] N. Forman, S. Pal, D. Rizzolo, and M. Winkel. Uniform control of local times of spectrally positive stable processes. Ann. Appl. Probab., 28(4):2592–2634, 2018.
• [11] A. Göing-Jaeschke and M. Yor. A survey and some generalizations of Bessel processes. Bernoulli, 9(2):313–349, 2003.
• [12] J.-M. Harrison and L.-A. Shepp. On skew Brownian motion. Ann. Probab., 9(2):309–313, 1981.
• [13] K. Kawazu and S. Watanabe. Branching processes with immigration and related limit theorems. Teor. Verojatnost. i Primenen., 16:34–51, 1971.
• [14] A. Lambert. The genealogy of continuous-state branching processes with immigration. Probab. Theory Related Fields, 122(1):42–70, 2002.
• [15] J. Lamperti. Semi-stable stochastic processes. Trans. Amer. Math. Soc., 104:62–78, 1962.
• [16] J.-F. Le Gall and M. Yor. Excursions browniennes et carrés de processus de Bessel. C. R. Acad. Sci. Paris Sér. I Math., 303(3):73–76, 1986.
• [17] Y. Le Jan. Markov paths, loops and fields, volume 2026 of Lecture Notes in Mathematics. Springer, Heidelberg, 2011. Lectures from the 38th Probability Summer School held in Saint-Flour, 2008, École d’Été de Probabilités de Saint-Flour. [Saint-Flour Probability Summer School].
• [18] A. Lejay. On the constructions of the skew Brownian motion. Probab. Surveys, 3:413–466, 2006.
• [19] Z.-h. Li. Branching processes with immigration and related topics. Front. Math. China, 1(1):73–97, 2006.
• [20] T. Lupu. Poisson ensembles of loops of one-dimensional diffusions. arXiv preprint arXiv:1302.3773, 2013.
• [21] M. B. Marcus and J. Rosen. Markov processes, Gaussian processes, and local times, volume 100 of Cambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge, 2006.
• [22] S. Pal. Wright–Fisher diffusion with negative mutation rates. Ann. Probab., 41(2):503–526, 2013.
• [23] J. Pitman and M. Yor. A decomposition of Bessel bridges. Z. Wahrsch. Verw. Gebiete, 59(4):425–457, 1982.
• [24] D. Revuz and M. Yor. Continuous martingales and Brownian motion, volume 293 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]. Springer-Verlag, Berlin, third edition, 1999.
• [25] L. C. G. Rogers and D. Williams. Diffusions, Markov processes, and martingales. Vol. 2. Cambridge Mathematical Library. Cambridge University Press, Cambridge, 2000. Itô calculus, Reprint of the second (1994) edition.
• [26] T. Shiga and S. Watanabe. Bessel diffusions as a one-parameter family of diffusion processes. Z. Wahrscheinlichkeitstheorie und Verw. Gebiete, 27:37–46, 1973.
• [27] V. A. Vatutin. A critical Galton–Watson branching process with immigration. Teor. Verojatnost. i Primenen., 22(3):482–497, 1977.
• [28] V. A. Vatutin and A. M. Zubkov. Branching processes. II. J. Soviet Math., 67(6):3407–3485, 1993. Probability theory and mathematical statistics, 1.
• [29] J. Warren. Branching processes, the Ray–Knight theorem, and sticky Brownian motion. In Séminaire de Probabilités, XXXI, volume 1655 of Lecture Notes in Math., pages 1–15. Springer, Berlin, 1997.
• [30] M. Winkel. Right inverses of nonsymmetric Lévy processes. Ann. Probab., 30(1):382–415, 2002.
• [31] M. Yor. Some aspects of Brownian motion. Part I: Some special functionals. Lectures in Math., ETH Zürich, Birkhäuser, 1992.