Skip to main content

Advertisement

SpringerLink
Log in
Menu
Find a journal Publish with us
Search
Cart
  1. Home
  2. Probability Theory and Related Fields
  3. Article
Cyclically stationary Brownian local time processes
Download PDF
Download PDF
  • Published: November 1996

Cyclically stationary Brownian local time processes

  • Jim Pitman1 

Probability Theory and Related Fields volume 106, pages 299–329 (1996)Cite this article

Summary.

Local time processes parameterized by a circle, defined by the occupation density up to time T of Brownian motion with constant drift on the circle, are studied for various random times T. While such processes are typically non-Markovian, their Laplace functionals are expressed by series formulae related to similar formulae for the Markovian local time processes subject to the Ray–Knight theorems for BM on the line, and for squares of Bessel processes and their bridges. For T the time that BM on the circle first returns to its starting point after a complete loop around the circle, the local time process is cyclically stationary, with same two-dimensional distributions, but not the same three-dimensional distributions, as the sum of squares of two i.i.d. cyclically stationary Gaussian processes. This local time process is the infinitely divisible sum of a Poisson point process of local time processes derived from Brownian excursions. The corresponding intensity measure on path space, and similar Lévy measures derived from squares of Bessel processes, are described in terms of a 4-dimensional Bessel bridge by Williams’ decomposition of Itô’s law of Brownian excursions.

Download to read the full article text

Working on a manuscript?

Avoid the common mistakes

Author information

Authors and Affiliations

  1. Department of Statistics, University of California, 367 Evans Hall # 3860, Berkeley, CA 94720-3860, USA, , , , , , US

    Jim Pitman

Authors
  1. Jim Pitman
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

Received: 28 June 1995

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Pitman, J. Cyclically stationary Brownian local time processes. Probab Theory Relat Fields 106, 299–329 (1996). https://doi.org/10.1007/s004400050066

Download citation

  • Issue Date: November 1996

  • DOI: https://doi.org/10.1007/s004400050066

Share this article

Anyone you share the following link with will be able to read this content:

Sorry, a shareable link is not currently available for this article.

Provided by the Springer Nature SharedIt content-sharing initiative

  • Mathematics Subject classification (1991):  60J55
  • 60J65
  • 60G10
Download PDF

Working on a manuscript?

Avoid the common mistakes

Advertisement

Search

Navigation

  • Find a journal
  • Publish with us

Discover content

  • Journals A-Z
  • Books A-Z

Publish with us

  • Publish your research
  • Open access publishing

Products and services

  • Our products
  • Librarians
  • Societies
  • Partners and advertisers

Our imprints

  • Springer
  • Nature Portfolio
  • BMC
  • Palgrave Macmillan
  • Apress
  • Your US state privacy rights
  • Accessibility statement
  • Terms and conditions
  • Privacy policy
  • Help and support

167.114.118.210

Not affiliated

Springer Nature

© 2023 Springer Nature