Electronic Communications in Probability

Explicit formula for the density of local times of Markov jump processes

Ruojun Huang, Daniel Kious, Vladas Sidoravicius, and Pierre Tarrès

Full-text: Open access

Abstract

In this note we show a simple formula for the joint density of local times, last exit tree and cycling numbers of continuous-time Markov chains on finite graphs, which involves the modified Bessel function of the first type.

Article information

Source
Electron. Commun. Probab., Volume 23 (2018), paper no. 90, 7 pp.

Dates
Received: 26 March 2018
Accepted: 13 November 2018
First available in Project Euclid: 15 December 2018

Permanent link to this document
https://projecteuclid.org/euclid.ecp/1544843113

Digital Object Identifier
doi:10.1214/18-ECP194

Mathematical Reviews number (MathSciNet)
MR3896828

Zentralblatt MATH identifier
07023476

Subjects
Primary: 60J55: Local time and additive functionals 60J75: Jump processes

Keywords
Markov jump process density of local times last exit trees cycling numbers modified Bessel function

Rights
Creative Commons Attribution 4.0 International License.

Citation

Huang, Ruojun; Kious, Daniel; Sidoravicius, Vladas; Tarrès, Pierre. Explicit formula for the density of local times of Markov jump processes. Electron. Commun. Probab. 23 (2018), paper no. 90, 7 pp. doi:10.1214/18-ECP194. https://projecteuclid.org/euclid.ecp/1544843113


Export citation

References

  • [1] Bogacheva, L. and Ratanov, N. (2011). Occupation time distributions for the telegraph process. Stochastic Processes and their Applications, 121, 1816–1844.
  • [2] Brydges, D., van der Hofstad, R. and König, W. (2007). Joint density of the local times of continuous-time Markov chains. Ann. Probab. 35(4), 1307–1332.
  • [3] Keane, M. S. and Rolles, S. W. W. (2000). Edge-reinforced random walk on finite graphs. Infinite dimensional stochastic analysis (Amsterdam, 1999), Verh. Afd. Natuurkd. 1. Reeks. K. Ned. Akad. Wet.. 52, 217–234.
  • [4] Kovchegov, Y., Meredith, N. and Nir, E. (2010). Occupation times and Bessel densities. Statistics and Probability Letters. 80(2),104–110.
  • [5] Le Jan, Y. (2011). Markov paths, loops and fields. Lectures from the 38th Probability Summer School held in Saint-Flour, 2008. Lecture Notes in Mathematics, 2026, Springern Heidelberg.
  • [6] Le Jan, Y. (2018). On Markovian random networks. Preprint. arXiv:1802.01032.
  • [7] Luttinger, J. M. (1983). The asymptotic evaluation of a class of path integrals. II. J. Math. Phys. 24, 2070–2073.
  • [8] March, P. and Sznitman, A.-S. (1987). Some connections between excursion theory and the discrete Schrödinger equation with random potentials. Probab. Theory Related Fields. 75(1), 11–53.
  • [9] Marcus, M. B. and Rosen, J. (2006). Markov processes, Gaussian processes, and local times. Cambridge University Press, Cambridge studies in advanced mathematics 100, Cambridge 2006, x + 620 pp.
  • [10] Merkl, F. and Rolles, S. W. W. and Tarrès, P. (2016). Convergence of vertex-reinforced jump process to an extension of the supersymmetric hyperbolic nonlinear sigma model preprint arXiv:1612.05409. To appear, Probab. Theory Relat. Fields.
  • [11] Pedler, P.J. (1971). Occupation times for two-state Markov chains. J. Appl. Prob. 8, 381–390.