Electronic Journal of Probability

A Connection between Gaussian Processes and Markov Processes

Nathalie Eisenbaum

Full-text: Open access


The Green function of a transient symmetric Markov process can be interpreted as the covariance of a centered Gaussian process. This relation leads to several fruitful identities in law. Symmetric Markov processes and their associated Gaussian process both benefit from these connections. Therefore it is of interest to characterize the associated Gaussian processes. We present here an answer to that question.

Article information

Electron. J. Probab., Volume 10 (2005), paper no. 6, 202-215.

Accepted: 4 March 2005
First available in Project Euclid: 1 June 2016

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

This work is licensed under aCreative Commons Attribution 3.0 License.


Eisenbaum, Nathalie. A Connection between Gaussian Processes and Markov Processes. Electron. J. Probab. 10 (2005), paper no. 6, 202--215. doi:10.1214/EJP.v10-238. https://projecteuclid.org/euclid.ejp/1464816807

Export citation


  • Bapat R. B. (1989), Infinite divisibility of multivariate gammadistribution and M-matrices, Sankhya,Ser.A 51, 73-78.
  • Bass R., Eisenbaum N. and Shi Z. (2000),The most visited sites of symmetric stable processes, Proba. Theory and Relat. Fields 116 (3), 391-404.
  • Dellacherie C. and Meyer P.A. (1987), Probabilités et Potentiel -Théorie du potentiel associée à une résolvante,Hermann, Paris.
  • Dynkin E. B. (1983), Local times and quantum fields. Seminar on Stochastic Processes,Birkhauser Boston 82, 69-84.
  • Eisenbaum N. (1995), Une version sans conditionnement duThéorème d'isomorphisme de Dynkin. Séminaire deProbabilités XXIX. LN1613,266-289.
  • Eisenbaum N. and Kaspi H., A characterization of the infinitely divisiblesquared Gaussian processes. To appear in the Annals of Probab.
  • Eisenbaum N., Kaspi H., Marcus M.B., Rosen J. and Shi Z. (2000), ARay-Knight theorem for symmetric Markov processes,Annals of Probab. 28,4, 1781-1796.
  • Griffiths R. C. (1984), Characterization of infinitely divisiblemultivariate gamma distributions, Jour. Multivar. Anal. 15, 12-20.
  • Le Jan Y. (1982), Dual Markovian semi-groups and processes Functional analysis in Markov processes (Katata/Kyoto, 1981),Springer, Berlin, LN923, 47-75.
  • Marcus M.B. and Rosen J.(1992),Sample path properties of thelocal times of strongly symmetric Markov processes via Gaussianprocesses, Annals of Probab..20,1603-1684.
  • Marcus M.B. and Rosen J. (2001), Gaussian processes andthe local times of symmetric Lévy processes, Lévy processes - Theory and Applications. Editors: O. Barnsdorff-Nielsen, T. Mikosch and S. Resnick,67-88.
  • Vere-Jones D. (1967), The infinite divisibility of a bivariate gammadistribution,Sankhlya, Ser.A 29, 412-422.