Electronic Journal of Probability

A Connection between Gaussian Processes and Markov Processes

Nathalie Eisenbaum

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Abstract

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

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

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

Permanent link to this document
https://projecteuclid.org/euclid.ejp/1464816807

Digital Object Identifier
doi:10.1214/EJP.v10-238

Mathematical Reviews number (MathSciNet)
MR2120243

Zentralblatt MATH identifier
1081.60015

Rights
This work is licensed under aCreative Commons Attribution 3.0 License.

Citation

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


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References

  • 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.