Markovian loop soups: permanental processes and isomorphism theorems

Patrick J. Fitzsimmons (UCSD)
Jay S. Rosen (CUNY)

Abstract


We construct  loop soups for general Markov processes without transition densities and show that the associated permanental process is equal in distribution to the loop soup local time. This is used to establish isomorphism theorems connecting the local time of the original process with the associated permanental process. Further properties of the loop measure are studied.

Full Text: Download PDF | View PDF online (requires PDF plugin)

Pages: 1-30

Publication Date: July 5, 2014

DOI: 10.1214/EJP.v19-3255

References

  • Blumenthal, R. M.; Getoor, R. K. Markov processes and potential theory. Pure and Applied Mathematics, Vol. 29 Academic Press, New York-London 1968 x+313 pp. MR0264757
  • Bertoin, Jean. Random fragmentation and coagulation processes. Cambridge Studies in Advanced Mathematics, 102. Cambridge University Press, Cambridge, 2006. viii+280 pp. ISBN: 978-0-521-86728-3; 0-521-86728-2 MR2253162
  • Y. Chang, Multi-occupation field generates the Borel sigma-field of loops. arXiv:1309.1558
  • C. Dellacherie, and P.-A. Meyer, (1978). Probabilities et Potential. Paris: Hermann.
  • Dellacherie, Claude; Meyer, Paul-Andre. Probabilities and potential. B. Theory of martingales. Translated from the French by J. P. Wilson. North-Holland Mathematics Studies, 72. North-Holland Publishing Co., Amsterdam, 1982. xvii+463 pp. ISBN: 0-444-86526-8 MR0745449
  • Dynkin, E. B. Minimal excessive measures and functions. Trans. Amer. Math. Soc. 258 (1980), no. 1, 217--244. MR0554330
  • Dynkin, E. B. Local times and quantum fields. Seminar on stochastic processes, 1983 (Gainesville, Fla., 1983), 69--83, Progr. Probab. Statist., 7, Birkhauser Boston, Boston, MA, 1984. MR0902412
  • Dynkin, E. B. Gaussian and non-Gaussian random fields associated with Markov processes. J. Funct. Anal. 55 (1984), no. 3, 344--376. MR0734803
  • Eisenbaum, Nathalie; Kaspi, Haya. On permanental processes. Stochastic Process. Appl. 119 (2009), no. 5, 1401--1415. MR2513113
  • Fitzsimmons, Pat; Pitman, Jim; Yor, Marc. Markovian bridges: construction, Palm interpretation, and splicing. Seminar on Stochastic Processes, 1992 (Seattle, WA, 1992), 101--134, Progr. Probab., 33, Birkhauser Boston, Boston, MA, 1993. MR1278079
  • Getoor, R. K. Some remarks on continuous additive functionals. Ann. Math. Statist 38 1967 1655--1660. MR0216573
  • Le Jan, Yves. Markov loops and renormalization. Ann. Probab. 38 (2010), no. 3, 1280--1319. MR2675000
  • Le Jan, Yves. Markov paths, loops and fields. Lectures from the 38th Probability Summer School held in Saint-Flour, 2008. Lecture Notes in Mathematics, 2026. Ecole d'ete de Probabilites de Saint-Flour. [Saint-Flour Probability Summer School] Springer, Heidelberg, 2011. viii+124 pp. ISBN: 978-3-642-21215-4 MR2815763
  • Y. Le Jan, M. B. Marcus and J. Rosen, Permanental fields, loop soups and continuous additive functionals, Ann. Probab., to appear. arxiv.org/pdf/1209.1804.pdf
  • Y. Le Jan, M. B. Marcus and J. Rosen, Intersection local times, loop soups and permanental Wick powers. arxiv.org/pdf/1308.2701.pdf
  • Kingman, J. F. C. Poisson processes. Oxford Studies in Probability, 3. Oxford Science Publications. The Clarendon Press, Oxford University Press, New York, 1993. viii+104 pp. ISBN: 0-19-853693-3 MR1207584
  • Lawler, Gregory F.; Limic, Vlada. Random walk: a modern introduction. Cambridge Studies in Advanced Mathematics, 123. Cambridge University Press, Cambridge, 2010. xii+364 pp. ISBN: 978-0-521-51918-2 MR2677157
  • Lawler, Gregory F. Conformally invariant processes in the plane. Mathematical Surveys and Monographs, 114. American Mathematical Society, Providence, RI, 2005. xii+242 pp. ISBN: 0-8218-3677-3 MR2129588
  • Lawler, Gregory F.; Trujillo Ferreras, Jose A. Random walk loop soup. Trans. Amer. Math. Soc. 359 (2007), no. 2, 767--787 (electronic). MR2255196
  • Lawler, Gregory F.; Werner, Wendelin. The Brownian loop soup. Probab. Theory Related Fields 128 (2004), no. 4, 565--588. MR2045953
  • T. Lupu, Poissonian ensembles of loops of one-dimensional diffusions. arxiv.org/pdf/1302.3773.pdf
  • T. Lupu, From loop clusters of parameter 1 øver 2 to the Gaussian free field. arxiv.org/pdf/1402.0298v2.pdf
  • Marcus, Michael B.; Rosen, Jay. Markov processes, Gaussian processes, and local times. Cambridge Studies in Advanced Mathematics, 100. Cambridge University Press, Cambridge, 2006. x+620 pp. ISBN: 978-0-521-86300-1; 0-521-86300-7 MR2250510
  • Sharpe, Michael. General theory of Markov processes. Pure and Applied Mathematics, 133. Academic Press, Inc., Boston, MA, 1988. xii+419 pp. ISBN: 0-12-639060-6 MR0958914
  • K. Symanzik, Euclidean quantum field theory, In Local Quantum Theory, (R. Jost, ed.). pp. 152--226. Acad. Press, New York, (1967).
  • Sznitman, Alain-Sol. Topics in occupation times and Gaussian free fields. Zurich Lectures in Advanced Mathematics. European Mathematical Society (EMS), Zurich, 2012. viii+114 pp. ISBN: 978-3-03719-109-5 MR2932978
  • Vere-Jones, D. Alpha-permanents and their applications to multivariate gamma, negative binomial and ordinary binomial distributions. New Zealand J. Math. 26 (1997), no. 1, 125--149. MR1450811


Creative Commons License
This work is licensed under a Creative Commons Attribution 3.0 License.