Isomorphism Theorems: Markov Processes, Gaussian Processes and Beyond

  • Jay RosenEmail author
Part of the Lecture Notes in Mathematics book series (LNM, volume 2143)


We begin with some introductory material on Gaussian processes and Markov processes and then study in turn the isomorphism theorems of Dynkin, Eisenbaum and the generalized second Ray-Knight theorem. In each case we give a proof and a sample application. We then introduce loop soups and permanental processes, the ingredients we use to develop an isomorphism theorem for non-symmetric Markov processes. Along the way we gain new insight into the reason that Gaussian processes appear in the isomorphism theorems in the symmetric case. Having developed some general material on Poisson processes in Sect. 7, we make use of it in the next two sections. Section 8 contains an excursion theory proof of the generalized second Ray-Knight theorem, and Sect. 9 explains a similar theorem for random interlacements. Up till this point proofs use the method of moments. In Sect. 10 we explain how to prove these theorems using the method of Laplace transforms.


Dynkin isomorphism Gaussian processes Local times Loop soups Markov processes Random interlacements Ray-Knight theorems 



Research was supported by grants from the National Science Foundation.


  1. 1.
    R.F. Bass, N. Eisenbaum, Z. Shi, The most visited sites of symmetric stable processes. Probab. Theory Relat. Fields 116, 391–404 (2000)zbMATHMathSciNetCrossRefGoogle Scholar
  2. 2.
    J. Bertoin, Random Fragmentation and Coagulation Processes (Cambridge University Press, Cambridge, 2006)zbMATHCrossRefGoogle Scholar
  3. 3.
    R.M. Blumenthal, Excursions of Markov Processes (Birkhauser, Boston, 1992)zbMATHCrossRefGoogle Scholar
  4. 4.
    C. Dellacherie, P.-A. Meyer, Probabilities et Potential, Chapitres XII a XVI (Hermann, Paris, 1987)Google Scholar
  5. 5.
    C. Dellacherie, B. Maisonneuve, P.-A. Meyer, Probabilities et Potential, Chapitres XVII a XXIV (Hermann, Paris, 1992)Google Scholar
  6. 6.
    J. Ding, J. Lee, Y. Peres, Cover times, blanket times, and majorizing measures. Ann. Math. 175(3), 1409–1471 (2012); conference version at STOC (2011)Google Scholar
  7. 7.
    J. Ding, Asymptotics of cover times via Gaussian free fields: Bounded-degree graphs and general trees. Ann. Probab. 42(2), 464–496Google Scholar
  8. 8.
    J. Ding, On cover times for 2D lattices. Electron. J. Probab. 17(45), 1–18 (2013)Google Scholar
  9. 9.
    J. Ding, O. Zeitouni, A sharp estimate for cover times on binary trees. Stoch. Process. Appl. 122(5), 2117–2133 (2012)zbMATHMathSciNetCrossRefGoogle Scholar
  10. 10.
    J. Ding, J. Lee, Y. Peres, Cover times, blanket times, and majorizing measures. Ann. Math. 175(3), 1409–1471 (2012); conference version at STOC (2011)Google Scholar
  11. 11.
    E.B. Dynkin, Local times and quantum fields, in Seminar on Stochastic Processes. Progress in Probability, vol. 7 (Birkhäuser, Boston, 1983), pp. 64–84Google Scholar
  12. 12.
    E.B. Dynkin, Gaussian and non-Gaussian random fields associated with Markov processes. J. Funct. Anal. 55, 344–376 (1984)zbMATHMathSciNetCrossRefGoogle Scholar
  13. 13.
    N. Eisenbaum, H. Kaspi, On permanental processes. Stoch. Process. Appl. 119, 1401–1415 (2009)zbMATHMathSciNetCrossRefGoogle Scholar
  14. 14.
    N. Eisenbaum, H. Kaspi, M. Marcus, J. Rosen, Z. Shi, A Ray-Knight theorem for symmetric Markov processes. Ann. Probab. 28, 1781–1796 (2000)zbMATHMathSciNetCrossRefGoogle Scholar
  15. 15.
    P. Fitzsimmons, J. Rosen, Markovian loop soups: Permanental processes and isomorphism theorems. Electron. J. Probab. 19(60), 1–30 (2014).
  16. 16.
    P. Fitzsimmons, J. Pitman, M. Yor, Markovian bridges: Construction, Palm interpretation, and splicing. Seminar on Stochastic Processes, ed. by E. Cinlar, K.L. Chung, M.J. Sharpe (Birkhäuser, Boston, 1993), pp. 101–134Google Scholar
  17. 17.
    J. F.C. Kingman, Poisson Processes. Oxford Studies in Probability (Clarendon Press, Oxford, 2002)Google Scholar
  18. 18.
    G. Lawler, V. Limic, Random Walk: A Modern Introduction (Cambridge University Press, Cambridge, 2009)Google Scholar
  19. 19.
    G. Lawler, J. Trujillo Ferreis, Random walk loop soup. TAMS 359, 565–588 (2007)CrossRefGoogle Scholar
  20. 20.
    G. Lawler, W. Werner, The Brownian loop soup. Probab. Theory Relat. Fields 44, 197–217 (2004)MathSciNetGoogle Scholar
  21. 21.
    Y. Le Jan, Markov loops and renormalization. Ann. Probab. 38, 1280–1319 (2010)zbMATHMathSciNetCrossRefGoogle Scholar
  22. 22.
    Y. Le Jan, Markov paths, loops and fields, in École d’Été de Probabilités de Saint-Flour XXXVIII - 2008. Lecture Notes in Mathematics, vol. 2026 (Springer, Berlin, 2011)Google Scholar
  23. 23.
    Y. Le Jan, M.B. Marcus, J. Rosen, Permanental fields, loop soups and continuous additive functionals. Ann. Probab. 43, 44–84 (2015).
  24. 24.
    Y. Le Jan, M.B. Marcus, J. Rosen, Intersection local times, loop soups and permanental Wick powers. (2013)
  25. 25.
    M.B. Marcus, J. Rosen, Markov Processes, Gaussian Processes and Local Times (Cambridge University Press, Cambridge, 2006)zbMATHCrossRefGoogle Scholar
  26. 26.
    M.B. Marcus, J. Rosen, A sufficient condition for the continuity of permanental processes with applications to local times of Markov processes. Ann. Probab. 41, 671–698 (2013).
  27. 27.
    M.B. Marcus, J. Rosen, Continuity conditions for a class of second order permanental chaoses, in High Dimensional Probability VI: The Banff Volume. Progress in Probability, vol. 66 (Springer, Basel, 2013), pp. 229–245Google Scholar
  28. 28.
    L.C.G. Rogers, D. Williams, Diffusions, Markov Processes, and Martingales. Ito Calculus, vol. 2 (Cambridge University Press, Cambridge, 2000)Google Scholar
  29. 29.
    A.-S. Sznitman, An isomorphism theorem for random interlacements, Electron. Commun. Probab. 17(9), 19 (2012)Google Scholar
  30. 30.
    D. Vere-Jones, Alpha-permanents. N. Z. J. Math. 26, 125–149 (1997)zbMATHMathSciNetGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  1. 1.Department of Mathematics, College of Staten IslandCity University of New YorkStaten IslandUSA

Personalised recommendations