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Markov snakes and superprocesses
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  • Published: December 1995

Markov snakes and superprocesses

  • E. B. Dynkin1 &
  • S. E. Kuznetsov2 

Probability Theory and Related Fields volume 103, pages 433–473 (1995)Cite this article

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  • 7 Citations

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Summary

We suggest the name Markov snakes for a class of path-valued Markov processes introduced recently by J.-F. Le Gall in connection with the theory of branching measure-valued processes. Le Gall applied this class to investigate path properties of superdiffusions and to approach probabilistically partial differential equations involving a nonlinear operator Δv−v 2. We establish an isomorphism theorem which allows to translate results on continuous superprocesses into the language of Markov snakes and vice versa. By using this theorem, we get limit theorems for discrete Markov snakes.

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References

  1. Aldous, D.: The continuum random tree. I. Ann. Probab.19, 1–28 (1991)

    Google Scholar 

  2. Aldous, D.: The continuum random tree. II. In: Barlow, M.T., Bingham, N.H. (eds.), Stochastic analysis, pp. 23–70. Cambridge. Cambridge University Press 1991

    Google Scholar 

  3. Aldous, D.: The continuum random tree. III. Ann. Probab.21, 248–289 (1993)

    Google Scholar 

  4. Blumenthal, R.M.: Excursions of Markov processes. Boston: Birkhäuser 1992

    Google Scholar 

  5. Dawson, D.A.: Measure-valued Markov processes. École d'Été de Probabilités de Saint Flour, 1991 (Lect. Notes in Math. Vol. 1541) Berlin Heidelberg New York: Springer 1993

    Google Scholar 

  6. Dawson, D.A., Perkins, E.A.: Historical processes, Memoirs of AMS, No 45493 (1991)

  7. Durrett, R., Kesten, H., Waymire, E.: On weighted heights of random trees. J. Theoret. Probab.4, 223–237 (1991)

    Google Scholar 

  8. Dynkin, E.B.: Representation for functionals of superprocesses by multiple stochastic integrals, with applications to self-intersection local times. Astérisque157–158, 147–171 (1988)

    Google Scholar 

  9. Dynkin, E.B.: Superprocesses and their linear additive functionals. Trans. Amer. Math. Soc.314, 255–282 (1989).

    Google Scholar 

  10. Dynkin, E.B.: Three classes of infinite dimensional diffusions. J. Funct. Anal.86, 75–110 (1989)

    Google Scholar 

  11. Dynkin, E.B.: A probabilistic approach to one class of nonlinear differential equations. Probab. Theory Rel. Fields89, 89–115 (1991)

    Google Scholar 

  12. Dynkin, E.B.: Branching particle systems and superprocesses. Ann. Probab.19, 1157–1194 (1991)

    Google Scholar 

  13. Dynkin, E.B.: Path processes and historical processes. Probab. Theory. Rel. Fields90, 89–115 (1991)

    Google Scholar 

  14. Dynkin, E.B.: Additive functionals of superdiffusion processes, Random walks, Brownian Motion and Interacting Particle Systems Progress in Probability (Rick Durrett, Harry Kesten, eds.), vol. 28. Birkhäuser, Boston, Basel, Berlin, 1991.

    Google Scholar 

  15. Dynkin, E.B.: Superdiffusions and parabolic nonlinear differential equations. Ann. Probab.20, 942–962 (1992).

    Google Scholar 

  16. Dynkin, E.B.: On regularity of superprocesses. Probab. Theory Rel. Fields95, 263–281 (1993)

    Google Scholar 

  17. Dynkin, E.B.: Superprocesses and partial differential equations. Ann. Probab.21, 1185–1262 (1993)

    Google Scholar 

  18. Dynkin, E.B.: An introduction to branching measure-valued processes, CRM Monograph Series, Vol. 6. Providence, RI: American Mathematical Society 1994

    Google Scholar 

  19. Dynkin, E.B., Getoor, R.K.: Additive functionals and Entrance Laws. J. Funct. Anal.62, 221–265 (1985)

    Google Scholar 

  20. Dynkin, E.B., Kuznetsov, S.E., Skorokhod, A.V.: Branching measure-valued processes. Probab. Theory Rel. Fields99, 55–96 (1994)

    Google Scholar 

  21. Feller, W.: An introduction to probability theory and its applications, Vol. II, New York London Sidney: Wiley 1966

    Google Scholar 

  22. Fitzsimmons, P.J.: Construction and regularity of measure-valued Markov branching processes. Israel J. Math.64, 337–361 (1988)

    Google Scholar 

  23. Harris, T.E.: First passage and recurrence distributions. Trans. Amer. Math. Soc.73, 471–486 (1956)

    Google Scholar 

  24. Itô, K., McKean, H.: Diffusion processes and their sample paths. Berlin Heidelberg New York: Springer 1965

    Google Scholar 

  25. Kesten, H.: Branching random walk with a critical branching part, Preprint 1994

  26. Kuznetsov, S.E.: Regularity properties of a supercritical superprocess. In: Freidlin (ed.), The Dynkin Festschrift, Markov processes and their Applications, pp. 221–236. Boston: Birkhäuser 1994

    Google Scholar 

  27. Le Gall, J.F.: Brownian excursions, trees and measure-valued branching processes. Ann. Probab.19, 1399–1439 (1991)

    Google Scholar 

  28. Le Gall, J.F.: A class of path-valued Markov processes and its applications to superprocesses. Probab. Theory Relat. Fields95, 25–46 (1993)

    Google Scholar 

  29. Le Gall, J.F.: Les solutions positives de Δu=u 2 dans le disque unité. C.R. Acad. Sci. Paris, Sér. I317, 873–878 (1993)

    Google Scholar 

  30. Le Gall, J.F.: Hitting probabilities and potential theory for the Brownian path-valued process. Ann. Institut Fourier44 (to appear)

  31. Le Gall, J.F.: The uniform random tree in a Brownian excursion. Probab. Theory Rel. Fields96, 369–383 (1993)

    Google Scholar 

  32. Le Gall, J.F.: A path-valued Markov process and its connections with partial differential equations. Proc. 1st European congress of mathematicians. Boston: Birkhäuser 1993

    Google Scholar 

  33. Le Gall, J.F.: The Brownian snake and solutions of Δu=u 2 in a domain. Perprint 1994

  34. Revuz, D., Yor, M.: Continuous martingales and Brownian motion. Berlin Heidelberg New York. Springer 1991

    Google Scholar 

  35. Rogers, L.C.G., Williams, D.: Diffusions, Markov processes and martingales. Vol. 2: Itô calculus. New York: Wiley 1987

    Google Scholar 

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Author information

Authors and Affiliations

  1. Department of Mathematics, Cornell University, White Hall, 14853-7901, Ithaca, NY, USA

    E. B. Dynkin

  2. Central Econ.-Math. Institute, Russian Academy of Sciences, 32 Krasikowa, 117418, Moscow, Russia

    S. E. Kuznetsov

Authors
  1. E. B. Dynkin
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  2. S. E. Kuznetsov
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Additional information

Partially supported by National Science Foundation Grant DMS-9301315 and by The US Army Research Office through the Mathematical Sciences Institute at Cornell University

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Dynkin, E.B., Kuznetsov, S.E. Markov snakes and superprocesses. Probab. Th. Rel. Fields 103, 433–473 (1995). https://doi.org/10.1007/BF01246334

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  • Received: 05 May 1994

  • Revised: 23 November 1994

  • Issue Date: December 1995

  • DOI: https://doi.org/10.1007/BF01246334

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Mathematics Subject Classification (1991)

  • 60J80
  • 60J55
  • 60J25
  • 60J65
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