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.
Article PDF
Similar content being viewed by others
References
Aldous, D.: The continuum random tree. I. Ann. Probab.19, 1–28 (1991)
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
Aldous, D.: The continuum random tree. III. Ann. Probab.21, 248–289 (1993)
Blumenthal, R.M.: Excursions of Markov processes. Boston: Birkhäuser 1992
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
Dawson, D.A., Perkins, E.A.: Historical processes, Memoirs of AMS, No 45493 (1991)
Durrett, R., Kesten, H., Waymire, E.: On weighted heights of random trees. J. Theoret. Probab.4, 223–237 (1991)
Dynkin, E.B.: Representation for functionals of superprocesses by multiple stochastic integrals, with applications to self-intersection local times. Astérisque157–158, 147–171 (1988)
Dynkin, E.B.: Superprocesses and their linear additive functionals. Trans. Amer. Math. Soc.314, 255–282 (1989).
Dynkin, E.B.: Three classes of infinite dimensional diffusions. J. Funct. Anal.86, 75–110 (1989)
Dynkin, E.B.: A probabilistic approach to one class of nonlinear differential equations. Probab. Theory Rel. Fields89, 89–115 (1991)
Dynkin, E.B.: Branching particle systems and superprocesses. Ann. Probab.19, 1157–1194 (1991)
Dynkin, E.B.: Path processes and historical processes. Probab. Theory. Rel. Fields90, 89–115 (1991)
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.
Dynkin, E.B.: Superdiffusions and parabolic nonlinear differential equations. Ann. Probab.20, 942–962 (1992).
Dynkin, E.B.: On regularity of superprocesses. Probab. Theory Rel. Fields95, 263–281 (1993)
Dynkin, E.B.: Superprocesses and partial differential equations. Ann. Probab.21, 1185–1262 (1993)
Dynkin, E.B.: An introduction to branching measure-valued processes, CRM Monograph Series, Vol. 6. Providence, RI: American Mathematical Society 1994
Dynkin, E.B., Getoor, R.K.: Additive functionals and Entrance Laws. J. Funct. Anal.62, 221–265 (1985)
Dynkin, E.B., Kuznetsov, S.E., Skorokhod, A.V.: Branching measure-valued processes. Probab. Theory Rel. Fields99, 55–96 (1994)
Feller, W.: An introduction to probability theory and its applications, Vol. II, New York London Sidney: Wiley 1966
Fitzsimmons, P.J.: Construction and regularity of measure-valued Markov branching processes. Israel J. Math.64, 337–361 (1988)
Harris, T.E.: First passage and recurrence distributions. Trans. Amer. Math. Soc.73, 471–486 (1956)
Itô, K., McKean, H.: Diffusion processes and their sample paths. Berlin Heidelberg New York: Springer 1965
Kesten, H.: Branching random walk with a critical branching part, Preprint 1994
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
Le Gall, J.F.: Brownian excursions, trees and measure-valued branching processes. Ann. Probab.19, 1399–1439 (1991)
Le Gall, J.F.: A class of path-valued Markov processes and its applications to superprocesses. Probab. Theory Relat. Fields95, 25–46 (1993)
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)
Le Gall, J.F.: Hitting probabilities and potential theory for the Brownian path-valued process. Ann. Institut Fourier44 (to appear)
Le Gall, J.F.: The uniform random tree in a Brownian excursion. Probab. Theory Rel. Fields96, 369–383 (1993)
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
Le Gall, J.F.: The Brownian snake and solutions of Δu=u 2 in a domain. Perprint 1994
Revuz, D., Yor, M.: Continuous martingales and Brownian motion. Berlin Heidelberg New York. Springer 1991
Rogers, L.C.G., Williams, D.: Diffusions, Markov processes and martingales. Vol. 2: Itô calculus. New York: Wiley 1987
Author information
Authors and Affiliations
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