Small Diffusion and Fast Dying Out Asymptotics for Superprocesses as Non-Hamiltonian Quasiclassics for Evolution Equations

Vassili N. Kolokoltsov (Nottingham Trent University)


The small diffusion and fast dying out asymptotics is calculated for nonlinear equations of a class of superprocesses on manifolds, and the corresponding logarithmic limit of the solution is shown to be given by a solution of a certain problem of calculus of variations with a non-additive (and non-integral) functional.

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Pages: 1-16

Publication Date: August 15, 2001

DOI: 10.1214/EJP.v6-94


  1. S. Albeverio, A. Hilbert and V.N. Kolokoltsov, Sur le comportement asymptotique du noyau associé á une diffusion dégénéré, C.R. Math. Rep. Acad. Sci. Canada 22, (2000) 151-159. Math. Reviews number not available
  2. A. Etheridge. An Introduction to Superprocesses. University Lecture Series 20, AMS, Providence, Rhode Island 2000. Math. Reviews number not available
  3. E. Joergensen, Construction of the Brownian motion and the Ornstein-Uhlenbeck Process in a Riemannian manifold, Z. Wahrscheinlichkeitstheorie Verw. Gebiete 44, (1978) 71-87. 80c:60094
  4. A. Klenke, Clustering and invariant measures for spatial branching models with infinite variance, Ann. Prob. 26, (1998) 1057-1087. 99i:60160
  5. V. Kolokoltsov, Semiclassical Analysis for Diffusions and Stochastic Processes. Springer Lecture Notes in Mathematics, v. 1724, 2000. 2001f:58073
  6. V. Kolokoltsov. On Linear, Additive, and Homogeneous Operators. In: V. Maslov and S. Samborski (Eds.) Idempotent Analysis. Advances in Soviet Mathematics 13, (1992) 87-101. MR93k:47065
  7. V.N. Kolokoltsov and V.P. Maslov. Idempotent Analysis and its Applications. Kluwer Academic 1997. MR97d:49031
  8. V.P. Maslov. Perturbation Theory and Asymptotical Methods. Moscow, MGU 1965 (in Russian). French translation Paris, Dunod, 1972. Math. Reviews number not available.
  9. V.P. Maslov. Complex Markov chains and Feynman path integral. Moscow, Nauka, 1976 (in Russian). MR57:18574
  10. A. Puchalskii. Large deviation of semimartingales: a maxingale approach. Stochastics and Stochastics Reports 61, (1997) 141-243. MR98h:60033
  11. J. Smoller. Shock Waves and Reaction-Diffusion Equations. Springer-Verlag, 1983. MR84d:35002
  12. S.R. Varadhan. On the behavior of the fundamental solution of the heat equation with variable coefficients. Comm. Pure Appl. Math. 20, (1967) 431-455. MR34:8001
  13. S.R. Varadhan. Diffusion Processes in a Small Time Interval. Comm. Pure Appl. Math. 20, (1967) 659-685. MR36:970
  14. S.R. Varadhan. Scaling limits for interacting diffusions. Comm. Math. Phys. 135, (1991) 331-353. MR92e:60195

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