Electronic Communications in Probability

A Resummed Branching Process Representation for a Class of Nonlinear ODEs

Francesco Morandin

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Abstract

We study some probabilistic representations, based on branching processes, of a simple nonlinear differential equation, i.e. $u'=\lambda u(au^R-1)$. The first approach is basically the same used by Le Jan and Sznitman for 3-d Navier-Stokes equations, which need small initial data to work. In our much simpler setting we are able to make this precise, finding all the cases where their method fails to give the solution. The second approach is based on a resummed representation, which we can prove to give all the solutions of the problem, even those with large initial data.

Article information

Source
Electron. Commun. Probab. Volume 10 (2005), paper no. 1, 1-6.

Dates
Accepted: 24 February 2005
First available in Project Euclid: 4 June 2016

Permanent link to this document
http://projecteuclid.org/euclid.ecp/1465058066

Digital Object Identifier
doi:10.1214/ECP.v10-1126

Mathematical Reviews number (MathSciNet)
MR2119148

Zentralblatt MATH identifier
1060.60085

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

Citation

Morandin, Francesco. A Resummed Branching Process Representation for a Class of Nonlinear ODEs. Electron. Commun. Probab. 10 (2005), paper no. 1, 1--6. doi:10.1214/ECP.v10-1126. http://projecteuclid.org/euclid.ecp/1465058066.


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References

  • K.B. Athreya P.E. Ney. Branching processes. Die Grundlehren der mathematischen\n Wissenschaften, Band 196 Springer-Verlag, New York-Heidelberg, 1972. xi+287 pp.
  • T.E. Harris. The theory of branching processes. Die Grundlehren der Mathematischen\n Wissenschaften, Band 119 Springer-Verlag, Berlin; Prentice-Hall, Inc., Englewood Cliffs,\n N.J. 1963. xiv+230 pp.
  • Y. Le Jan and A.S. Sznitman. Stochastic cascades and 3-dimensional Navier-Stokes\n equations. Probab. Theory Related Fields 109 (1997), no 3, 343-366.