Electronic Communications in Probability

Electronic Communications in Probability > Vol. 10 (2005) > Paper 1

A Resummed Branching Process Representation for a Class of Nonlinear ODEs

Francesco Morandin, Università degli Studi di Parma, Italy

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.

Full text: PDF

Pages: 1-6

Published on: February 24, 2005

Bibliography

K.B. Athreya P.E. Ney. Branching processes. Die Grundlehren der mathematischen Wissenschaften, Band 196 Springer-Verlag, New York-Heidelberg, 1972. xi+287 pp. Math. Review (51 #9242)
T.E. Harris. The theory of branching processes. Die Grundlehren der Mathematischen Wissenschaften, Band 119 Springer-Verlag, Berlin; Prentice-Hall, Inc., Englewood Cliffs, N.J. 1963. xiv+230 pp. Math. Review (29 #664)
Y. Le Jan and A.S. Sznitman. Stochastic cascades and 3-dimensional Navier-Stokes equations. Probab. Theory Related Fields 109 (1997), no 3, 343-366. Math. Review 98j:35144

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Electronic Communications in Probability. ISSN: 1083-589X