A probabilistic representation for the solutions to some non-linear PDEs using pruned branching trees

Abstract The solutions to a large class of non-linear parabolic PDE s are given in terms of expectations of suitable functionals of a tree of branching particles. A sufficient, and in some cases necessary, condition is given for the integrability of the stochastic representation, using a comparison scalar PDE . In cases where the representation fails to be integrable, a sequence of pruned trees is constructed, producing approximate stochastic representations that in some cases converge, globally in time, to the solution of the original PDE .

[1]  H. Bunke,et al.  T. E. Harris, The Theory of Branching Processes (Die Grundlehren der mathematischen Wissenschaften, Band 119). XVI + 230 S. m. 6 Fig. Berlin/Göttingen/Heidelberg 1963. Springer‐Verlag. Preis geb. DM 36,— , 1965 .

[2]  Anton Wakolbinger,et al.  Length of Galton-Watson trees and blow-up of semilinear systems , 1998 .

[3]  Uniqueness for a class of one-dimensional stochastic PDEs using moment duality , 1999, math/9903194.

[4]  H. McKean.  Application of brownian motion to the equation of kolmogorov-petrovskii-piskunov , 1975 .

[5]  A. Skorokhod.  Branching Diffusion Processes , 1964 .

[6]  Andrew M. Stuart,et al.  Waveform relaxation as a dynamical system , 1997, Math. Comput..

[7]  Alain-Sol Sznitman,et al.  Cascades aléatoires et équations de Navier-Stokes , 1997 .

[8]  T. E. Harris,et al.  The Theory of Branching Processes. , 1963 .

[9]  N. Ikeda,et al.  Branching Markov Processes III , 1968 .

[10]  Germany,et al.  Amorphous thin-film growth: Theory compared with experiment , 1999, cond-mat/9912249.

[11]  On Itô's complex measure condition , 2003 .

[12]  Alain-Sol Sznitman,et al.  Stochastic cascades and 3-dimensional Navier–Stokes equations , 1997 .

[13]  R. Temam.  Navier-Stokes Equations and Nonlinear Functional Analysis , 1987 .

[14]  S. Montgomery-Smith.  Finite time blow up for a Navier-Stokes like equation , 1999, math/9911223.

[15]  Edward C. Waymire,et al.  Probability & incompressible Navier-Stokes equations: An overview of some recent developments , 2005, math/0511266.

[16]  A Resummed Branching Process Representation for a Class of Nonlinear ODEs , 2005 .

[17]  R. Temam.  Infinite Dimensional Dynamical Systems in Mechanics and Physics Springer Verlag , 1993 .

[18]  J. Craggs.  Applied Mathematical Sciences , 1973 .

[19]  R. Bhattacharya,et al.  Majorizing kernels and stochastic cascades with applications to incompressible Navier-Stokes equations , 2003 .

[20]  N. Ikeda,et al.  On Branching Markov Processes , 1965 .

[21]  D. Blömker,et al.  Thin-film-growth models: roughness and correlation functions , 2002, European Journal of Applied Mathematics.

[22]  A probabilistic representation of solutions of the incompressible Navier-Stokes equations in R3 , 2004, math/0412034.

[23]  Yuri Bakhtin.  Existence and Uniqueness of Stationary Solutions for 3D Navier–Stokes System with Small Random Forcing via Stochastic Cascades , 2006 .