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

Blömker, D.; Romito, M.; Tribe, R.

doi:10.1016/j.anihpb.2006.02.001

Blömker, D. ; Romito, M. ; Tribe, R.

Annales de l'I.H.P. Probabilités et statistiques, Tome 43 (2007) no. 2, pp. 175-192.

EuDML MR Zbl

DOI : 10.1016/j.anihpb.2006.02.001

@article{AIHPB_2007__43_2_175_0,
     author = {Bl\"omker, D. and Romito, M. and Tribe, R.},
     title = {A probabilistic representation for the solutions to some non-linear {PDEs} using pruned branching trees},
     journal = {Annales de l'I.H.P. Probabilit\'es et statistiques},
     pages = {175--192},
     publisher = {Elsevier},
     volume = {43},
     number = {2},
     year = {2007},
     doi = {10.1016/j.anihpb.2006.02.001},
     mrnumber = {2303118},
     zbl = {1119.60060},
     language = {en},
     url = {http://www.numdam.org/articles/10.1016/j.anihpb.2006.02.001/}
}

TY  - JOUR
AU  - Blömker, D.
AU  - Romito, M.
AU  - Tribe, R.
TI  - A probabilistic representation for the solutions to some non-linear PDEs using pruned branching trees
JO  - Annales de l'I.H.P. Probabilités et statistiques
PY  - 2007
SP  - 175
EP  - 192
VL  - 43
IS  - 2
PB  - Elsevier
UR  - http://www.numdam.org/articles/10.1016/j.anihpb.2006.02.001/
DO  - 10.1016/j.anihpb.2006.02.001
LA  - en
ID  - AIHPB_2007__43_2_175_0
ER  -

%0 Journal Article
%A Blömker, D.
%A Romito, M.
%A Tribe, R.
%T A probabilistic representation for the solutions to some non-linear PDEs using pruned branching trees
%J Annales de l'I.H.P. Probabilités et statistiques
%D 2007
%P 175-192
%V 43
%N 2
%I Elsevier
%U http://www.numdam.org/articles/10.1016/j.anihpb.2006.02.001/
%R 10.1016/j.anihpb.2006.02.001
%G en
%F AIHPB_2007__43_2_175_0

Blömker, D.; Romito, M.; Tribe, R. A probabilistic representation for the solutions to some non-linear PDEs using pruned branching trees. Annales de l'I.H.P. Probabilités et statistiques, Tome 43 (2007) no. 2, pp. 175-192. doi : 10.1016/j.anihpb.2006.02.001. http://www.numdam.org/articles/10.1016/j.anihpb.2006.02.001/

Bibliographie
Cité par

[1] K.B. Athreya, P.E. Ney, Branching Processes, Die Grundlehren der Mathematischen Wissenschaften, Band 196, Springer-Verlag, New York-Heidelberg, 1972. | Zbl

[2] S. Athreya, R. Tribe, Uniqueness for a class of one-dimensional stochastic PDEs using moment duality, Ann. Probab. 28 (4) (2000) 1711-1734. | MR | Zbl

[3] Y. Bakhtin, Existence and uniqueness of stationary solutions for 3D Navier-Stokes system with small random forcing via stochastic cascades, J. Statist. Phys. 122 (2) (2006) 351-360. | Zbl

[4] R. Bhattacharya, L. Chen, S. Dobson, R. Guenther, Ch. Orum, M. Ossiander, E. Thomann, E. Waymire, Majorizing kernels and stochastic cascades with applications to incompressible Navier-Stokes equations, Trans. Amer. Math. Soc. 355 (2003) 5003-5040. | Zbl

[5] M. Bjørhus, A.M. Stuart, Waveform relaxation as a dynamical system, Math. Comput. 66 (219) (1997) 1101-1117. | MR | Zbl

[6] L. Chen, S. Dobson, R. Guenther, Ch. Orum, M. Ossiander, E. Thomann, E. Waymire, On Itô's complex measure condition, in: Probability, Statistics and their Applications: Papers in Honor of Rabi Bhattacharya, IMS Lecture Notes, vol. 41, 2003, pp. 65-80. | MR | Zbl

[7] D. Blömker, C. Gugg, M. Raible, Thin-film-growth models: Roughness and correlation functions, Eur. J. Appl. Math. 13 (4) (2002) 385-402. | MR | Zbl

[8] T.E. Harris, The Theory of Branching Processes, Die Grundlehren der Mathematischen Wissenschaften, Band 119, Springer-Verlag, Berlin, 1963. | MR | Zbl

[9] N. Ikeda, M. Nagasawa, S. Watanabe, Branching Markov processes. I, J. Math. Kyoto Univ. 8 (1968) 233-278. | MR | Zbl

[10] Y. Le Jan, A.S. Sznitman, Stochastic cascades and 3-dimensional Navier-Stokes equations, Probab. Theory Related Fields 109 (3) (1997) 343-366. | Zbl

[11] Y. Le Jan, A.S. Sznitman, Cascades aléatoires et équations de Navier-Stokes, C. R. Acad. Sci. Paris Sér. I Math. 324 (7) (1997) 823-826. | Zbl

[12] J.A. López-Mimbela, A. Wakolbinger, Length of Galton-Watson trees and blow-up of semilinear systems, J. Appl. Probab. 35 (4) (1998) 802-811. | Zbl

[13] H.P. Mckean, Application of Brownian motion to the equation of Kolmogorov-Petrovskii-Piskunov, Comm. Pure Appl. Math. 28 (3) (1975) 323-331. | Zbl

[14] S. Montgomery-Smith, Finite time blow-up for a Navier-Stokes like equation, Proc. Amer. Math. Soc. 129 (10) (2001) 3025-3029. | Zbl

[15] F. Morandin, A resummed branching process representation for a class of nonlinear ODEs, Electron. Comm. Probab. 10 (2005) 1-6. | MR | Zbl

[16] M. Ossiander, A probabilistic representation of the incompressible Navier-Stokes equation in $R^{3}$ , Probab. Theory Relat. Fields 133 (2) (2005) 267-298. | Zbl

[17] M. Raible, S.G. Mayr, S.J. Linz, M. Moske, P. Hänggi, K. Samwer, Amorphous thin film growth: Theory compared with experiment, Europhys. Lett. 50 (2000) 61-67.

[18] A.V. Skorokhod, Branching diffusion processes, Theor. Probab. Appl. 9 (1964) 445-449. | Zbl

[19] R. Temam, Infinite-Dimensional Dynamical Systems in Mechanics and Physics, Applied Mathematical Sciences, vol. 68, Springer-Verlag, New York, 1988. | MR | Zbl

[20] R. Temam, Navier-Stokes Equations and Nonlinear Functional Analysis, CBMS-NSF Regional Conference Series in Applied Mathematics, vol. 41, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1983. | Zbl

[21] E. Waymire, Probability and incompressible Navier-Stokes equations: An overview of some recent developments, Probab. Surveys 2 (2005) 1-32.

Cité par Sources :