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

D. Blömker; M. Romito; R. Tribe

Annales de l'I.H.P. Probabilités et statistiques (2007)

  • Volume: 43, Issue: 2, page 175-192
  • ISSN: 0246-0203

How to cite

top

Blömker, D., Romito, M., and 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 43.2 (2007): 175-192. <http://eudml.org/doc/77930>.

@article{Blömker2007,
author = {Blömker, D., Romito, M., Tribe, R.},
journal = {Annales de l'I.H.P. Probabilités et statistiques},
keywords = {nonlinear PDEs; stochastic representation; branching processes; pruned trees},
language = {eng},
number = {2},
pages = {175-192},
publisher = {Elsevier},
title = {A probabilistic representation for the solutions to some non-linear PDEs using pruned branching trees},
url = {http://eudml.org/doc/77930},
volume = {43},
year = {2007},
}

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
PB - Elsevier
VL - 43
IS - 2
SP - 175
EP - 192
LA - eng
KW - nonlinear PDEs; stochastic representation; branching processes; pruned trees
UR - http://eudml.org/doc/77930
ER -

References

top
  1. [1] K.B. Athreya, P.E. Ney, Branching Processes, Die Grundlehren der Mathematischen Wissenschaften, Band 196, Springer-Verlag, New York–Heidelberg, 1972. Zbl0259.60002
  2. [2] S. Athreya, R. Tribe, Uniqueness for a class of one-dimensional stochastic PDEs using moment duality, Ann. Probab.28 (4) (2000) 1711-1734. Zbl1044.60048MR1813840
  3. [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. Zbl1089.76012
  4. [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. Zbl1031.35115
  5. [5] M. Bjørhus, A.M. Stuart, Waveform relaxation as a dynamical system, Math. Comput.66 (219) (1997) 1101-1117. Zbl0870.65061MR1415796
  6. [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. Zbl1046.60056MR1999415
  7. [7] D. Blömker, C. Gugg, M. Raible, Thin-film-growth models: Roughness and correlation functions, Eur. J. Appl. Math.13 (4) (2002) 385-402. Zbl1020.82014MR1925258
  8. [8] T.E. Harris, The Theory of Branching Processes, Die Grundlehren der Mathematischen Wissenschaften, Band 119, Springer-Verlag, Berlin, 1963. Zbl0117.13002MR163361
  9. [9] N. Ikeda, M. Nagasawa, S. Watanabe, Branching Markov processes. I, J. Math. Kyoto Univ.8 (1968) 233-278. Zbl0233.60068MR232439
  10. [10] Y. Le Jan, A.S. Sznitman, Stochastic cascades and 3-dimensional Navier–Stokes equations, Probab. Theory Related Fields109 (3) (1997) 343-366. Zbl0888.60072
  11. [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. Zbl0876.35083
  12. [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. Zbl0933.60085
  13. [13] H.P. McKean, Application of Brownian motion to the equation of Kolmogorov–Petrovskii–Piskunov, Comm. Pure Appl. Math.28 (3) (1975) 323-331. Zbl0316.35053
  14. [14] S. Montgomery-Smith, Finite time blow-up for a Navier–Stokes like equation, Proc. Amer. Math. Soc.129 (10) (2001) 3025-3029. Zbl0970.35100
  15. [15] F. Morandin, A resummed branching process representation for a class of nonlinear ODEs, Electron. Comm. Probab.10 (2005) 1-6. Zbl1060.60085MR2119148
  16. [16] M. Ossiander, A probabilistic representation of the incompressible Navier–Stokes equation in R 3 , Probab. Theory Relat. Fields133 (2) (2005) 267-298. Zbl1077.35107
  17. [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. [18] A.V. Skorokhod, Branching diffusion processes, Theor. Probab. Appl.9 (1964) 445-449. Zbl0264.60058
  19. [19] R. Temam, Infinite-Dimensional Dynamical Systems in Mechanics and Physics, Applied Mathematical Sciences, vol. 68, Springer-Verlag, New York, 1988. Zbl0662.35001MR953967
  20. [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. Zbl0522.35002
  21. [21] E. Waymire, Probability and incompressible Navier–Stokes equations: An overview of some recent developments, Probab. Surveys2 (2005) 1-32. Zbl1189.76424

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.