In this work, we review part of the results obtained in [12] for computing cancellations for dispersive PDEs with random initial data. The idea is to get a new combinatorial perspective on the cancellations discovered by Deng and Hani (see [16]) in the context of Wave Turbulence when one wants to derive rigorously wave-kinetic equations. This new perspective is based on decorated trees developed for low regularity schemes, together with a well-chosen arborification map that rewrites these trees into linear combinations of words. With this new combinatorial basis, one develops graphical rules to compute cancellations.
Yvain Bruned. Decorated trees, arborification for cancellations in wave turbulence. Séminaire Laurent Schwartz — EDP et applications (2024-2025), Talk no. 16, 15 p.. doi: 10.5802/slsedp.180
@article{SLSEDP_2024-2025____A7_0,
author = {Yvain Bruned},
title = {Decorated trees, arborification for cancellations in wave~turbulence},
journal = {S\'eminaire Laurent Schwartz {\textemdash} EDP et applications},
note = {talk:16},
pages = {1--15},
year = {2024-2025},
publisher = {Institut des hautes \'etudes scientifiques & Centre de math\'ematiques Laurent Schwartz, \'Ecole polytechnique},
doi = {10.5802/slsedp.180},
language = {en},
url = {https://proceedings.centre-mersenne.org/articles/10.5802/slsedp.180/}
}
TY - JOUR AU - Yvain Bruned TI - Decorated trees, arborification for cancellations in wave turbulence JO - Séminaire Laurent Schwartz — EDP et applications N1 - talk:16 PY - 2024-2025 SP - 1 EP - 15 PB - Institut des hautes études scientifiques & Centre de mathématiques Laurent Schwartz, École polytechnique UR - https://proceedings.centre-mersenne.org/articles/10.5802/slsedp.180/ DO - 10.5802/slsedp.180 LA - en ID - SLSEDP_2024-2025____A7_0 ER -
%0 Journal Article %A Yvain Bruned %T Decorated trees, arborification for cancellations in wave turbulence %J Séminaire Laurent Schwartz — EDP et applications %Z talk:16 %D 2024-2025 %P 1-15 %I Institut des hautes études scientifiques & Centre de mathématiques Laurent Schwartz, École polytechnique %U https://proceedings.centre-mersenne.org/articles/10.5802/slsedp.180/ %R 10.5802/slsedp.180 %G en %F SLSEDP_2024-2025____A7_0
[1] Y. Alama Bronsard, Y. Bruned, K. Schratz. Approximations of Dispersive PDEs in the Presence of Low-Regularity Randomness. Found. Comput. Math. 24, (2024), 1819–1869. | DOI | Zbl
[2] C. Bellingeri, Y. Bruned. Symmetries for the gKPZ equation via multi-indices. | arXiv | Zbl
[3] Y. Bruned, A. Chandra, I. Chevyrev, M. Hairer. Renormalising SPDEs in regularity structures. J. Eur. Math. Soc. (JEMS), 23, no. 3, (2021), 869-947. | DOI | Zbl
[4] B. Bringmann Y. Deng, A. Nahmod, H. Yue . Invariant Gibbs measures for the three dimensional cubic nonlinear wave equation. Invent. Math. 236, (2024), 1133–1411. | DOI | Zbl
[5] Y. Bruned, V. Dotsenko. Chain rule symmetry for singular SPDEs. | arXiv | Zbl
[6] Y. Bruned, F. Gabriel, M. Hairer, L. Zambotti, Geometric stochastic heat equations. J. Amer. Math. Soc. (JAMS) 35, no. 1, (2022), 1-80. | DOI | Zbl
[7] Y. Bruned, M. Gerencsér, U. Nadeem Quasi-generalised KPZ equation. | arXiv | Zbl
[8] Y. Bruned, M. Hairer, L. Zambotti. Algebraic renormalisation of regularity structures. Invent. Math. 215, no. 3, (2019), 1039–1156. | DOI | Zbl
[9] Y. Bruned. Singular KPZ Type Equations. 205 pages, PhD thesis, Université Pierre et Marie Curie - Paris VI, 2015. tel-01306427
[10] Y. Bruned. Derivation of normal forms for dispersive PDEs via arborification. | arXiv | Zbl
[11] Y. Bruned, K. Schratz. Resonance based schemes for dispersive equations via decorated trees. Forum of Mathematics, Pi, 10, (2022), E2. | DOI | Zbl
[12] Y. Bruned, L. Tolomeo Cancellations for dispersive PDEs with random initial data. | arXiv | Zbl
[13] J. C. Butcher. An algebraic theory of integration methods. Math. Comp. 26, (1972), 79–106. | DOI | Zbl
[14] A. Connes, D. Kreimer. Hopf algebras, renormalization and noncommutative geometry. Comm. Math. Phys. 199, no. 1, (1998), 203–242. | DOI | Zbl
[15] Y. Deng, Z. Hani. Full derivation of the wave kinetic equation. Invent. math. 233, (2023), 543-724. | Zbl | DOI
[16] Y. Deng, Z. Hani. Derivation of the wave kinetic equation: Full range of scaling laws. | arXiv | Zbl
[17] Y. Deng, Z. Hani. Long time justification of wave turbulence theory. | arXiv
[18] Y. Deng, Z. Hani, X. Ma. Long time derivation of Boltzmann equation from hard sphere dynamics. | arXiv | Zbl
[19] J. Ecalle, B. Vallet. The arborification-coarborification transform: analytic, combinatorial, and algebraic aspects. Ann. Fac. Sci. Toulouse (6) 13, no. 3, (2004), 575–657. | DOI | Zbl
[20] F. Fauvet and F. Menous, Ecalle’s arborification coarborification transforms and Connes Kreimer Hopf algebra, Annales Sc. de l’Ecole Normale Sup. 50, no. 1, (2017), 39-83. | DOI | Zbl
[21] M. Gerencsér. Nondivergence form quasilinear heat equations driven by space-time white noise. Annales de l’Institut Henri Poincaré, Analyse non linéaire 37, no. 3. (2020), 663–682 | Zbl | DOI
[22] Z. Guo, S. Kwon, T. Oh. Poincaré-Dulac normal form reduction for unconditional well-posedness of the periodic cubic NLS. Comm. Math. Phys. 322, no. 1, (2013), 19–48. | DOI | MR | Zbl
[23] M. Hairer. Solving the KPZ equation. Ann. Math 178, no. 2, (2013), 559–664. | DOI | Zbl
[24] M. Hairer. A theory of regularity structures. Invent. Math. 198, no. 2, (2014), 269–504. | DOI | Zbl
[25] M. Hairer, E. Pardoux. A Wong-Zakai theorem for stochastic PDEs. J. Math. Soc. Japan, 67, no. 4, (2015), 1551–1604. | DOI | Zbl
[26] M. Hairer, J. Quastel. A class of growth models rescaling to KPZ. Forum of Mathematics, Pi, 6, (2018), E3. | DOI | Zbl
Cited by Sources: