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Hamiltonian ODE, homogenization, and symplectic topology

Published online by Cambridge University Press:  10 May 2024

Albert Fathi
Affiliation:
Georgia Institute of Technology
Philip J. Morrison
Affiliation:
University of Texas, Austin
Tere M-Seara
Affiliation:
Universitat Politècnica de Catalunya, Barcelona
Sergei Tabachnikov
Affiliation:
Pennsylvania State University
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Summary

This article is based on a course the author gave in the Fall of 2018 at UC Berkeley, in connection with the MSRI program Hamiltonian systems, from topology to applications through analysis. In this article we explore the connection between the Hamiltonian ODEs and Hamilton–Jacobi PDEs, and give an overview of some of the existing techniques for the question of homogenization. We also discuss stochastic formulations of several classical problems in symplectic geometry.

Type
Chapter
Information
Hamiltonian Systems
Dynamics, Analysis, Applications
, pp. 297 - 367
Publisher: Cambridge University Press
Print publication year: 2024

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References

Anantharaman, N., “On the zero-temperature or vanishing viscosity limit for certain Markov processes arising from Lagrangian dynamics”, J. Eur. Math. Soc. (JEMS) 6:2 (2004), 207276.CrossRefGoogle Scholar
Arnold, V. I., Mathematical methods of classical mechanics, Graduate Texts in Mathematics 60, Springer, 1978.Google Scholar
Bernard, P., “The Lax–Oleinik semi-group: a Hamiltonian point of view”, Proc. Roy. Soc. Edinburgh Sect. A 142:6 (2012), 11311177.CrossRefGoogle Scholar
Bernard, P., “Semi-concave singularities and the Hamilton–Jacobi equation”, Regul. Chaotic Dyn. 18:6 (2013), 674685.CrossRefGoogle Scholar
Bernardi, O. and Cardin, F., “On C0-variational solutions for Hamilton–Jacobi equations”, Discrete Contin. Dyn. Syst. 31:2 (2011), 385406.CrossRefGoogle Scholar
Cannarsa, P. and Sinestrari, C., Semiconcave functions, Hamilton–Jacobi equations, and optimal control, Progress in Nonlinear Differential Equations and their Applications 58, Birkhäuser, Boston, 2004.Google Scholar
Chaperon, M., “Familles génératrices”, 1990. Cours donné a l’école d’été Erasmus de Samos.Google Scholar
Evans, L. C., Partial differential equations, 2nd ed., Graduate Studies in Mathematics 19, American Mathematical Society, Providence, RI, 2010.Google Scholar
Evans, L. C. and Souganidis, P. E., “Differential games and representation formulas for solutions of Hamilton–Jacobi–Isaacs equations”, Indiana Univ. Math. J. 33:5 (1984), 773797.CrossRefGoogle Scholar
Fathi, A., “Weak KAM theorem in Lagrangian dynamics”, preliminary version, number 10, 2008, tinyurl.com/FathiKAM.Google Scholar
Golé, C., Symplectic twist maps, Advanced Series in Nonlinear Dynamics 18, World Scientific Publishing Co., River Edge, NJ, 2001.Google Scholar
Herman, M.-R., “Sur la conjugaison différentiable des difféomorphismes du cercle à des rotations”, Inst. Hautes Études Sci. Publ. Math. 49 (1979), 5233.CrossRefGoogle Scholar
Hofer, H. and Zehnder, E., Symplectic invariants and Hamiltonian dynamics, Birkhäuser, Basel, 1994.CrossRefGoogle Scholar
Katok, A. and Hasselblatt, B., Introduction to the modern theory of dynamical systems, Encyclopedia of Mathematics and its Applications 54, Cambridge University Press, 1995.Google Scholar
Kosygina, E., Rezakhanlou, F., and Varadhan, S. R. S., “Stochastic homogenization of Hamilton–Jacobi–Bellman equations”, Comm. Pure Appl. Math. 59:10 (2006), 14891521.CrossRefGoogle Scholar
Lax, P. D., Functional analysis, Wiley, New York, 2002.Google Scholar
Lions, P. L., Papanicolaou, G., and Varadhan, S. R. S., “Homogenization of Hamilton–Jacobi equations”, unpublihsed, 1996, https://tinyurl.com/HomogenizationHamiltonJacobi.Google Scholar
Matic, I., Homogenizations and large deviations, Ph.D. thesis, University of California, Berkeley, 2010, https://www.proquest.com/docview/749077874.Google Scholar
McDuff, D. and Salamon, D., Introduction to symplectic topology, 3rd ed., Oxford Graduate Texts in Mathematics 27, Oxford University Press, 2017.Google Scholar
Pelayo, A. and Rezakhanlou, F., “Poincaré–Birkhoff theorems in random dynamics”, Trans. Amer. Math. Soc. 370:1 (2018), 601639.CrossRefGoogle Scholar
Pelayo, A. and Rezakhanlou, F., “The random Arnold conjecture: a new probabilistic Conley– Zehnder theory for symplectic”, preprint, 2023. arXiv 2306.1558év2Google Scholar
Rezakhanlou, F., “Continuum limit for some growth models, II”, Ann. Probab. 29:3 (2001), 13291372.CrossRefGoogle Scholar
Rezakhanlou, F., “Lectures on large deviation principle”, https:math.berkeley.edu/~rezakhan/LD.pdf.Google Scholar
Rezakhanlou, F., “Lectures on symplectic geometry”, https:math.berkeley.edu/~rezakhan/Symplectic.pdf.Google Scholar
Rezakhanlou, F., “Stochastic growth and KPZ equation”, https:math.berkeley.edu/~rezakhan/SGKPZ.pdf.Google Scholar
Rezakhanlou, F. and Tarver, J. E., “Homogenization for stochastic Hamilton–Jacobi equations”, Arch. Ration. Mech. Anal. 151:4 (2000), 277309.CrossRefGoogle Scholar
Roos, V., Variational and viscosity solutions of the Hamilton–Jacobi equation, Ph.D. thesis, Université Paris sciences et lettres, 2017, https://theses.hal.science/tel-01635263.Google Scholar
Roos, V., “Variational and viscosity operators for the evolutionary Hamilton–Jacobi equation”, Commun. Contemp. Math. 21:4 (2019), 1850018, 76.CrossRefGoogle Scholar
Souganidis, P. E., “Stochastic homogenization of Hamilton–Jacobi equations and some applications”, Asymptot. Anal. 20:1 (1999), 111.Google Scholar
Sznitman, A.-S., Brownian motion, obstacles and random media, Springer, 1998.CrossRefGoogle Scholar
Viterbo, C., “Symplectic homogenization”, J. Éc. polytech. Math. 10 (2023), 67140.CrossRefGoogle Scholar
Wayne, C. E., “An introduction to KAM theory”, pp. 329 in Dynamical systems and probabilistic methods in partial differential equations (Berkeley, CA, 1994), edited by Deift, P. et al., Lectures in Appl. Math. 31, Amer. Math. Soc., Providence, RI, 1996.Google Scholar

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