Discrete and Continuous Dynamical Systems - Series A (DCDS-A)

An infinite-dimensional weak KAM theory via random variables
Pages: 6167 - 6185, Issue 11, November 2016

doi:10.3934/dcds.2016069      Abstract        References        Full text (479.1K)           Related Articles

Diogo Gomes - 4700 King Abdullah University of Science and Technology (KAUST), CEMSE Division, Thuwal 23955-6900, Saudi Arabia (email)
Levon Nurbekyan - 4700 King Abdullah University of Science and Technology (KAUST), CEMSE Division, Thuwal 23955-6900, Saudi Arabia (email)

1 L. Ambrosio, N. Gigli and G. Savaré, Gradient Flows in Metric Spaces and in the Space of Probability Measures, $2^{nd}$ edition, Birkhäuser Verlag, Basel, 2008.       
2 V. I. Arnol'd, Proof of a theorem of A. N. Kolmogorov on the preservation of conditionally periodic motions under a small perturbation of the Hamiltonian, Uspehi Mat. Nauk, 18 (1963), 13-40.       
3 V. I. Arnol'd, Mathematical Models of Classical Mechanics, (Translated from the Russian by K. Vogtmann and A. Weinstein) Springer-Verlag, New York-Heidelberg, 1978.       
4 V. I. Arnol'd, V. V. Kozlov and A. I. Neĭshtadt, Dynamical Systems. III, (Translated from the Russian by A. Iacob) Springer-Verlag, Berlin, 1988.       
5 E. Asplund, Fréchet differentiability of convex functions, Acta Math., 121 (1968), 31-47.       
6 M. Bardi and I. Capuzzo-Dolcetta, Optimal control and viscosity solutions of Hamilton-Jacobi-Bellman equations, Birkhäuser Boston Inc., Boston, MA, 1997.       
7 G. Barles, Solutions de Viscosité des Équations de Hamilton-Jacobi, (French) [Viscosity Solutions of Hamilton-Jacobi Equations], Springer-Verlag, Paris, 1994.       
8 P. Bernard, Existence of $C^{1,1}$ critical sub-solutions of the Hamilton-Jacobi equation on compact manifolds}, Ann. Sci. École Norm. Sup. (4), 40 (2007), 445-452.       
9 P. Bernard, Smooth critical sub-solutions of the Hamilton-Jacobi equation, Math. Res. Lett., 14 (2007), 503-511.       
10 P. Bernard, The Lax-Oleinik semi-group: A Hamiltonian point of view, Proc. Roy. Soc. Edinburgh Sect. A, 142 (2012), 1131-1177.       
11 U. Bessi, Chaotic motions for a version of the Vlasov equation, SIAM J. Math. Anal., 44 (2012), 2496-2525.       
12 U. Bessi, The Aubry set for a version of the Vlasov equation, NoDEA Nonlinear Differential Equations Appl., 20 (2013), 1411-1452.       
13 U. Bessi, A time-step approximation scheme for a viscous version of the Vlasov equation, Adv. Math., 266 (2014), 17-83.       
14 U. Bessi, Viscous Aubry-Mather theory and the Vlasov equation, Discrete Contin. Dyn. Syst., 34 (2014), 379-420.       
15 W. Braun and K. Hepp, The Vlasov dynamics and its fluctuations in the $1/N$ limit of interacting classical particles, Comm. Math. Phys., 56 (1977), 101-113.       
16 P. Cardaliaguet, Notes on Mean Field Games (from P. L. Lions' lectures at Collège de France), 2012. Available from: https://www.ceremade.dauphine.fr/~cardalia/MFG20130420.pdf.
17 M. G. Crandall and P. L. Lions, Hamilton-Jacobi equations in infinite dimensions. II. Existence of viscosity solutions, J. Funct. Anal., 65 (1986), 368-405.       
18 R. L. Dobrušin, Vlasov equations, (Russian) Funktsional. Anal. i Prilozhen., 13 (1979), 48-58, 96.       
19 A. Fathi, Weak KAM Theorem in Lagrangian Dynamics, (to appear in Cambridge Studies in Advanced Mathematics).
20 A. Fathi, Solutions {KAM} faibles conjuguées et barrières de Peierls, (French) [Weakly conjugate KAM solutions and Peierls's barriers], C. R. Acad. Sci. Paris Sér. I Math., 325 (1997), 649-652.       
21 A. Fathi, Théorème KAM faible et théorie de Mather sur les systèmes lagrangiens, (French) [A weak KAM theorem and Mather's theory of Lagrangian systems], C. R. Acad. Sci. Paris Sér. I Math., 324 (1997), 1043-1046.       
22 A. Fathi, Sur la convergence du semi-groupe de Lax-Oleinik, (French) [Convergence of the Lax-Oleinik semigroup], C. R. Acad. Sci. Paris Sér. I Math., 327 (1998), 267-270.       
23 A. Fathi, On existence of smooth critical subsolutions of the Hamilton-Jacobi equation, Publ. Mat. Urug., 12 (2011), 87-98.       
24 A. Fathi, A. Giuliani and A. Sorrentino, Uniqueness of invariant Lagrangian graphs in a homology or a cohomology class, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5), 8 (2009), 659-680.       
25 A. Fathi and E. Maderna, Weak KAM theorem on non compact manifolds, NoDEA Nonlinear Differential Equations Appl., 14 (2007), 1-27.       
26 A. Fathi and A. Siconolfi, Existence of $C^1$ critical subsolutions of the Hamilton-Jacobi equation, Invent. Math., 155 (2004), 363-388.       
27 A. Fathi and A. Siconolfi, PDE aspects of Aubry-Mather theory for quasiconvex Hamiltonians, Calc. Var. Partial Differential Equations, 22 (2005), 185-228.       
28 W. H. Fleming and H. M. Soner, Controlled Markov Processes and Viscosity Solutions, Springer-Verlag, New York, 1993.       
29 W. Gangbo and A. Tudorascu, Lagrangian dynamics on an infinite-dimensional torus; a weak KAM theorem, Adv. Math., 224 (2010), 260-292.       
30 W. Gangbo and A. Tudorascu, A weak KAM theorem; from finite to infinite dimension, in Optimal transportation, geometry and functional inequalities, Ed. Norm., Pisa, (2010), 45-72.       
31 W. Gangbo and A. Tudorascu, Weak KAM theory on the Wasserstein torus with multidimensional underlying space, Comm. Pure Appl. Math., 67 (2014), 408-463.       
32 D. Gomes, Hamilton Jacobi Equations, Viscosity Solutions and Asymptotics of Hamiltonian Systems, Ph.D thesis, University of California at Berkeley, 2000.       
33 D. Gomes, Viscosity solutions of Hamilton-Jacobi equations, and asymptotics for Hamiltonian systems, Calc. Var. Partial Differential Equations, 14 (2002), 345-357.       
34 D. Gomes, Perturbation theory for viscosity solutions of Hamilton-Jacobi equations and stability of Aubry-Mather sets, SIAM J. Math. Anal., 35 (2003), 135-147.       
35 D. Gomes, Regularity theory for Hamilton-Jacobi equations, J. Differential Equations, 187 (2003), 359-374.       
36 D. Gomes and L. Nurbekyan, On the minimizers of calculus of variations problems in Hilbert spaces, Calc. Var. Partial Differential Equations, 52 (2015), 65-93.       
37 H. Ishii, Viscosity solutions for a class of Hamilton-Jacobi equations in Hilbert spaces, J. Funct. Anal., 105 (1992), 301-341.       
38 A. N. Kolmogorov, On conservation of conditionally periodic motions for a small change in Hamilton's function, (Russian) Dokl. Akad. Nauk SSSR (N.S.), 98 (1954), 527-530.       
39 P. L. Lions, Lectures on Mean Field Games. Available from: http://www.college-de-france.fr/site/en-pierre-louis-lions/index.htm.
40 P. L. Lions, G. Papanicolau and S. R. S. Varadhan, Homogenization of Hamilton-Jacobi equations, unpublished, (1987). Available from: https://www.scribd.com/doc/146255460/Lions-Papanicolaou-Varadhan-Homogenization-of-Hamilton-Jacobi-Equations.
41 V. P. Maslov, Equations of the self-consistent field, in Current problems in mathematics, (Russian) Akad. Nauk SSSR Vsesojuz. Inst. Nau\v cn. i Tehn. Informacii, Moscow, (1978), 153-234.       
42 G. J. Minty, On the monotonicity of the gradient of a convex function, Pacific J. Math., 14 (1964), 243-247.       
43 L. Nurbekyan, Weak KAM Theory on the $d$-infinite Dimensional Torus, Ph.D thesis, Instituto Superior Técnico, 2012.
44 D. Tataru, Viscosity solutions of Hamilton-Jacobi equations with unbounded nonlinear terms, J. Math. Anal. Appl., 163 (1992), 345-392.       

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