American Institute of Mathematical Sciences

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November  2016, 36(11): 6167-6185. doi: 10.3934/dcds.2016069

An infinite-dimensional weak KAM theory via random variables

Diogo Gomes 1, and  Levon Nurbekyan 1,

1. 

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

Received  August 2015 Revised  July 2016 Published  August 2016

We develop several aspects of the infinite-dimensional Weak KAM theory using a random variables' approach. We prove that the infinite-dimensional cell problem admits a viscosity solution that is a fixed point of the Lax-Oleinik semigroup. Furthermore, we show the existence of invariant minimizing measures and calibrated curves defined on $\mathbb{R}$.
Keywords: Hamilton-Jacobi equations, Dynamical systems, voscosity solutions., weak KAM theory.
Mathematics Subject Classification: Primary: 37K05, 37K55; Secondary: 49L20, 49L2.
Citation: Diogo Gomes, Levon Nurbekyan. An infinite-dimensional weak KAM theory via random variables. Discrete and Continuous Dynamical Systems, 2016, 36 (11) : 6167-6185. doi: 10.3934/dcds.2016069
References:
[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. doi: 10.1007/978-3-642-61551-1.

[5]

E. Asplund, Fréchet differentiability of convex functions, Acta Math., 121 (1968), 31-47. doi: 10.1007/BF02391908.

[6]

M. Bardi and I. Capuzzo-Dolcetta, Optimal control and viscosity solutions of Hamilton-Jacobi-Bellman equations, Birkhäuser Boston Inc., Boston, MA, 1997. doi: 10.1007/978-0-8176-4755-1.

[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. doi: 10.1016/j.ansens.2007.01.004.

[9]

P. Bernard, Smooth critical sub-solutions of the Hamilton-Jacobi equation, Math. Res. Lett., 14 (2007), 503-511. doi: 10.4310/MRL.2007.v14.n3.a14.

[10]

P. Bernard, The Lax-Oleinik semi-group: A Hamiltonian point of view, Proc. Roy. Soc. Edinburgh Sect. A, 142 (2012), 1131-1177. doi: 10.1017/S0308210511000059.

[11]

U. Bessi, Chaotic motions for a version of the Vlasov equation, SIAM J. Math. Anal., 44 (2012), 2496-2525. doi: 10.1137/110851225.

[12]

U. Bessi, The Aubry set for a version of the Vlasov equation, NoDEA Nonlinear Differential Equations Appl., 20 (2013), 1411-1452. doi: 10.1007/s00030-012-0216-8.

[13]

U. Bessi, A time-step approximation scheme for a viscous version of the Vlasov equation, Adv. Math., 266 (2014), 17-83. doi: 10.1016/j.aim.2014.07.023.

[14]

U. Bessi, Viscous Aubry-Mather theory and the Vlasov equation, Discrete Contin. Dyn. Syst., 34 (2014), 379-420. doi: 10.3934/dcds.2014.34.379.

[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. doi: 10.1007/BF01611497.

[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. doi: 10.1016/0022-1236(86)90026-1.

[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. doi: 10.1016/S0764-4442(97)84777-5.

[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. doi: 10.1016/S0764-4442(97)87883-4.

[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. doi: 10.1016/S0764-4442(98)80144-4.

[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. doi: 10.1007/s00030-007-2047-6.

[26]

A. Fathi and A. Siconolfi, Existence of $C^1$ critical subsolutions of the Hamilton-Jacobi equation, Invent. Math., 155 (2004), 363-388. doi: 10.1007/s00222-003-0323-6.

[27]

A. Fathi and A. Siconolfi, PDE aspects of Aubry-Mather theory for quasiconvex Hamiltonians, Calc. Var. Partial Differential Equations, 22 (2005), 185-228. doi: 10.1007/s00526-004-0271-z.

[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. doi: 10.1016/j.aim.2009.11.005.

[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. doi: 10.1002/cpa.21492.

[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. doi: 10.1007/s005260100106.

[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. doi: 10.1137/S0036141002405960.

[35]

D. Gomes, Regularity theory for Hamilton-Jacobi equations, J. Differential Equations, 187 (2003), 359-374. doi: 10.1016/S0022-0396(02)00013-X.

[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. doi: 10.1007/s00526-013-0705-6.

[37]

H. Ishii, Viscosity solutions for a class of Hamilton-Jacobi equations in Hilbert spaces, J. Funct. Anal., 105 (1992), 301-341. doi: 10.1016/0022-1236(92)90081-S.

[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. doi: 10.2140/pjm.1964.14.243.

[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. doi: 10.1016/0022-247X(92)90256-D.

show all references

References:
[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. doi: 10.1007/978-3-642-61551-1.

[5]

E. Asplund, Fréchet differentiability of convex functions, Acta Math., 121 (1968), 31-47. doi: 10.1007/BF02391908.

[6]

M. Bardi and I. Capuzzo-Dolcetta, Optimal control and viscosity solutions of Hamilton-Jacobi-Bellman equations, Birkhäuser Boston Inc., Boston, MA, 1997. doi: 10.1007/978-0-8176-4755-1.

[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. doi: 10.1016/j.ansens.2007.01.004.

[9]

P. Bernard, Smooth critical sub-solutions of the Hamilton-Jacobi equation, Math. Res. Lett., 14 (2007), 503-511. doi: 10.4310/MRL.2007.v14.n3.a14.

[10]

P. Bernard, The Lax-Oleinik semi-group: A Hamiltonian point of view, Proc. Roy. Soc. Edinburgh Sect. A, 142 (2012), 1131-1177. doi: 10.1017/S0308210511000059.

[11]

U. Bessi, Chaotic motions for a version of the Vlasov equation, SIAM J. Math. Anal., 44 (2012), 2496-2525. doi: 10.1137/110851225.

[12]

U. Bessi, The Aubry set for a version of the Vlasov equation, NoDEA Nonlinear Differential Equations Appl., 20 (2013), 1411-1452. doi: 10.1007/s00030-012-0216-8.

[13]

U. Bessi, A time-step approximation scheme for a viscous version of the Vlasov equation, Adv. Math., 266 (2014), 17-83. doi: 10.1016/j.aim.2014.07.023.

[14]

U. Bessi, Viscous Aubry-Mather theory and the Vlasov equation, Discrete Contin. Dyn. Syst., 34 (2014), 379-420. doi: 10.3934/dcds.2014.34.379.

[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. doi: 10.1007/BF01611497.

[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. doi: 10.1016/0022-1236(86)90026-1.

[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. doi: 10.1016/S0764-4442(97)84777-5.

[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. doi: 10.1016/S0764-4442(97)87883-4.

[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. doi: 10.1016/S0764-4442(98)80144-4.

[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. doi: 10.1007/s00030-007-2047-6.

[26]

A. Fathi and A. Siconolfi, Existence of $C^1$ critical subsolutions of the Hamilton-Jacobi equation, Invent. Math., 155 (2004), 363-388. doi: 10.1007/s00222-003-0323-6.

[27]

A. Fathi and A. Siconolfi, PDE aspects of Aubry-Mather theory for quasiconvex Hamiltonians, Calc. Var. Partial Differential Equations, 22 (2005), 185-228. doi: 10.1007/s00526-004-0271-z.

[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. doi: 10.1016/j.aim.2009.11.005.

[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. doi: 10.1002/cpa.21492.

[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. doi: 10.1007/s005260100106.

[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. doi: 10.1137/S0036141002405960.

[35]

D. Gomes, Regularity theory for Hamilton-Jacobi equations, J. Differential Equations, 187 (2003), 359-374. doi: 10.1016/S0022-0396(02)00013-X.

[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. doi: 10.1007/s00526-013-0705-6.

[37]

H. Ishii, Viscosity solutions for a class of Hamilton-Jacobi equations in Hilbert spaces, J. Funct. Anal., 105 (1992), 301-341. doi: 10.1016/0022-1236(92)90081-S.

[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. doi: 10.2140/pjm.1964.14.243.

[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. doi: 10.1016/0022-247X(92)90256-D.

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