Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

Request Permissions   Purchase Content 
 
 

 

A stochastic Evans-Aronsson problem


Authors: Diogo Gomes and Héctor Sánchez Morgado
Journal: Trans. Amer. Math. Soc. 366 (2014), 903-929
MSC (2010): Primary 49L99
DOI: https://doi.org/10.1090/S0002-9947-2013-05936-3
Published electronically: August 8, 2013
MathSciNet review: 3130321
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: In this paper the stochastic version of the Evans-Aronsson problem is studied. Both for the mechanical case and two dimensional problems we prove the existence of smooth solutions. We establish that the corresponding effective Lagrangian and Hamiltonian are smooth. We study the limiting behavior and the convergence of the effective Lagrangian and Hamiltonian, Mather measures and minimizers. We end the paper with applications to stationary mean-field games.


References [Enhancements On Off] (What's this?)

  • [HMC06] M.Y. Huang, R.P. Malhame and P.E. Caines.
    Large Population Stochastic Dynamic Games: Closed Loop McKean-Vlasov Systems and the Nash Certainty Equivalence Principle.
    Special issue in honor of the 65th birthday of Tyrone Duncan, Communications in Information and Systems. Vol. 6, Number 3, 2006, pp 221 - 252. MR 2346927 (2009f:91008)
  • [HMC07] M.Y. Huang, R.P. Malhame and P.E. Caines.
    Population Cost-Coupled LQG Problems with Non-uniform Agents: Individual-Mass Behavior and Decentralized - Nash Equilibria.
    IEEE Trans. on Automatic Control, Vol. 52. No. 9, September, 2007, pp 1560 - 1571. MR 2352434 (2008g:91047)
  • [D1] J. Dieudonné. Foundations of Modern Analysis. Vol. I, Academic Press, New York, 1969. MR 0349288 (50:1782)
  • [E1] L.C. Evans. Some new PDE methods for weak KAM theory. Calc. Var. Partial Differential Equations 17 (2003), no. 2, 159 - 177. MR 1986317 (2004e:37097)
  • [E2] L.C. Evans. Further PDE methods for weak KAM theory. Calc. Var. Partial Differential Equations 35 (2009), no. 4, 435 - 462. MR 2496651 (2011f:37111)
  • [Eva10] L. C. Evans,
    Adjoint and compensated compactness methods for Hamilton-Jacobi PDE,
    Arch. Ration. Mech. Anal., 197 (2010), 1053-1088. MR 2679366 (2011h:35042)
  • [G] D. Gomes. A stochastic analog of Aubry-Mather theory. Nonlinearity 15 (2002), 3, 581-603. MR 1901094 (2003b:37096)
  • [GISMY] D. Gomes, Iturriaga R., Sánchez-Morgado H., Y. Yu. Mather measures selected by an approximation scheme. Proc. Amer. Math. Soc. 138 (2010), no. 10, 3591-3601. MR 2661558 (2011h:37093)
  • [Ge1] O. Guéant. Mean field games equations with quadratic hamiltonian: a specific approach. working paper, 2011.
  • [I-SM] R. Iturriaga and H. Sánchez-Morgado. On the stochastic Aubry Mather theory. Bol. Soc. Mat. Mex. (3) vol 11 no. 1 (2005) 91-99. MR 2198592 (2006k:37166)
  • [LL06a] Jean-Michel Lasry and Pierre-Louis Lions.
    Jeux à champ moyen. I. Le cas stationnaire.
    C. R. Math. Acad. Sci. Paris, 343(9):619-625, 2006. MR 2269875 (2007m:91021)
  • [LL06b] Jean-Michel Lasry and Pierre-Louis Lions.
    Jeux à champ moyen. II. Horizon fini et contrôle optimal.
    C. R. Math. Acad. Sci. Paris, 343(10):679-684, 2006. MR 2271747 (2007m:91022)
  • [LL07a] Jean-Michel Lasry and Pierre-Louis Lions.
    Mean field games.
    Jpn. J. Math., 2(1):229-260, 2007. MR 2295621 (2008k:91034)
  • [LL07b] Jean-Michel Lasry and Pierre-Louis Lions.
    Mean field games.
    Cahiers de la Chaire Finance et Développement Durable, 2007. MR 2295621 (2008k:91034)
  • [LLG10a] Jean-Michel Lasry, Pierre-Louis Lions, and O. Guéant.
    Application of mean field games to growth theory.
    preprint, 2010.
  • [LLG10b] Jean-Michel Lasry, Pierre-Louis Lions, and O. Guéant.
    Mean field games and applications.
    Paris-Princeton lectures on Mathematical Finance, 2010. MR 2762362
  • [M] R. Mañé. Generic properties and problems of minimizing measures of Lagrangian systems. Nonlinearity 9 (1996), 273-310. MR 1384478 (97d:58118)
  • [Ma] J. Mather. Action minimizing invariant measure for positive definite Lagrangian systems. Math. Z. 207 (1991), 169-207. MR 1109661 (92m:58048)
  • [W] K. Wang. Action minimizing stochastic invariant measures for a class of Lagrangian systems. Comm. Pure App. Analysis. 7 no. 5 (2008). 1211-1223 MR 2410876 (2010a:37105)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2010): 49L99

Retrieve articles in all journals with MSC (2010): 49L99


Additional Information

Diogo Gomes
Affiliation: Departamento de Matemática, Instituto Superior Técnico, Av. Rovisco Pais, Lisboa, Portugal
Email: dgomes@math.ist.utl.pt

Héctor Sánchez Morgado
Affiliation: Instituto de Matemáticas, Universidad Nacional Autónoma de México, México DF 04510, México
Email: hector@matem.unam.mx

DOI: https://doi.org/10.1090/S0002-9947-2013-05936-3
Received by editor(s): November 9, 2011
Received by editor(s) in revised form: June 1, 2012
Published electronically: August 8, 2013
Additional Notes: The first author was partially supported by CAMGSD-LARSys through FCT Program POCTI-FEDER and by grants PTDC/MAT/114397/2009, UTAustin/MAT/0057/2008, and UTA-CMU/MAT/0007/2009
The second author is grateful to the Instituto Superior Técnico for its hospitality
Article copyright: © Copyright 2013 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

American Mathematical Society