# American Institute of Mathematical Sciences

February  2014, 34(2): 379-420. doi: 10.3934/dcds.2014.34.379

## Viscous Aubry-Mather theory and the Vlasov equation

 1 Dip. di Matematica, Università di Roma Tre, L.go S. Leonardo Murialdo 1, 00146, Roma, Italy

Received  November 2012 Revised  May 2013 Published  August 2013

The Vlasov equation models a group of particles moving under a potential $V$; moreover, each particle exerts a force, of potential $W$, on the other ones. We shall suppose that these particles move on the $p$-dimensional torus $T^p$ and that the interaction potential $W$ is smooth. We are going to perturb this equation by a Brownian motion on $T^p$; adapting to the viscous case methods of Gangbo, Nguyen, Tudorascu and Gomes, we study the existence of periodic solutions and the asymptotics of the Hopf-Lax semigroup.
Citation: Ugo Bessi. Viscous Aubry-Mather theory and the Vlasov equation. Discrete and Continuous Dynamical Systems, 2014, 34 (2) : 379-420. doi: 10.3934/dcds.2014.34.379
##### References:
 [1] L. Ambrosio, Lecture notes on optimal transport problems, in "Mathematical Aspects of Evolving Interfaces" (Funchal, 2000), Lecture Notes in Math., 1812, Springer, Berlin, (2003), 1-52. doi: 10.1007/978-3-540-39189-0_1. [2] N. Anantharaman, On the zero-temperature vanishing viscosity limit for certain Markov processes arising from Lagrangian dynamics, J. Eur. Math. Soc. (JEMS), 6 (2004), 207-276. [3] G. Birkhoff, "Lattice Theory," Third edition, AMS Colloquium Publ., Vol. XXV, AMS, Providence, R. I., 1967. [4] P. Cardialiaguet, "Notes on Mean Field Games," from P.-L. Lions' lectures at the Collège de France, mimeographed notes. [5] E. Carlen, "Lectures on Optimal Mass Transportation and Certain of its Applications," mimeographed notes, 2009. [6] G. Da Prato, "Introduction to Stochastic Analysis and Malliavin Calculus," Appunti. Scuola Normale Superiore di Pisa (Nuova Serie) [Lecture Notes. Scuola Normale Superiore di Pisa (New Series)], 6, Edizioni della Normale, Pisa, 2007. [7] R. L. Dobrušin, Vlasov equations, Functional Analysis and its Applications, 13 (1979), 45-58. [8] A. Fathi, "Weak KAM Theorem in Lagrangian Dynamics," Fourth preliminary version, mimeographed notes, Lyon, 2003. [9] W. H. Fleming, The Cauchy problem for a nonlinear first order PDE, Journal of Differential Equations, 5 (1969), 515-530. doi: 10.1016/0022-0396(69)90091-6. [10] A. Friedman, "Partial Differential Equations of Parabolic Type," Dover, New York, 1992. [11] 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. [12] W. Gangbo, T. Nguyen and A. Tudorascu, Hamilton-Jacobi equations in the Wasserstein space, Methods Appl. Anal., 15 (2008), 155-183. [13] D. Gomes, A stochastic analog of Aubry-Mather theory, Nonlinearity, 15 (2002), 581-603. doi: 10.1088/0951-7715/15/3/304. [14] T. Hida, "Brownian Motion," Applications of Mathematics, 11, Springer-Verlag, New York-Berlin, 1980. [15] J. N. Mather, Variational construction of connecting orbits, Ann. Inst. Fourier (Grenoble), 43 (1993), 1349-1386. doi: 10.5802/aif.1377. [16] M. Viana, "Stochastic Dynamics of Deterministic Systems," mimeographed notes, 2000. [17] C. Villani, "Topics in Optimal Transportation," Graduate Studies in Mathematics, 58, American Mathematical Society, Providence, RI, 2003. doi: 10.1007/b12016.

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##### References:
 [1] L. Ambrosio, Lecture notes on optimal transport problems, in "Mathematical Aspects of Evolving Interfaces" (Funchal, 2000), Lecture Notes in Math., 1812, Springer, Berlin, (2003), 1-52. doi: 10.1007/978-3-540-39189-0_1. [2] N. Anantharaman, On the zero-temperature vanishing viscosity limit for certain Markov processes arising from Lagrangian dynamics, J. Eur. Math. Soc. (JEMS), 6 (2004), 207-276. [3] G. Birkhoff, "Lattice Theory," Third edition, AMS Colloquium Publ., Vol. XXV, AMS, Providence, R. I., 1967. [4] P. Cardialiaguet, "Notes on Mean Field Games," from P.-L. Lions' lectures at the Collège de France, mimeographed notes. [5] E. Carlen, "Lectures on Optimal Mass Transportation and Certain of its Applications," mimeographed notes, 2009. [6] G. Da Prato, "Introduction to Stochastic Analysis and Malliavin Calculus," Appunti. Scuola Normale Superiore di Pisa (Nuova Serie) [Lecture Notes. Scuola Normale Superiore di Pisa (New Series)], 6, Edizioni della Normale, Pisa, 2007. [7] R. L. Dobrušin, Vlasov equations, Functional Analysis and its Applications, 13 (1979), 45-58. [8] A. Fathi, "Weak KAM Theorem in Lagrangian Dynamics," Fourth preliminary version, mimeographed notes, Lyon, 2003. [9] W. H. Fleming, The Cauchy problem for a nonlinear first order PDE, Journal of Differential Equations, 5 (1969), 515-530. doi: 10.1016/0022-0396(69)90091-6. [10] A. Friedman, "Partial Differential Equations of Parabolic Type," Dover, New York, 1992. [11] 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. [12] W. Gangbo, T. Nguyen and A. Tudorascu, Hamilton-Jacobi equations in the Wasserstein space, Methods Appl. Anal., 15 (2008), 155-183. [13] D. Gomes, A stochastic analog of Aubry-Mather theory, Nonlinearity, 15 (2002), 581-603. doi: 10.1088/0951-7715/15/3/304. [14] T. Hida, "Brownian Motion," Applications of Mathematics, 11, Springer-Verlag, New York-Berlin, 1980. [15] J. N. Mather, Variational construction of connecting orbits, Ann. Inst. Fourier (Grenoble), 43 (1993), 1349-1386. doi: 10.5802/aif.1377. [16] M. Viana, "Stochastic Dynamics of Deterministic Systems," mimeographed notes, 2000. [17] C. Villani, "Topics in Optimal Transportation," Graduate Studies in Mathematics, 58, American Mathematical Society, Providence, RI, 2003. doi: 10.1007/b12016.
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