# 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 & Continuous Dynamical Systems - A, 2014, 34 (2) : 379-420. doi: 10.3934/dcds.2014.34.379
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##### References:
 [1] L. Ambrosio, Lecture notes on optimal transport problems,, in, 1812 (2003), 1.  doi: 10.1007/978-3-540-39189-0_1.  Google Scholar [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.   Google Scholar [3] G. Birkhoff, "Lattice Theory,", Third edition, (1967).   Google Scholar [4] P. Cardialiaguet, "Notes on Mean Field Games,", from P.-L. Lions' lectures at the Collège de France, ().   Google Scholar [5] E. Carlen, "Lectures on Optimal Mass Transportation and Certain of its Applications,", mimeographed notes, (2009).   Google Scholar [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 (2007).   Google Scholar [7] R. L. Dobrušin, Vlasov equations,, Functional Analysis and its Applications, 13 (1979), 45.   Google Scholar [8] A. Fathi, "Weak KAM Theorem in Lagrangian Dynamics,", Fourth preliminary version, (2003).   Google Scholar [9] W. H. Fleming, The Cauchy problem for a nonlinear first order PDE,, Journal of Differential Equations, 5 (1969), 515.  doi: 10.1016/0022-0396(69)90091-6.  Google Scholar [10] A. Friedman, "Partial Differential Equations of Parabolic Type,", Dover, (1992).   Google Scholar [11] W. Gangbo and A. Tudorascu, Lagrangian dynamics on an infinite-dimensional torus; a weak KAM theorem,, Adv. Math., 224 (2010), 260.  doi: 10.1016/j.aim.2009.11.005.  Google Scholar [12] W. Gangbo, T. Nguyen and A. Tudorascu, Hamilton-Jacobi equations in the Wasserstein space,, Methods Appl. Anal., 15 (2008), 155.   Google Scholar [13] D. Gomes, A stochastic analog of Aubry-Mather theory,, Nonlinearity, 15 (2002), 581.  doi: 10.1088/0951-7715/15/3/304.  Google Scholar [14] T. Hida, "Brownian Motion,", Applications of Mathematics, 11 (1980).   Google Scholar [15] J. N. Mather, Variational construction of connecting orbits,, Ann. Inst. Fourier (Grenoble), 43 (1993), 1349.  doi: 10.5802/aif.1377.  Google Scholar [16] M. Viana, "Stochastic Dynamics of Deterministic Systems,", mimeographed notes, (2000).   Google Scholar [17] C. Villani, "Topics in Optimal Transportation,", Graduate Studies in Mathematics, 58 (2003).  doi: 10.1007/b12016.  Google Scholar
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