Geodesic
On the zero-temperature or vanishing viscosity limit for certain Markov processes arising from Lagrangian dynamics
Journal of the European Mathematical Society, Tome 6 (2004) no. 2, pp. 207-276 Cet article a éte moissonné depuis la source EMS Press

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We study the zero-temperature limit for Gibbs measures associated to Frenkel-Kontorova models on (R d) Z/Z d . We prove that equilibrium states concentrate on configurations of minimal energy, and, in addition, must satisfy a variational principle involving metric entropy and Lyapunov exponents, a bit like in the Ruelle-Pesin inequality. Then we transpose the result to certain continuous-time stationary stochastic processes associated to the viscous Hamilton-Jacobi equation. As the viscosity vanishes, the invariant measure of the process concentrates on the so-called Mather set of classical mechanics, and must, in addition, minimize the gap in the Ruelle-Pesin inequality.
DOI : 10.4171/jems/9
Classification : 76-XX, 49-XX, 60-XX, 00-XX
Keywords: Gibbs measures, Frenkel-Kontorova models, Viscous Hamilton-Jacobi equation, Mather set, Ruelle-Pesin inequality 
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     author = {Nalini Anantharaman},
     title = {On the zero-temperature or vanishing viscosity limit for certain {Markov} processes arising from {Lagrangian} dynamics},
     journal = {Journal of the European Mathematical Society},
     pages = {207--276},
     year = {2004},
     volume = {6},
     number = {2},
     doi = {10.4171/jems/9},
     url = {http://geodesic.mathdoc.fr/articles/10.4171/jems/9/}
}
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Nalini Anantharaman. On the zero-temperature or vanishing viscosity limit for certain Markov processes arising from Lagrangian dynamics. Journal of the European Mathematical Society, Tome 6 (2004) no. 2, pp. 207-276. doi: 10.4171/jems/9

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