On the zero-temperature or vanishing viscosity limit for certain Markov processes arising from Lagrangian dynamics

Abstract

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.