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

Anantharaman, Nalini. "On the zero-temperature or vanishing viscosity limit for certain Markov processes arising from Lagrangian dynamics." Journal of the European Mathematical Society 006.2 (2004): 207-276. <http://eudml.org/doc/277277>.

@article{Anantharaman2004,
abstract = {We study the zero-temperature limit for Gibbs measures associated to Frenkel–Kontorova models on $(\mathbb \{R\}^d)^\{\mathbb \{Z\}\}/\mathbb \{Z\}^d$. 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.},
author = {Anantharaman, Nalini},
journal = {Journal of the European Mathematical Society},
keywords = {Gibbs measures; Frenkel-Kontorova models; Viscous Hamilton-Jacobi equation; Mather set; Ruelle-Pesin inequality; Gibbs measures; Frenkel-Kontorova models; Viscous Hamilton-Jacobi equation, Mather set; Ruelle-Pesin inequality},
language = {eng},
number = {2},
pages = {207-276},
publisher = {European Mathematical Society Publishing House},
title = {On the zero-temperature or vanishing viscosity limit for certain Markov processes arising from Lagrangian dynamics},
url = {http://eudml.org/doc/277277},
volume = {006},
year = {2004},
}

TY - JOUR
AU - Anantharaman, Nalini
TI - On the zero-temperature or vanishing viscosity limit for certain Markov processes arising from Lagrangian dynamics
JO - Journal of the European Mathematical Society
PY - 2004
PB - European Mathematical Society Publishing House
VL - 006
IS - 2
SP - 207
EP - 276
AB - We study the zero-temperature limit for Gibbs measures associated to Frenkel–Kontorova models on $(\mathbb {R}^d)^{\mathbb {Z}}/\mathbb {Z}^d$. 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.
LA - eng
KW - Gibbs measures; Frenkel-Kontorova models; Viscous Hamilton-Jacobi equation; Mather set; Ruelle-Pesin inequality; Gibbs measures; Frenkel-Kontorova models; Viscous Hamilton-Jacobi equation, Mather set; Ruelle-Pesin inequality
UR - http://eudml.org/doc/277277
ER -