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

Journal of the European Mathematical Society (2004)

- Volume: 006, Issue: 2, page 207-276
- ISSN: 1435-9855

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topAnantharaman, 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 -

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