1 5 M ay 2 01 7 POINCARÉ AND LOGARITHMIC SOBOLEV CONSTANTS FOR METASTABLE MARKOV CHAINS VIA CAPACITARY INEQUALITIES
نویسنده
چکیده
We investigate the metastable behaviour of reversible Markov chains on countable infinite state spaces. Based on a definition of metastable sets, we compute precisely the mean exit time from a metastable set. Under additional size and regularity properties of metastable sets, we establish asymptotic sharp estimates on the Poincaré and logarithmic Sobolev constant. The main ingredient are capacitary inequalities along the lines of V. Maz’ya relating regularity properties of harmonic functions and capacities. We exemplify the usefulness of our approach in the context of the random field Curie-Weiss model. CONTENTS
منابع مشابه
Logarithmic Sobolev, Isoperimetry and Transport Inequalities on Graphs
In this paper, we study some functional inequalities (such as Poincaré inequality, logarithmic Sobolev inequality, generalized Cheeger isoperimetric inequality, transportation-information inequality and transportation-entropy inequality) for reversible nearest-neighbor Markov processes on connected finite graphs by means of (random) path method. We provide estimates of the involved constants.
متن کاملLog-sobolev, Isoperimetry and Transport Inequalities on Graphs
In this paper, we study some functional inequalities (such as Poincaré inequalities, logarithmic Sobolev inequalities, generalized Cheeger isoperimetric inequalities, transportation-information inequalities and transportation-entropy inequalities) for reversible nearest-neighbor Markov processes on a connected finite graph by means of (random) path method. We provide estimates of the involved c...
متن کاملThe Eyring-Kramers formula for Poincaré and logarithmic Sobolev inequalities
The topic of this thesis is a diffusion process on a potential landscape which is given by a smooth Hamiltonian function in the regime of small noise. The work provides a new proof of the Eyring-Kramers formula for the Poincaré inequality of the associated generator of the diffusion. The Poincaré inequality characterizes the spectral gap of the generator and establishes the exponential rate of ...
متن کاملConvex Sobolev inequalities and spectral gap Inégalités de Sobolev convexes et trou spectral
This note is devoted to the proof of convex Sobolev (or generalized Poincaré) inequalities which interpolate between spectral gap (or Poincaré) inequalities and logarithmic Sobolev inequalities. We extend to the whole family of convex Sobolev inequalities results which have recently been obtained by Cattiaux [11] and Carlen and Loss [10] for logarithmic Sobolev inequalities. Under local conditi...
متن کاملFunctional inequalities and uniqueness of the Gibbs measure — from log-Sobolev to Poincaré
In a statistical mechanics model with unbounded spins, we prove uniqueness of the Gibbs measure under various assumptions on finite volume functional inequalities. We follow Royer’s approach ([11]) and obtain uniqueness by showing convergence properties of a Glauber-Langevin dynamics. The result was known when the measures on the box [−n, n] (with free boundary conditions) satisfied the same lo...
متن کامل