Abstract
We obtain estimates on the exponential rate of decay of the relative entropy from equilibrium for Markov processes with a non-local infinitesimal generator. We adapt some of the ideas coming from the Bakry-Emery approach to this setting. In particular, we obtain volume-independent lower bounds for the Glauber dynamics of interacting point particles and for various classes of hardcore models.
Highlights
The study of contractivity and hypercontractivity of Markov semigroups has received a tremendous impulse from seminal paper [1], which has introduced the so-called Γ2approach, and has originated a number of developments in different directions
In [4] and [5] we have addressed the problem of going beyond spectral gap estimates for non-local operators, looking for estimates on the exponential rate of decay of the relative entropy from equilibrium
In the case of diffusion operators, a strictly positive exponential rate is equivalent to the validity of a logarithmic Sobolev inequality
Summary
The study of contractivity and hypercontractivity of Markov semigroups has received a tremendous impulse from seminal paper [1], which has introduced the so-called Γ2approach, and has originated a number of developments in different directions (see e.g. [9, 12, 14]). For Brownian diffusions in a convex potential, the Γ2-approach provides a short and elegant proof of the fact that lower bounds on the Hessian of the potential translate into lower bounds for both the spectral gap and the logarithmic Sobolev constant How much these ideas can be adapted to non-local operators, such as generators of discrete Markov chains, is not yet fully understood. This paper improves substantially the results mentioned above; we obtain, high temperature estimates on the best constant in the entropy inequality for Glauber-type dynamics of interacting systems. The main example concerns interacting point particles, where estimates on the spectral gap, as well as constants for other functional inequalities, have been obtained with various techniques [2, 11, 16]. The rest of the paper is devoted to specific examples
Sign-in/Register to view highlights & summary 
Published Version (
Free)
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call