Convergence to equilibrium and functional inequalities
The ergodic theorem states that under certain conditions the time average of a stochastic process converges to its ensemble average. In this research area the goal is to quantify this convergence which can be done with the help of functional inequalities like the Poincaré inequality or the logarithmic Sobolev inequality. The constant appearing in those inequalities determines the speed of the convergence of the time average to the ensemble average. I am particularly interested in the dependence of that constant on parameters appearing in the underlying physical model.
Further reading:
Analysis and Geometry of Markov Diffusion Operators, Authors: D. Barry, I. Gentil, M. Ledoux. Introduction in Equilibrium dynamics of continuous unbounded spin systems, Dissertation of Georg Menz (2010). LSI for Kawasaki dynamics with weak interaction, Communications in Mathematical Physics 307, 817-860, (2011) Uniform logarithmic Sobolev inequalities for conservative spin systems with super-quadratic single-site potential, with Otto, F., Annals of Probability 41 (3B), 2182-2224 (2013). Approximate tensorization of entropy at high temperature. (with Caputo, P. and Tetali, P.), Ann. Fac. Sci. Toulouse Math. (6) 24, no. 4, 691-716 (2015).