Functional inequalities such as the Poincaré and log-Sobolev inequalities quantify convergence to equilibrium in continuous-time Markov chains (and more generally Markov processes) by linking properties of the generator to variance and entropy decay. However, in certain applications, such as coarse-graining problems arising in molecular dynamics, it becomes necessary to study entropy decay with respect to a reference measure that is not the steady state. In such settings, the dynamics are typically non-reversible (i.e. not of gradient-flow type), and the classical functional inequality framework tied to equilibrium does not directly apply.

In this talk, I will introduce a generalisation of the log-Sobolev inequality with respect to arbitrary probability measures on a finite state space. This generalisation retains key features of the classical inequality while exhibiting properties relevant for coarse-graining applications, including continuity with respect to the reference measure and explicitly computable lower bounds. As an application, we derive quantitative error bounds for coarse-graining of finite Markov chains.

This talks is based on joint work with Bastian Hilder and Patrick van Meurs.