Geometric Functional Inequalities for Markov Processes: An Invitation via the Langevin Diffusion

Abstract

Functional inequalities provide a language for reasoning about the long-time properties of ergodic Markov processes in terms of their macroscopic geometry. This article offers an informal introduction to functional inequalities for readers who are more familiar with probabilistic tools based around couplings, Lyapunov techniques, and similar. The main ideas are developed through the overdamped Langevin diffusion, where various functionals related to this convergence admit particularly transparent expressions, and for which the key associated inequalities -- namely, the Poincaré and logarithmic Sobolev inequalities -- are especially well-studied. We explain the qualitative content of these inequalities, what they imply about convergence to equilibrium, how they transform and compose under natural operations, and how they interface with classical probabilistic approaches. Related inequalities are discussed more briefly, and an annotated guide is provided for readers wishing to enter the wider literature.