Non-reversible Lifts of Reversible Diffusions and their Convergence to Equilibrium

Abstract

This thesis is concerned with convergence to stationarity of non-reversible Markov processes with degenerate noise, known as hypocoercivity. We develop a framework that allows to derive rates of convergence towards the invariant measure in <em>L</em>² for a large class of such dynamics, which we term second-order lifts of reversible diffusions. These are motivated by the concept of lifts of Markov chains that was introduced to understand acceleration due to non-reversibility in Markov chain Monte Carlo methods. The six works forming the basis for this thesis are included in the appendix as Chapters A–F. <br/>

The development of this framework is the main content of Chapters A, B and C. The concept of second-order lifts of reversible diffusions is motivated and introduced in Chapter A, providing a structural relationship between many non-reversible dynamics with degenerate noise and simple, reversible diffusions that can be exploited to derive a direct lower bound on the <em>L</em>²-relaxation time. We give a first demonstration of how it can also be used to obtain upper bounds on the relaxation time. In Chapter B, we extend this framework to processes on Riemannian manifolds with boundary, and prove the quantitative divergence lemma. The latter is a statement purely on the generator of the underlying reversible diffusion, and is a crucial ingredient in the proof of upper bounds on the relaxation time using the approach of second-order lifts. We conclude in Chapter C by turning these ideas into a general framework. It allows to derive quantitative bounds on rates of convergence to stationarity for second-order lifts by proving a flow Poincaré inequality, a time-averaged Poincaré inequality along trajectories of the associated transition semigroup, under assumptions that are simple to verify in practice. Chapter D compares our approach to hypocoercivity with the highly influential one developed by Dolbeault, Mouhot and Schmeiser, uncovering several structural similarities. <br/>

The works underlying Chapters E and F focus on two classes of self-interacting processes, namely self-repellent random walks and self-repelling diffusions. We show that these are second-order lifts of a suitable Ornstein-Uhlenbeck process whose invariant probability measure corresponds to that of the environment process. These processes are closely related to Event Chain Monte Carlo methods and models for polymer growth. For both classes, we show that convergence rates to stationarity can be obtained using the framework of second-order lifts and the results of Chapter C. <br/>

We begin with some background on convergence to equilibrium of Markov processes and examples motivating acceleration through non-reversibility in Chapter 1. The main result, the hypocoercivity framework based on second-order lifts and the flow Poincaré inequality, is presented in Chapter 2, focussing on the convergence of Langevin dynamics to highlight the main ideas. Finally, Chapter 3 presents the contributions of the individual projects and some open questions.