Functional inequalities and convergence of stochastic processes

By: Material type: TextTextPublication details: Institute of Science and Technology Austria 2024Online resources:
Contents:
Abstract
Acknowledgements
About the Author
List of Collaborators and Publications
Table of Contents
1 Introduction
2 Local Conditions for Global Convergence of Gradient Flows and Proximal Point Sequences in Metric Spaces
3 Contractive coupling rates and curvature lower bounds for Markov chains
4 Improved Convergence of Score-Based Diffusion Models via Prediction-Correction
5 Heat flow, log-concavity, and Lipschitz transport maps
6 L^∞-optimal transport of anisotropic log-concave measures and exponential convergence in Fisher's infinitesimal model
Bibliography
Summary: This thesis deals with the study of stochastic processes and their ergodicity properties. The variety of problems encountered calls for a set of different approaches, ranging from classical to modern ones: a special place is held by probabilistic methods based on couplings, by functional inequalities, and by the theory of gradient flows in the space of measures. The material is organized as follows. Chapter 1 contains the introduction to this thesis, starting with a general presentation of some of the relevant topics. Section 1.1 is dedicated to the theory of gradient flows in metric spaces, and introduces the first contribution of this thesis [DSMP24], which is presented in detail in Chapter 2. Section 1.2 moves to the topic of curvature of Markov chains, concluding with a brief description of our second contribution [Ped23], which is included in Chapter 3. Section 1.3 discusses applications of stochastic processes to the theory of sampling, in particular the recent framework of score-based diffusion models, and our contribution [PMM24], which is contained in Chapter 4. Section 1.4 discusses some related problems, concerning the regularization properties of the heat flow. It serves as a motivation for the work [BP24], which we report in Chapter 5. Finally, Section 1.5 discusses the last contribution of this thesis, which can be found in Chapter 6. It deals with the convergence to equilibrium of a particular stochastic model from quantitative genetics: this is established via some functional inequalities, which we prove with probabilistic arguments based on couplings.
List(s) this item appears in: ISTA Thesis | New Arrivals October 2025

Thesis

Abstract

Acknowledgements

About the Author

List of Collaborators and Publications

Table of Contents

1 Introduction

2 Local Conditions for Global Convergence of Gradient Flows and Proximal Point Sequences in Metric Spaces

3 Contractive coupling rates and curvature lower bounds for Markov chains

4 Improved Convergence of Score-Based Diffusion Models via Prediction-Correction

5 Heat flow, log-concavity, and Lipschitz transport maps

6 L^∞-optimal transport of anisotropic log-concave measures and exponential convergence in Fisher's infinitesimal model

Bibliography

This thesis deals with the study of stochastic processes and their ergodicity properties. The variety of problems encountered calls for a set of different approaches, ranging from classical to modern ones: a special place is held by probabilistic methods based on couplings, by functional inequalities, and by the theory of gradient flows in the space of measures. The material is organized as follows. Chapter 1 contains the introduction to this thesis, starting with a general presentation of some of the relevant topics. Section 1.1 is dedicated to the theory of gradient flows in metric spaces, and introduces the first contribution of this thesis [DSMP24], which is presented in detail in Chapter 2. Section 1.2 moves to the topic of curvature of Markov chains, concluding with a brief description of our second contribution [Ped23], which is included in Chapter 3. Section 1.3 discusses applications of stochastic processes to the theory of sampling, in particular the recent framework of score-based diffusion models, and our contribution [PMM24], which is contained in Chapter 4. Section 1.4 discusses some related problems, concerning the regularization properties of the heat flow. It serves as a motivation for the work [BP24], which we report in Chapter 5. Finally, Section 1.5 discusses the last contribution of this thesis, which can be found in Chapter 6. It deals with the convergence to equilibrium of a particular stochastic model from quantitative genetics: this is established via some functional inequalities, which we prove with probabilistic arguments based on couplings.

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