Bolier A., den Hollander F. Metastability: A Potential-Theoretic Approach

Springer International Publishing, Switzerland, 2015. — 578 p. — (Grundlehren der mathematischen Wissenschaften 351) — ISBN: 3319247751.This monograph provides a concise presentation of a mathematical approach to metastability, a wide-spread phenomenon in the dynamics of non-linear systems - physical, chemical, biological or economic - subject to the action of temporal random forces typically referred to as noise, based on potential theory of reversible Markov processes.
The authors shed new light on the metastability phenomenon as a sequence of visits of the path of the process to different metastable sets, and focuses on the precise analysis of the respective hitting probabilities and hitting times of these sets.
The theory is illustrated with many examples, ranging from finite-state Markov chains, finite-dimensional diffusions and stochastic partial differential equations, via mean-field dynamics with and without disorder, to stochastic spin-flip and particle-hop dynamics and probabilistic cellular automata, unveiling the common universal features of these systems with respect to their metastable behaviour.
The monograph will serve both as comprehensive introduction and as reference for graduate students and researchers interested in metastability.Background and Motivation
Aims and Scopes
Markov Processes
Some Basic Notions from Probability Theory
Markov Processes in Discrete Time
Markov Processes in Continuous Time
Large Deviations
Potential Theory
Metastability
Key Definitions and Basic Properties
Basic Techniques
Applications: Diffusions with Small Noise
Discrete Reversible Diffusions
Diffusion Processes with Gradient Drift
Stochastic Partial Differential Equations
Applications: Coarse-Graining in Large Volumes at Positive Temperatures
The Curie-Weiss Model
The Curie-Weiss Model with a Random Magnetic Field: Discrete Distributions
The Curie-Weiss Model with Random Magnetic Field: Continuous Distributions
Applications: Lattice Systems in Small Volumes at Low Temperatures
Abstract Set-Up and Metastability in the Zero-Temperature Limit
Glauber Dynamics
Kawasaki Dynamics
Applications: Lattice Systems in Large Volumes at Low Temperatures
Glauber Dynamics
Kawasaki Dynamics
Applications: Lattice Systems in Small Volumes at High Densities
The Zero-Range Process
Challenges
ChallengesWithin Metastability
Challenges Beyond Metastability