Duality for Markov Processes: A Lie Algebraic Approach (Grundlehren der mathematischen Wissenschaften, 365) - Hardcover

Giardinà, Cristian; Redig, Frank

Hardcover

ISBN 10: 3032040981 ISBN 13: 9783032040985

Publisher: Springer, 2026

Synopsis

This monograph studies duality in interacting particle systems, a topic combining probability theory, statistical physics, Lie algebras, and orthogonal polynomials. It offers the first comprehensive account of duality theory in the context of interacting particle systems.

Using a Lie algebraic framework, the book demonstrates how dualities arise in families of systems linked to algebraic representations. The exposition centers on three key processes: independent random walks, the inclusion process, and the exclusion process—associated with the Heisenberg, su(1,1), and su(2) algebras, respectively. From these three basic cases, several new processes and their duality relations are derived. Additional models, such as the Brownian energy process, the KMP model and the Kac model, are also discussed, along with topics like the hydrodynamic limit and non-equilibrium behavior. Further, integrable systems associated to the su(1,1) algebra are studied and their non-equilibrium steady states are computed.
 
Intentionally accessible and self-contained, this book is aimed at graduate-level researchers and also serves as a comprehensive introduction to the duality of Markov processes and beyond.

About this title

About the Author

Cristian Giardinà is a professor of mathematics at Modena and Reggio Emilia University. His research activity centers on probability theory, statistical physics, and interacting particle systems. He has been working on duality properties of Markov processes, the hydrodynamic limit of open systems, large deviations and cloning algorithms, and the Ising model on random graphs. The overall aim is to understand how complex macroscopic behavior arises from microscopic models. He has published over 90 scientific papers, and two monographs on “Free boundary problems in PDE and particle systems” and “Perspectives on spin glasses”. 

Frank Redig is a professor of probability theory at the Delft Institute of Applied Mathematics. His research focuses on Markov process theory, Gibbs measures, non-equilibrium statistical physics, and interacting particle systems.

From the Back Cover

This monograph studies duality in interacting particle systems, a topic combining probability theory, statistical physics, Lie algebras, and orthogonal polynomials. It offers the first comprehensive account of duality theory in the context of interacting particle systems.

 

Using a Lie algebraic framework, the book demonstrates how dualities arise in families of systems linked to algebraic representations. The exposition centers on three key processes: independent random walks, the inclusion process, and the exclusion process—associated with the Heisenberg, su(1,1), and su(2) algebras, respectively. From these three basic cases, several new processes and their duality relations are derived. Additional models, such as the Brownian energy process, the KMP model and the Kac model, are also discussed, along with topics like the hydrodynamic limit and non-equilibrium behavior. Further, integrable systems associated to the su(1,1) algebra are studied and their non-equilibrium steady states are computed.

 

Intentionally accessible and self-contained, this book is aimed at graduate-level researchers and also serves as a comprehensive introduction to the duality of Markov processes and beyond.

Edition details

• Publisher: Springer
• Publication date: 2026
• Language: English
• ISBN 10: 3032040981
• ISBN 13: 9783032040985
• Binding: Hardcover
• Number of pages: 564