**EMS Zurich Lectures in Advanced Mathematics**

Volume: 16;
2012;
122 pp;
Softcover

MSC: Primary 60; 82;
**Print ISBN: 978-3-03719-109-5
Product Code: EMSZLEC/16**

List Price: $36.00

Individual Member Price: $28.80

# Topics in Occupation Times and Gaussian Free Fields

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*Alain-Sol Sznitman*

A publication of the European Mathematical Society

This book grew out of a graduate course at ETH Zürich during the spring 2011 term. It explores various links between such notions as occupation times of Markov chains, Gaussian free fields, Poisson point processes of Markovian loops, and random interlacements, which have been the object of intensive research over the last few years. These notions are developed in the convenient setup of finite weighted graphs endowed with killing measures.

This book first discusses elements of continuous-time Markov chains, Dirichlet forms, potential theory, together with some consequences for Gaussian free fields. Next, isomorphism theorems and generalized Ray–Knight theorems, which relate occupation times of Markov chains to Gaussian free fields, are presented. Markovian loops are constructed and some of their key properties derived. The field of occupation times of Poisson point processes of Markovian loops is investigated. Of special interest are its connection to the Gaussian free field, and a formula of Symanzik. Finally, links between random interlacements and Markovian loops are discussed, and some further connections with Gaussian free fields are mentioned.

A publication of the European Mathematical Society (EMS). Distributed within the Americas by the American Mathematical Society.

#### Table of Contents

# Table of Contents

## Topics in Occupation Times and Gaussian Free Fields

#### Readership

Graduate students and research mathematicians interested in occupation times, Gaussian free fields, Markovian loops, and random interlacements.