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3044 results in Probability theory and stochastic processes

Page 50 of 153

Random Walks and Heat Kernels on Graphs
  • Martin T. Barlow
  • Published online:
    09 March 2017
    Print publication:
    23 February 2017
  • This introduction to random walks on infinite graphs gives particular emphasis to graphs with polynomial volume growth. It offers an overview of analytic methods, starting with the connection between random walks and electrical resistance, and then proceeding to study the use of isoperimetric and Poincaré inequalities. The book presents rough isometries and looks at the properties of a graph that are stable under these transformations. Applications include the 'type problem': determining whether a graph is transient or recurrent. The final chapters show how geometric properties of the graph can be used to establish heat kernel bounds, that is, bounds on the transition probabilities of the random walk, and it is proved that Gaussian bounds hold for graphs that are roughly isometric to Euclidean space. Aimed at graduate students in mathematics, the book is also useful for researchers as a reference for results that are hard to find elsewhere.

  • Preface

  • Martin T. Barlow, University of British Columbia, Vancouver
  • Book:
    Random Walks and Heat Kernels on Graphs
    Published online:
    09 March 2017
    Print publication:
    23 February 2017, pp ix-xii

  • Summary

    The topic of random walks on graphs is a vast one, and has close connections with many other areas of probability, as well as analysis, geometry, and algebra. In the probabilistic direction, a random walk on a graph is just a reversible or symmetric Markov chain, and many results on random walks on graphs also hold for more general Markov chains. However, pursuing this generalisation too far leads to a loss of concrete interest, and in this text the context will be restricted to random walks on graphs where each vertex has a finite number of neighbours.

    Even with these restrictions, there are many topics which this book does not cover – in particular the very active field of relaxation times for finite graphs. This book is mainly concerned with infinite graphs, and in particular those which have polynomial volume growth. The main topic is the relation between geometric properties of the graph and asymptotic properties of the random walk. A particular emphasis is on properties which are stable under minor perturbations of the graph – for example the addition of a number of diagonal edges to the Euclidean lattice Z2. The precise definition of ‘minor perturbation’ is given by the concept of a rough isometry, or quasi-isometry. One example of a property which is stable under these perturbations is transience; this stability is proved in Chapter 2 using electrical networks. A considerably harder theorem, which is one of the main results of this book, is that the property of satisfying Gaussian heat kernel bounds is also stable under rough isometries.

    The second main theme of this book is deriving bounds on the transition density of the random walk, or the heat kernel, from geometric information on the graph. Once one has these bounds, many properties of the random walk can then be obtained in a straightforward fashion.

    Chapter 1 gives the basic definition of graphs and random walks, as well as that of rough isometry. Chapter 2 explores the close connections between random walks and electrical networks.

Non-homogeneous Random Walks
  • Lyapunov Function Methods for Near-Critical Stochastic Systems
  • Mikhail Menshikov, Serguei Popov, Andrew Wade
  • Published online:
    02 February 2017
    Print publication:
    22 December 2016
  • Stochastic systems provide powerful abstract models for a variety of important real-life applications: for example, power supply, traffic flow, data transmission. They (and the real systems they model) are often subject to phase transitions, behaving in one way when a parameter is below a certain critical value, then switching behaviour as soon as that critical value is reached. In a real system, we do not necessarily have control over all the parameter values, so it is important to know how to find critical points and to understand system behaviour near these points. This book is a modern presentation of the 'semimartingale' or 'Lyapunov function' method applied to near-critical stochastic systems, exemplified by non-homogeneous random walks. Applications treat near-critical stochastic systems and range across modern probability theory from stochastic billiards models to interacting particle systems. Spatially non-homogeneous random walks are explored in depth, as they provide prototypical near-critical systems.

Page 50 of 153