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3023 results in Probability theory and stochastic processes

Page 93 of 152

  • 9 - Intersections and self-intersections of Brownian paths

  • Peter Mörters, University of Bath, Yuval Peres
  • Book:
    Brownian Motion
    Published online:
    05 June 2012
    Print publication:
    25 March 2010, pp 255-289

  • Summary

    In this chapter we study multiple points of d-dimensional Brownian motion. We shall see, for example, in which dimensions the Brownian path has double points and explore how many double points there are. This chapter also contains some of the highlights of the book: a proof that planar Brownian motion has points of infinite multiplicity, the intersection equivalence of Brownian motion and percolation limit sets, and the surprising dimension-doubling theorem of Kaufman.

    Intersection of paths: Existence and Hausdorff dimension

    Existence of intersections

    Suppose that {B1 (t): t ≥ 0} and {B2(t): t ≥ 0} are two independent d-dimensional Brownian motions started in arbitrary points. The question we ask in this section is, in which dimensions the ranges, or paths, of the two motions have a nontrivial intersection, in other words whether there exist times t1, t1 > 0 such that B1(t1) = B2(t2). As this question is easy if d = 1 we assume d ≥ 2 throughout this section.

    We have developed the tools to decide this question in Chapter 4 and Chapter 8. Keeping the path {B1(t): t ≥ 0} fixed, we have to decide whether it is a polar set for the second Brownian motion. By Kakutani's theorem, Theorem 8.20, this question depends on its capacity with respect to the potential kernel. As the capacity is again related to Hausdorff measure and dimension, the results of Chapter 4 are crucial in the proof of the following result.

Ergodicity for Infinite Dimensional Systems
  • G. Da Prato, J. Zabczyk
  • Published online:
    24 March 2010
    Print publication:
    16 May 1996
  • This book is devoted to the asymptotic properties of solutions of stochastic evolution equations in infinite dimensional spaces. It is divided into three parts: Markovian dynamical systems; invariant measures for stochastic evolution equations; invariant measures for specific models. The focus is on models of dynamical processes affected by white noise, which are described by partial differential equations such as the reaction-diffusion equations or Navier–Stokes equations. Besides existence and uniqueness questions, special attention is paid to the asymptotic behaviour of the solutions, to invariant measures and ergodicity. Some of the results found here are presented for the first time. For all whose research interests involve stochastic modelling, dynamical systems, or ergodic theory, this book will be an essential purchase.

Stochastic Equations in Infinite Dimensions
  • Guiseppe Da Prato, Jerzy Zabczyk
  • Published online:
    21 March 2010
    Print publication:
    03 December 1992
  • The aim of this book is to give a systematic and self-contained presentation of basic results on stochastic evolution equations in infinite dimensional, typically Hilbert and Banach, spaces. These are a generalization of stochastic differential equations as introduced by Itô and Gikham that occur, for instance, when describing random phenomena that crop up in science and engineering, as well as in the study of differential equations. The book is divided into three parts. In the first the authors give a self-contained exposition of the basic properties of probability measure on separable Banach and Hilbert spaces, as required later; they assume a reasonable background in probability theory and finite dimensional stochastic processes. The second part is devoted to the existence and uniqueness of solutions of a general stochastic evolution equation, and the third concerns the qualitative properties of those solutions. Appendices gather together background results from analysis that are otherwise hard to find under one roof. The book ends with a comprehensive bibliography that will contribute to the book's value for all working in stochastic differential equations.

Stopping Time Techniques for Analysts and Probabilists
  • L. Egghe
  • Published online:
    20 March 2010
    Print publication:
    27 September 1984
  • This book considers convergence of adapted sequences of real and Banach space-valued integrable functions, emphasizing the use of stopping time techniques. Not only are highly specialized results given, but also elementary applications of these results. The book starts by discussing the convergence theory of martingales and sub-( or super-) martingales with values in a Banach space with or without the Radon-Nikodym property. Several inequalities which are of use in the study of the convergence of more general adapted sequence such as (uniform) amarts, mils and pramarts are proved and sub- and superpramarts are discussed and applied to the convergence of pramarts. Most of the results have a strong relationship with (or in fact are characterizations of) topological or geometrical properties of Banach spaces. The book will interest research and graduate students in probability theory, functional analysis and measure theory, as well as proving a useful textbook for specialized courses on martingale theory.

Markov Processes and Related Problems of Analysis
  • E. B. Dynkin
  • Published online:
    18 March 2010
    Print publication:
    23 September 1982
  • The theory of Markov Processes has become a powerful tool in partial differential equations and potential theory with important applications to physics. Professor Dynkin has made many profound contributions to the subject and in this volume are collected several of his most important expository and survey articles. The content of these articles has not been covered in any monograph as yet. This account is accessible to graduate students in mathematics and operations research and will be welcomed by all those interested in stochastic processes and their applications.

Page 93 of 152