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Some Topics in Probability and Analysis
Richard F. Gundy
A co-publication of the AMS and CBMS.
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CBMS Regional Conference Series in Mathematics
1989; 49 pp; softcover
Number: 70
ISBN-10: 0-8218-0721-8
ISBN-13: 978-0-8218-0721-7
List Price: US$22
Member Price: US$17.60
All Individuals: US$17.60
Order Code: CBMS/70
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This book is based on lectures presented by the author at DePaul University in July 1986. The lectures cover three main topics. The first is local time theory for Brownian motion and some geometrical inequalities for harmonic functions in the upper half-plane \(R^{n+1}_+\). The author sketches a proof of the inequalities obtained by Barlow and Yor for the maximal local time functional. The second topic concerns a probabilistic treatment of Riesz transforms in \(R^{n+1}_+\), and semimartingale inequalities. The author proves semimartingale inequalities of the type usually obtained for martingales. The final topic centers on a discussion of the Ornstein-Uhlenbeck semigroup and P. A. Meyer's extension of the Riesz inequalities for the infinite-dimensional version of this semigroup. One of the major results of the book is the establishment of inequalities for the density of the area integral.

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Table of Contents

  • The Barlow-Yor inequalities
  • The density of the area integral
  • Norm inequalities for \(D\)
  • Local estimates
  • Terminal expectations and singular integrals
  • The Ornstein-Uhlenbeck semigroup
  • P. A. Meyer's Riesz transform inequalities
  • The Ornstien-Uhlenbeck process for the Rademacher functions
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