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Resistance Forms, Quasisymmetric Maps and Heat Kernel Estimates
Jun Kigami, Kyoto University, Japan

Memoirs of the American Mathematical Society
2012; 132 pp; softcover
Volume: 216
ISBN-10: 0-8218-5299-X
ISBN-13: 978-0-8218-5299-6
List Price: US$71
Individual Members: US$42.60
Institutional Members: US$56.80
Order Code: MEMO/216/1015
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Assume that there is some analytic structure, a differential equation or a stochastic process for example, on a metric space. To describe asymptotic behaviors of analytic objects, the original metric of the space may not be the best one. Every now and then one can construct a better metric which is somehow "intrinsic" with respect to the analytic structure and under which asymptotic behaviors of the analytic objects have nice expressions. The problem is when and how one can find such a metric.

In this paper, the author considers the above problem in the case of stochastic processes associated with Dirichlet forms derived from resistance forms.

The author's main concerns are the following two problems:

(I) When and how to find a metric which is suitable for describing asymptotic behaviors of the heat kernels associated with such processes.

(II) What kind of requirement for jumps of a process is necessary to ensure good asymptotic behaviors of the heat kernels associated with such processes.

Table of Contents

  • Introduction
Part 1. Resistance forms and heat kernels
  • Topology associated with a subspace of functions
  • Basics on resistance forms
  • The Green function
  • Topologies associated with resistance forms
  • Regularity of resistance forms
  • Annulus comparable condition and local property
  • Trace of resistance form
  • Resistance forms as Dirichlet forms
  • Transition density
Part 2. Quasisymmetric metrics and volume doubling measures
  • Semi-quasisymmetric metrics
  • Quasisymmetric metrics
  • Relations of measures and metrics
  • Construction of quasisymmetric metrics
Part 3. Volume doubling measures and heat kernel estimates
  • Main results on heat kernel estimates
  • Example: the \(\alpha\)-stable process on \(\mathbb{R}\)
  • Basic tools in heat kernel estimates
  • Proof of Theorem 15.6
  • Proof of Theorems 15.10, 15.11, and 15.13
Part 4. Random Sierpinski gaskets
  • Generalized Sierpinski gasket
  • Random Sierpinski gasket
  • Resistance forms on Random Sierpinski gaskets
  • Volume doubling property
  • Homogeneous case
  • Introducing randomness
  • Bibliography
  • Assumptions, conditions and properties in parentheses
  • List of notations
  • Index
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