Memoirs of the American Mathematical Society 2012; 132 pp; softcover Volume: 216 ISBN10: 082185299X ISBN13: 9780821852996 List Price: US$71 Individual Members: US$42.60 Institutional Members: US$56.80 Order Code: MEMO/216/1015
 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 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  Semiquasisymmetric 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
