# Resistance Forms, Quasisymmetric Maps and Heat Kernel Estimates

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*Jun Kigami*

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

# Table of Contents

## Resistance Forms, Quasisymmetric Maps and Heat Kernel Estimates

- Chapter 1. Introduction 18 free
- Part 1. Resistance forms and heat kernels 714 free
- Chapter 2. Topology associated with a subspace of functions 916
- Chapter 3. Basics on resistance forms 1320
- Chapter 4. The Green function 1724
- Chapter 5. Topologies associated with resistance forms 2128
- Chapter 6. Regularity of resistance forms 2532
- Chapter 7. Annulus comparable condition and local property 2734
- Chapter 8. Trace of resistance form 3138
- Chapter 9. Resistance forms as Dirichlet forms 3542
- Chapter 10. Transition density 3946

- Part 2. Quasisymmetric metrics and volume doubling measures 4754
- Part 3. Volume doubling measures and heat kernel estimates 6572
- Part 4. Random Sierpinski gaskets 9198
- Chapter 20. Generalized Sierpinski gasket 93100
- Chapter 21. Random Sierpinski gasket 99106
- Chapter 22. Resistance forms on Random Sierpinski gaskets 103110
- Chapter 23. Volume doubling property 109116
- Chapter 24. Homogeneous case 115122
- Chapter 25. Introducing randomness 121128
- Bibliography 123130
- Assumptions, Conditions and Properties in Parentheses 127134
- List of Notations 129136
- Index 131138 free