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Resistance forms, quasisymmetric maps and heat kernel estimates
About this Title
Jun Kigami, Graduate School of Informatics, Kyoto University, Kyoto 606-8501, Japan
Publication: Memoirs of the American Mathematical Society
Publication Year:
2012; Volume 216, Number 1015
ISBNs: 978-0-8218-5299-6 (print); 978-0-8218-8523-9 (online)
DOI: https://doi.org/10.1090/S0065-9266-2011-00632-5
Published electronically: May 20, 2011
Keywords: Resistance form,
Green function,
quasisymmetric map,
volume doubling property,
jump process,
heat kernel
MSC: Primary 30L10, 31E05, 60J35; Secondary 28A80, 43A99, 60G52
Table of Contents
Chapters
- 1. Introduction
1. Resistance forms and heat kernels
- 2. Topology associated with a subspace of functions
- 3. Basics on resistance forms
- 4. The Green function
- 5. Topologies associated with resistance forms
- 6. Regularity of resistance forms
- 7. Annulus comparable condition and local property
- 8. Trace of resistance form
- 9. Resistance forms as Dirichlet forms
- 10. Transition density
2. Quasisymmetric metrics and volume doubling measures
- 11. Semi-quasisymmetric metrics
- 12. Quasisymmetric metrics
- 13. Relations of measures and metrics
- 14. Construction of quasisymmetric metrics
3. Volume doubling measures and heat kernel estimates
- 15. Main results on heat kernel estimates
- 16. Example: the $\alpha$-stable process on $\mathbb {R}$
- 17. Basic tools in heat kernel estimates
- 18. Proof of Theorem
- 19. Proof of Theorems , and
4. Random Sierpinski gaskets
- 20. Generalized Sierpinski gasket
- 21. Random Sierpinski gasket
- 22. Resistance forms on Random Sierpinski gaskets
- 23. Volume doubling property
- 24. Homogeneous case
- 25. Introducing randomness
Abstract
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, we consider the above problem in the case of stochastic processes associated with Dirichlet forms derived from resistance forms. Our main concerns are following two problems:
(I) When and how can we 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?
Note that in general stochastic processes associated with Dirichlet forms have jumps, i. e. paths of such processes are not continuous.
The answer to (I) is for measures to have volume doubling property with respect to the resistance metric associated with a resistance form. Under volume doubling property, a new metric which is quasisymmetric with respect to the resistance metric is constructed and the Li-Yau type diagonal sub-Gaussian estimate of the heat kernel associated with the process using the new metric is shown.
About the question (II), we will propose a condition called annulus comparable condition, (ACC) for short. This condition is shown to be equivalent to the existence of a good diagonal heat kernel estimate.
As an application, asymptotic behaviors of the traces of $1$-dimensional $\alpha$-stable processes are obtained.
In the course of discussion, considerable numbers of pages are spent on the theory of resistance forms and quasisymmetric maps.
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