eBook ISBN:  9780821885239 
Product Code:  MEMO/216/1015.E 
List Price:  $71.00 
MAA Member Price:  $63.90 
AMS Member Price:  $42.60 
eBook ISBN:  9780821885239 
Product Code:  MEMO/216/1015.E 
List Price:  $71.00 
MAA Member Price:  $63.90 
AMS Member Price:  $42.60 

Book DetailsMemoirs of the American Mathematical SocietyVolume: 216; 2012; 132 ppMSC: Primary 30; 31; 60; Secondary 28; 43
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

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. Semiquasisymmetric 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


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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.

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. Semiquasisymmetric 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