| eBook ISBN: | 978-0-8218-8523-9 |
| Product Code: | MEMO/216/1015.E |
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| AMS Member Price: | $42.60 |
| eBook ISBN: | 978-0-8218-8523-9 |
| Product Code: | MEMO/216/1015.E |
| List Price: | $71.00 |
| MAA Member Price: | $63.90 |
| AMS Member Price: | $42.60 |
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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.
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Table of Contents
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Chapters
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1. Introduction
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1. Resistance forms and heat kernels
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2. Topology associated with a subspace of functions
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3. Basics on resistance forms
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4. The Green function
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5. Topologies associated with resistance forms
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6. Regularity of resistance forms
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7. Annulus comparable condition and local property
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8. Trace of resistance form
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9. Resistance forms as Dirichlet forms
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10. Transition density
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2. Quasisymmetric metrics and volume doubling measures
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11. Semi-quasisymmetric metrics
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12. Quasisymmetric metrics
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13. Relations of measures and metrics
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14. Construction of quasisymmetric metrics
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3. Volume doubling measures and heat kernel estimates
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15. Main results on heat kernel estimates
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16. Example: the $\alpha $-stable process on $\mathbb {R}$
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17. Basic tools in heat kernel estimates
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18. Proof of Theorem
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19. Proof of Theorems , and
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4. Random Sierpinski gaskets
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20. Generalized Sierpinski gasket
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21. Random Sierpinski gasket
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22. Resistance forms on Random Sierpinski gaskets
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23. Volume doubling property
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24. Homogeneous case
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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. 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
