Resistance forms, quasisymmetric maps and heat kernel estimates
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...
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| Format: | E-Book Buch |
| Sprache: | Englisch |
| Veröffentlicht: |
Providence, Rhode Island
American Mathematical Society
2011
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| Ausgabe: | 1 |
| Schriftenreihe: | Memoirs of the American Mathematical Society |
| Schlagworte: | |
| ISBN: | 082185299X, 9780821852996 |
| ISSN: | 0065-9266, 1947-6221 |
| Online-Zugang: | Volltext |
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| Zusammenfassung: | 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
In the course of discussion, considerable numbers of pages are spent on the theory of resistance forms and quasisymmetric maps. |
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| Bibliographie: | "March 2012, volume 216, number 1015 (first of 4 numbers)"--T.p Includes bibliography (p. 123-125) and index |
| ISBN: | 082185299X 9780821852996 |
| ISSN: | 0065-9266 1947-6221 |
| DOI: | 10.1090/S0065-9266-2011-00632-5 |