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Proceedings of the London Mathematical Society Advance Access originally published online on February 2, 2007
Proceedings of the London Mathematical Society 2007 94(3):672-694; doi:10.1112/plms/pdl025
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© 2007 London Mathematical Society

Heat kernel fluctuations for a resistance form with non-uniform volume growth

D. A. Croydon

Department of Statistics
University of Warwick Coventry
CV4 7AL
United Kingdom
d.a.croydon{at}warwick.ac.uk

Received 12 August 2005. Revision received 16 May 2006.

In this article, we consider the problem of estimating the heat kernel on measure-metric spaces equipped with a resistance form. Such spaces admit a corresponding resistance metric that reflects the conductivity properties of the set. In this situation, it has been proved that when there is uniform polynomial volume growth with respect to the resistance metric the behaviour of the on-diagonal part of the heat kernel is completely determined by this rate of volume growth. However, recent results have shown that for certain random fractal sets, there are global and local (point-wise) fluctuations in the volume as r -> 0 and so these uniform results do not apply. Motivated by these examples, we present global and local on-diagonal heat kernel estimates when the volume growth is not uniform, and demonstrate that when the volume fluctuations are non-trivial, there will be non-trivial fluctuations of the same order (up to exponents) in the short-time heat kernel asymptotics. We also provide bounds for the off-diagonal part of the heat kernel. These results apply to deterministic and random self-similar fractals, and metric space dendrites (the topological analogues of graph trees).


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