On singularity of energy measures for symmetric diffusions with full off-diagonal heat kernel estimates II: Some borderline examples

We present a concrete family of fractals, which we call the (two-dimensional) thin scale irregular Sierpi\'{n}ski gaskets and each of which is equipped with a canonical strongly local regular symmetric Dirichlet form. We prove that any fractal \(K\) in this family satisfies the full off-diagona...

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Bibliographic Details
Published in:arXiv.org 2021-11
Main Author: Kajino, Naotaka
Format: Article
Language:English
Subjects:
Decay rate
Dirichlet problem
Estimates
Fractals
Gaskets
Kernels
Singularities
Spacetime
Time
Online Access:Get full text
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Summary:We present a concrete family of fractals, which we call the (two-dimensional) thin scale irregular Sierpi\'{n}ski gaskets and each of which is equipped with a canonical strongly local regular symmetric Dirichlet form. We prove that any fractal \(K\) in this family satisfies the full off-diagonal heat kernel estimates with some space-time scale function \(\Psi_{K}\) and the singularity of the associated energy measures with respect to the canonical volume measure (uniform distribution) on \(K\), and also that the decay rate of \(r^{-2}\Psi_{K}(r)\) to \(0\) as \(r\downarrow 0\) can be made arbitrarily slow by suitable choices of \(K\). These results together support the energy measure singularity dichotomy conjecture [Ann. Probab. 48 (2020), no. 6, 2920--2951, Conjecture 2.15] stating that, if the full off-diagonal heat kernel estimates with space-time scale function \(\Psi\) satisfying \(\lim_{r\downarrow 0}r^{-2}\Psi(r)=0\) hold for a strongly local regular symmetric Dirichlet space with complete metric, then the associated energy measures are singular with respect to the reference measure of the Dirichlet space.
ISSN:2331-8422