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

2021/08/04 by Kajino, Naotaka · 2 citations
#28A80 #31C25 #35K08 #60G30 (Primary) 31E05 #60J60 (Secondary) #FOS: Mathematics #Probability (math.PR)

paper · doi:10.48550/arxiv.2108.02027

Abstract

We present a concrete family of fractals, which we call the (two-dimensional) thin scale irregular Sierpiński 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 Ψ_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^-2Ψ_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 Ψ satisfying lim_{r\downarrow 0}r^-2Ψ(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.

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