Explicit formulas for heat kernels on diamond fractals

Abstract

This paper provides explicit pointwise formulas for the heat kernel on compact metric measure spaces that belong to a $(\mathbb{N}\times \mathbb{N})$-parameter family of fractals which are regarded as projective limits of metric measure graphs and do not satisfy the volume doubling property. The formulas are applied to obtain uniform continuity estimates of the heat kernel and to derive an expression of the fundamental solution of the free Schrödinger equation. The results also open up the possibility to approach infinite dimensional spaces based on this model.

19 pages, 7 figures

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• Regularized Laplacian determinants of self-similar fractals
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• Renormalization-Group Theory of the Heisenberg Model in d Dimensions
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• Phase Transitions of the Variety of Random-Field Potts Models
• Canonical diffusions on the pattern spaces of aperiodic Delone sets
• Hamiltonian systems, Toda lattices, Solitons, Lax Pairs on weighted Z-graded graphs
• Nematic Ordering in the Heisenberg Spin-Glass System in d=3 Dimensions
• Spin-s Spin-Glass Phases in the d=3 Ising Model
• Heat kernel analysis on diamond fractals
• Metastable Reverse-Phase Droplets within Ordered Phases: Renormalization-Group Calculation of Field and Temperature Dependence of Limiting Size