Abstract
For the canonical heat kernels p t (x, y) associated with Dirichlet forms on post-critically finite self-similar fractals, e.g. the transition densities (heat kernels) of Brownian motion on affine nested fractals, the non-existence of the limit \({\lim_{t\downarrow 0}t^{d_{s}/2}p_{t}(x,x)}\) is established for a “generic” (in particular, almost every) point x, where d s denotes the spectral dimension. Furthermore the same is proved for any point x in the case of the d-dimensional standard Sierpinski gasket with d ≥ 2 and the N-polygasket with N ≥ 3 odd, e.g. the pentagasket (N = 5) and the heptagasket (N = 7).
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The author was supported by the Japan Society for the Promotion of Science (JSPS Research Fellow PD (20 · 6088)).
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Kajino, N. On-diagonal oscillation of the heat kernels on post-critically finite self-similar fractals. Probab. Theory Relat. Fields 156, 51–74 (2013). https://doi.org/10.1007/s00440-012-0420-9
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DOI: https://doi.org/10.1007/s00440-012-0420-9
Keywords
- Post-critically finite self-similar fractals
- Affine nested fractals
- Dirichlet form
- Heat kernel
- Oscillation
- Short time asymptotics