Abstract
We study upper estimates of the martingale dimension d m of diffusion processes associated with strong local Dirichlet forms. By applying a general strategy to self-similar Dirichlet forms on self-similar fractals, we prove that d m = 1 for natural diffusions on post-critically finite self-similar sets and that d m is dominated by the spectral dimension for the Brownian motion on Sierpinski carpets.
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Dedicated to Professor Leonard Gross on the occasion of his 80th birthday.
This research was partially supported by KAKENHI (21740094, 24540170).
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Hino, M. Upper estimate of martingale dimension for self-similar fractals. Probab. Theory Relat. Fields 156, 739–793 (2013). https://doi.org/10.1007/s00440-012-0442-3
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DOI: https://doi.org/10.1007/s00440-012-0442-3
Keywords
- Martingale dimension
- Self-similar set
- Sierpinski carpet
- Dirichlet form
Mathematics Subject Classification (2000)
- 60G44
- 28A80
- 31C25
- 60J60