Summary
We study the homogenization problem on nested fractals. LetX t be the continuous time Markov chain on the pre-nested fractal given by puttingi.i.d. random resistors on each cell. It is proved that under some conditions,\(\alpha ^{ - n} X_{t_E^n t} \) converges in law to a constant time change of the Brownian motion on the fractal asn→∞, where α is the contraction rate andt E is a time scale constant. As the Brownian motion on fractals is not a semi-martingale, we need a different approach from the well-developed martingale method.
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Dedicated to Professor Masatoshi Fukushima on his 60th birthday
Research partially supported by the Yukawa Foundation
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Kumagai, T., Kusuoka, S. Homogenization on nested fractals. Probab. Th. Rel. Fields 104, 375–398 (1996). https://doi.org/10.1007/BF01213686
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DOI: https://doi.org/10.1007/BF01213686
Mathematics Subject Classification (1991)
- 31C25
- 47A35
- 60J60