Asymptotically one-dimensional diffusions on the Sierpinski gasket and theabc-gaskets

Hattori, Kohshi; Hattori, Tetsuya; Watanabe, Hiroshi

doi:10.1007/bf01204955

Cited by 26 publications

(29 citation statements)

References 8 publications

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“…9 and 10 for the diffusion on Sierpifiski gasket, and generalized for scale-irregular fractals, ~7) and pursued on Sierpifiski carpet in the context of resistor networksJ 2, 3) In what follows, we study the asymptotically one-dimensional diffusions on abc-gaskets along the line of ref. 9 where this work was announced.…”

Section: Introductionmentioning

confidence: 87%

“…In Section 4, we give the estimate on the generating function for numbers of steps of anisotropic random walks on the abcgaskets that are sufficient for the construction of the asymptotically onedimensional diffusion along the line of ref. 9. The algebraic part of our proof of Proposition 4.2 is computer-aided because of the routine of rather lengthy calculations.…”

Section: Introductionmentioning

confidence: 98%

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Anisotropic random walks and asymptotically one-dimensional diffusion onabc-gaskets

Hattori

Watanabe

1997

J Stat Phys

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Asymptotically one-dimensional diffusion processes are studied on the class of fractals called abc-gaskets. The class is a set of certain variants of the Sierpiflski gasket containing infinitely many fractals without any nondegenerate fixed point of renormalization maps. While the "standard" method of constructing diffusions on the Sierpifiski gasket and on nested fractals relies on the existence of a nondegenerate fixed point and hence it is not applicable to all abc-gaskets, the asymptotically one-dimensional diffusion is constructed on any abc-gasket by means of an unstable degenerate fixed point. To this end, the generating functions for numbers of steps of anisotropic random walks on the abc-gaskets are analyzed, along the line of the authors' previous studies. In addition, a general stategy of handling random walk sequences with more than one parameter for the construction of asymptotically one-dimensional diffusion is proposed.

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Section: Introductionmentioning

confidence: 87%

Section: Introductionmentioning

confidence: 98%

Anisotropic random walks and asymptotically one-dimensional diffusion onabc-gaskets

Hattori

Watanabe

1997

J Stat Phys

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“…It is well-known that a regular strongly local Dirichlet form associates with a continuous diffusion process [5]. In fact, this probability counter part of E (a 0 ,b 0 ) had been studied by Hattori et al [9] as an asymptotically one-dimensional diffusion processes on the SG. To conclude this section, we give a brief discussion of their study in comparison with our consideration.…”

Section: Proof Of Theorem 12mentioning

confidence: 99%

On a recursive construction of Dirichlet form on the Sierpiński gasket

Lau

Qiu

2019

Journal of Mathematical Analysis and Applications

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Let Γ n denote the n-th level Sierpiński graph of the Sierpiński gasket K. We consider, for any given conductance (a 0 , b 0 , c 0 ) on Γ 0 , the Dirchlet form E on K obtained from a recursive construction of compatible sequence of conductances (a n , b n , c n ) on Γ n , n ≥ 0. We prove that there is a dichotomy situation: either a 0 = b 0 = c 0 and E is the standard Dirichlet form, or a 0 > b 0 = c 0 (or the two symmetric alternatives), and E is a non-self-similar Dirichlet form independent of a 0 , b 0 . The second situation has also been studied in [9, 10] as a one-dimensional asymptotic diffusion process on the Sierpiński gasket. For the spectral property, we give a sharp estimate of the eigenvalue distribution of the associated Laplacian, which improves a similar result in [10].

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“…We are also thankful for the anonymous feedback pointing out the references [19,22,23,35,36] and raising the questions of defining ALD processes on Hanoi attractors and of extending this processes to more general objects out of these examples.…”

Section: Acknowledgesmentioning

confidence: 99%

“…Such processes were first treated in the context of abc-gaskets in [23], and studied later on Hambly and Kumagai in [19] for some particular nested fractals.…”

Section: Conclusion and Open Problemsmentioning

confidence: 99%