### Abstract

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 Sierpiński gasket containing infinitely many fractals without any nondegenerate fixed point of renormalization maps. While the "standard" method of constructing diffusions on the Sierpiński 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.

Original language | English |
---|---|

Pages (from-to) | 105-128 |

Number of pages | 24 |

Journal | Journal of Statistical Physics |

Volume | 88 |

Issue number | 1-2 |

Publication status | Published - 1997 Jul |

Externally published | Yes |

### Fingerprint

### Keywords

- Branching process
- Diffusion process
- Finitely ramified fractal
- Random walk
- Renormalization group

### ASJC Scopus subject areas

- Mathematical Physics
- Physics and Astronomy(all)
- Statistical and Nonlinear Physics

### Cite this

*Journal of Statistical Physics*,

*88*(1-2), 105-128.

**Anisotropic random walks and asymptotically one-dimensional diffusion on abc-gaskets.** / Hattori, Tetsuya; Watanabe, Hiroshi.

Research output: Contribution to journal › Article

*Journal of Statistical Physics*, vol. 88, no. 1-2, pp. 105-128.

}

TY - JOUR

T1 - Anisotropic random walks and asymptotically one-dimensional diffusion on abc-gaskets

AU - Hattori, Tetsuya

AU - Watanabe, Hiroshi

PY - 1997/7

Y1 - 1997/7

N2 - 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 Sierpiński gasket containing infinitely many fractals without any nondegenerate fixed point of renormalization maps. While the "standard" method of constructing diffusions on the Sierpiński 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.

AB - 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 Sierpiński gasket containing infinitely many fractals without any nondegenerate fixed point of renormalization maps. While the "standard" method of constructing diffusions on the Sierpiński 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.

KW - Branching process

KW - Diffusion process

KW - Finitely ramified fractal

KW - Random walk

KW - Renormalization group

UR - http://www.scopus.com/inward/record.url?scp=0031184715&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0031184715&partnerID=8YFLogxK

M3 - Article

AN - SCOPUS:0031184715

VL - 88

SP - 105

EP - 128

JO - Journal of Statistical Physics

JF - Journal of Statistical Physics

SN - 0022-4715

IS - 1-2

ER -