### Abstract

We study large deviations for Brownian motion on the Sierpinski gasket in the short time limit. Because of the subtle oscillation of hitting times of the process, no large deviation principle can hold. In fact, our result shows that there is an infinity of different large deviation principles for different subsequences, with different (good) rate functions. Thus, instead of taking the time scaling ε→0, we prove that the large deviations hold for ε_{n}
^{z}≡(25)^{n}z as n→∞ using one parameter family of rate functions I^{z}(z∈[25,1)). As a corollary, we obtain Strassen-type laws of the iterated logarithm.

Original language | English (US) |
---|---|

Pages (from-to) | 225-235 |

Number of pages | 11 |

Journal | Stochastic Processes and their Applications |

Volume | 85 |

Issue number | 2 |

State | Published - Feb 1 2000 |

### Fingerprint

### Keywords

- 60F10
- 60J60
- 60J80
- Branching process
- Diffusion
- Fractal
- Large deviation
- Sierpinski gasket

### ASJC Scopus subject areas

- Statistics, Probability and Uncertainty
- Mathematics(all)
- Statistics and Probability

### Cite this

*Stochastic Processes and their Applications*,

*85*(2), 225-235.

**Large deviations for Brownian motion on the Sierpinski gasket.** / Arous, Gerard Ben; Kumagai, Takashi.

Research output: Contribution to journal › Article

*Stochastic Processes and their Applications*, vol. 85, no. 2, pp. 225-235.

}

TY - JOUR

T1 - Large deviations for Brownian motion on the Sierpinski gasket

AU - Arous, Gerard Ben

AU - Kumagai, Takashi

PY - 2000/2/1

Y1 - 2000/2/1

N2 - We study large deviations for Brownian motion on the Sierpinski gasket in the short time limit. Because of the subtle oscillation of hitting times of the process, no large deviation principle can hold. In fact, our result shows that there is an infinity of different large deviation principles for different subsequences, with different (good) rate functions. Thus, instead of taking the time scaling ε→0, we prove that the large deviations hold for εn z≡(25)nz as n→∞ using one parameter family of rate functions Iz(z∈[25,1)). As a corollary, we obtain Strassen-type laws of the iterated logarithm.

AB - We study large deviations for Brownian motion on the Sierpinski gasket in the short time limit. Because of the subtle oscillation of hitting times of the process, no large deviation principle can hold. In fact, our result shows that there is an infinity of different large deviation principles for different subsequences, with different (good) rate functions. Thus, instead of taking the time scaling ε→0, we prove that the large deviations hold for εn z≡(25)nz as n→∞ using one parameter family of rate functions Iz(z∈[25,1)). As a corollary, we obtain Strassen-type laws of the iterated logarithm.

KW - 60F10

KW - 60J60

KW - 60J80

KW - Branching process

KW - Diffusion

KW - Fractal

KW - Large deviation

KW - Sierpinski gasket

UR - http://www.scopus.com/inward/record.url?scp=0013298664&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0013298664&partnerID=8YFLogxK

M3 - Article

AN - SCOPUS:0013298664

VL - 85

SP - 225

EP - 235

JO - Stochastic Processes and their Applications

JF - Stochastic Processes and their Applications

SN - 0304-4149

IS - 2

ER -