Fluctuation of the transition density for Brownian motion on random recursive Sierpinski gaskets

B. M. Hambly; Takashi Kumagai

doi:10.1016/S0304-4149(00)00073-9

Fluctuation of the transition density for Brownian motion on random recursive Sierpinski gaskets

B. M. Hambly^*, Takashi Kumagai

^*Corresponding author for this work

Research output: Contribution to journal › Article › peer-review

8 Citations (Scopus)

Abstract

We consider a class of random recursive Sierpinski gaskets and examine the short-time asymptotics of the on-diagonal transition density for a natural Brownian motion. In contrast to the case of divergence form operators in Rⁿ or regular fractals we show that there are unbounded fluctuations in the leading order term. Using the resolvent density we are able to explicitly describe the fluctuations in time at typical points in the fractal and, by considering the supremum and infimum of the on-diagonal transition density over all points in the fractal, we can also describe the fluctuations in space.

Original language	English
Pages (from-to)	61-85
Number of pages	25
Journal	Stochastic Processes and their Applications
Volume	92
Issue number	1
DOIs	https://doi.org/10.1016/S0304-4149(00)00073-9
Publication status	Published - 2001 Mar
Externally published	Yes

Keywords

60J65
General branching process
Heat kernel
Laplace operator
Primary 60J60
Random recursive fractals
Resolvent density
Secondary 60J25

ASJC Scopus subject areas

Statistics and Probability
Modelling and Simulation
Applied Mathematics

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10.1016/S0304-4149(00)00073-9

Cite this

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title = "Fluctuation of the transition density for Brownian motion on random recursive Sierpinski gaskets",

abstract = "We consider a class of random recursive Sierpinski gaskets and examine the short-time asymptotics of the on-diagonal transition density for a natural Brownian motion. In contrast to the case of divergence form operators in Rn or regular fractals we show that there are unbounded fluctuations in the leading order term. Using the resolvent density we are able to explicitly describe the fluctuations in time at typical points in the fractal and, by considering the supremum and infimum of the on-diagonal transition density over all points in the fractal, we can also describe the fluctuations in space.",

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AU - Hambly, B. M.

AU - Kumagai, Takashi

N1 - Funding Information: Research partly supported by a Royal Society Industry Fellowship at BRIMS, Hewlett-Packard and Grant-in-Aid for Scientific Research (B)(2) 10440029.

PY - 2001/3

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N2 - We consider a class of random recursive Sierpinski gaskets and examine the short-time asymptotics of the on-diagonal transition density for a natural Brownian motion. In contrast to the case of divergence form operators in Rn or regular fractals we show that there are unbounded fluctuations in the leading order term. Using the resolvent density we are able to explicitly describe the fluctuations in time at typical points in the fractal and, by considering the supremum and infimum of the on-diagonal transition density over all points in the fractal, we can also describe the fluctuations in space.

AB - We consider a class of random recursive Sierpinski gaskets and examine the short-time asymptotics of the on-diagonal transition density for a natural Brownian motion. In contrast to the case of divergence form operators in Rn or regular fractals we show that there are unbounded fluctuations in the leading order term. Using the resolvent density we are able to explicitly describe the fluctuations in time at typical points in the fractal and, by considering the supremum and infimum of the on-diagonal transition density over all points in the fractal, we can also describe the fluctuations in space.

KW - 60J65

KW - General branching process

KW - Heat kernel

KW - Laplace operator

KW - Primary 60J60

KW - Random recursive fractals

KW - Resolvent density

KW - Secondary 60J25

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Fluctuation of the transition density for Brownian motion on random recursive Sierpinski gaskets

Abstract

Keywords

ASJC Scopus subject areas

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