The Annals of Probability

Brownian motion on a random recursive Sierpinski gasket

B. M. Hambly

Full-text: Open access

Abstract

We introduce a random recursive fractal based on the Sierpinski gasket and construct a diffusion upon the fractal via a Dirichlet form. This form and its symmetrizing measure are determined by the electrical resistance of the fractal. The effective resistance provides a metric with which to discuss the properties of the fractal and the diffusion. The main result is to obtain uniform upper and lower bounds for the transition density of the Brownian motion on the fractal in terms of this metric. The bounds are not tight as there are logarithmic corrections due to the randomness in the environment, and the behavior of the shortest paths in the effective resistance metric is not well understood. The results are deduced from the study of a suitable general branching process.

Article information

Source
Ann. Probab., Volume 25, Number 3 (1997), 1059-1102.

Dates
First available in Project Euclid: 18 June 2002

Permanent link to this document
https://projecteuclid.org/euclid.aop/1024404506

Digital Object Identifier
doi:10.1214/aop/1024404506

Mathematical Reviews number (MathSciNet)
MR1457612

Zentralblatt MATH identifier
0895.60081

Subjects
Primary: 60J60: Diffusion processes [See also 58J65] 60J25: Continuous-time Markov processes on general state spaces 60J65: Brownian motion [See also 58J65]

Keywords
Brownian motion random fractal Dirichlet form spectral dimension transition density estimate general branching process

Citation

Hambly, B. M. Brownian motion on a random recursive Sierpinski gasket. Ann. Probab. 25 (1997), no. 3, 1059--1102. doi:10.1214/aop/1024404506. https://projecteuclid.org/euclid.aop/1024404506


Export citation

References

  • 1 ASMUSSEN, S. and HERING, K. 1984. Branching Processes. Birkhauser, Boston. ¨
  • 2 BARLOW, M. T. and BASS, R. F. 1989. Construction of Brownian motion on the Sierpinski carpet. Ann. Inst. H. Poincare 25 225 257. ´
  • 3 BARLOW, M. T. and BASS, R. F. 1992. Transition densities for Brownian motion on the Sierpinski carpet. Probab. Theory Related Fields 91 307 330.
  • 4 BARLOW, M. T. and PERKINS, E. A. 1988. Brownian motion on the Sierpinski gasket. Probab. Theory Related Fields 79 543 624.
  • 5 DEKKING, F. M. and GRIMMETT, G. R. 1988. Superbranching processes and projections of random Cantor sets. Probab. Theory Related Fields 78 335 355.
  • 6 FALCONER, K. J. 1986. Random graphs. Math. Proc. Cambridge Philos. Soc. 100 559 582.
  • 7 FALCONER, K. J. 1990. Fractal Geometry: Mathematical Foundations and Applications. Wiley, Chichester.
  • 8 FITZSIMMONS, P. J., HAMBLY, B. M. and KUMAGAI, T. 1994. Transition density estimates for Brownian motion on affine nested fractals. Comm. Math. Phys. 165 595 620.
  • 9 FUKUSHIMA, M. 1980. Dirichlet Forms and Markov Processes. North-Holland, Amsterdam.
  • 10 FUKUSHIMA, M. 1992. Dirichlet forms, diffusion processes and spectral dimensions for nested fractals. In Ideas and Methods in Mathematical Analysis, Stochastics, and Z Applications. In Memory of R. Hoegh-Krohn S. Albeverio, J. E. Fenstad, H. Holden. and T. Lindstrom, eds. 1 151 161. Cambridge Univ. Press.
  • 11 GRAF, S. 1987. Statistically self-similar fractals. Probab. Theory Related Fields 74 357 392.
  • 12 GRAF, S., MAULDIN, R. D. and WILLIAMS, S. C. 1988. The exact Hausdorff dimension in random recursive constructions. Mem. Amer. Math. Soc. 381.
  • 13 HAMBLY, B. M. 1992. On the limiting distribution of a supercritical branching process in a random environment. J. Appl. Probab. 29 499 518.
  • 14 HAMBLY, B. M. 1992. Brownian motion on a homogeneous random fractal. Probab. Theory Related Fields 94 1 38.
  • 15 HATTORI, T. 1994. Asymptotically one-dimensional diffusions on scale irregular gaskets. Preprint.
  • 16 JAGERS, P. 1975. Branching Processes with Biological Applications. Wiley, Chichester.
  • 17 KIGAMI, J. 1992. Harmonic calculus on p.c.f. self-similar sets. Trans. Amer. Math. Soc. 335 721 755.
  • 18 KIGAMI, J. 1995. Hausdorff dimension of self-similar sets and shortest path metrics. J. Math. Soc. Japan. 43 381 404.
  • 19 KIGAMI, J. 1994. Effective resistances for harmonic structures on P.C.F self-similar sets. Math. Proc. Cambridge Philos. Soc. 115 291 303.
  • 20 KIGAMI, J. 1995. Harmonic calculus on limits of networks and its application to dendrites. J. Funct. Anal. 128 48 86.
  • 21 KUMAGAI, T. 1993. Estimates of transition densities for Brownian motion on nested fractals. Probab. Theory Related Fields 96 205 224.
  • 22 KUSUOKA, S. 1993. Diffusion processes on nested fractals. Statistical Mechanics and Fractals. Lecture Notes in Math. 1567. Springer, Berlin.
  • 23 LINDSTROM, T. 1990. Brownian motion on nested fractals. Mem. Amer. Math. Soc. 420.
  • 24 MARCUS, M. B. and ROSEN, J. 1992. Sample path properties of the local times of strongly symmetric Markov processes via Gaussian processes. Ann. Probab. 20 1603 1684.
  • 25 MAULDIN, R. D. and WILLIAMS, S. C. 1986. Random recursive constructions: asymptotic geometric and topological properties. Trans. Amer. Math. Soc. 295 325 346.
  • 26 METZ, V. 1995. Renormalization of finitely ramified fractals. Proc. Roy. Soc. Edinburgh Sect. A 125 1085 1104.
  • 27 MILOSEVIC, S., STASSINOPOULOS, D. and STANLEY, H. E. 1988. Asymptotic behaviour of the spectral dimension at the fractal to lattice crossover. J. Phys. A 21 1477 1482.
  • 28 TAYLOR, S. J. 1987. On the measure theory of random fractals. Math. Proc. Cambridge Philos. Soc. 100 345 367.