Probability Theory and Related Fields

, Volume 101, Issue 3, pp 421–433 | Cite as

Brownian motion on the continuum tree

  • W. B. Krebs
Article

Summary

We construct Brownian motion on a continuum tree, a structure introduced as an asymptotic limit to certain families of finite trees. We approximate the Dirichlet form of Brownian motion on the continuum tree by adjoining one-dimensional Brownian excursions. We study the local times of the resulting diffusion. Using time-change methods, we find explicit expressions for certain hitting probabilities and the mean occupation density of the process.

Mathematics Subject Classification

60J65 31C25 60J55 

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References

  1. 1.
    Albeverio, S., Røckner, M.: Classical Dirichlet forms on topological vector spaces-closability and a Cameron-Martin formula. J. Funct. Anal.88, 395–436 (1990)Google Scholar
  2. 2.
    Aldous, D.J.: The continuum random tree I. Ann. probab.19, 1–28 (1991)Google Scholar
  3. 3.
    Aldous, D.J.: The continuum random tree II: an overview. In: Stochastic Anal. (eds.) Cambridge University Press, Cambridge, New York: Barlow, M.T., Bingham, N.H. 1992Google Scholar
  4. 4.
    Aldous, D.J.: The continuum random tree III. Ann. Probab.21 248–289 (1993)Google Scholar
  5. 5.
    Barlow, M., Bass, R.: The construction of Brownian motion on the Sierpinski carpet. Ann. Inst. Henri Poincaré.25, 225–257 (1989)Google Scholar
  6. 6.
    Barlow, M., Bass, R.: Local times for Brownian motion on the Sierpinski carpet. Probab. Theory Relat. Fields.85, 91–104 (1990)Google Scholar
  7. 7.
    Barlow, M., Bass, R.: Transition densities for Brownian motion on the Sierpinski carpet (1991)Google Scholar
  8. 8.
    Barlow, M.T., Perkins, E.A.: Brownian motion on the Sierpinski Gasket. Probab. Theory Relat. Fields.9, 543–623 (1987)Google Scholar
  9. 9.
    Fukushima, M.: Dirichlet forms and Markov processes. New York: North-Holland 1980Google Scholar
  10. 10.
    Hamby, B.M.: Brownian motion on a homogeneous random fractal. Probab. Theory Relat. Fields94, 1–38 (1992)Google Scholar
  11. 11.
    Lindstrøm, T.: Brownian motion on nested fractals. Memoirs am. Math. Soc.420, (1990)Google Scholar
  12. 12.
    Marcus, M.B., Pisier, G.: Random Fourier series with applications to harmonic analysis. (Ann. math. Studies, Vol. 101) Princeton: Princeton University Press.Google Scholar
  13. 13.
    Marcus, M.B., Rosen, J.: Sample path properties of the local times of strongly symmetric Markov processes via Gaussian processes. Ann. Probab.20, 1603–1684 (1992)Google Scholar
  14. 14.
    Robinson, D.W.: The Thermodynamic Pressure in Quantum Statistical Mechanics. Berlin Heidelberg New York: Springer, 1971Google Scholar
  15. 15.
    Sharpe, M. (1988) General Theory of Markov Processes. Academic Press, San Diego.Google Scholar
  16. 16.
    Silverstein, M.L. (1973) Dirichlet Spaces and Random Time Change. Illinois J. Math.16, pp. 1–72.Google Scholar

Copyright information

© Springer-Verlag 1995

Authors and Affiliations

  • W. B. Krebs
    • 1
  1. 1.Department of StatisticsFlorida State UniversityTallahasseeUSA

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