Skip to main content
SpringerLink
Log in

Diffusions on fractals

  • Chapter
  • First Online:
Lectures on Probability Theory and Statistics

Part of the book series: Lecture Notes in Mathematics ((LNMECOLE,volume 1690))

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 39.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 54.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  • [AO] S. Alexander and R. Orbach: Density of states on fractals: “fractons”. J. Physique (Paris) Lett. 43, L625–L631 (1982).

    Article  Google Scholar 

  • [AT] W.N. Anderson and G.E. Trapp: Shorted operators, II, SIAM J. Appl. Math. 28, 60–71 (1975).

    Article  MathSciNet  MATH  Google Scholar 

  • [Ar] D.G. Aronson, Bounds on the fundamental solution of a parabolic equation. Bull. Amer. Math. Soc. 73, 890–896 (1967).

    Article  MathSciNet  MATH  Google Scholar 

  • [BS] C. Bandt and J. Stahnke: Self-similar sets 6. Interior distance on deterministic fractals. Preprint 1990.

    Google Scholar 

  • [Bar1] M.T. Barlow: Random walks, electrical resistance and nested fractals. In Asymptotic Problems in Probability Theory, ed. K.D. Elworthy, N. Ikeda, 131–157, Longman Scientific, Harlow UK, 1990.

    Google Scholar 

  • [Bar2] M.T. Barlow: Random walks and diffusions on fractals. Proc. Int. Congress Math. Kyoto 1990, 1025–1035. Springer, Tokyo 1991.

    MATH  Google Scholar 

  • [Bar3] M.T. Barlow: Harmonic analysis on fractal spaces. Séminaire Bourbaki Volume 1991/1992, Astérisque 206 (1992).

    Google Scholar 

  • [BB1] M.T. Barlow and R.F. Bass: The construction of Brownian motion on the Sierpinski carpet. Ann. Inst. H. Poincaré 25, 225–257 (1989).

    MathSciNet  MATH  Google Scholar 

  • [BB2] M.T. Barlow and R.F. Bass: Local times for Brownian motion on the Sierpinski carpet. Probab. Th. Rel. Fields, 85, 91–104 (1990).

    Article  MathSciNet  MATH  Google Scholar 

  • [BB3] M.T. Barlow and R.F. Bass: On the resistance of the Sierpinski carpet. Proc. R. Soc. London A. 431, 345–360 (1990).

    Article  MathSciNet  MATH  Google Scholar 

  • [BB4] M.T. Barlow and R.F. Bass: Transition densities for Brownian motion on the Sierpinski carpet. Probab. Th. Rel. Fields 91, 307–330 (1992).

    Article  MathSciNet  MATH  Google Scholar 

  • [BB5] M.T. Barlow and R.F. Bass: Coupling and Harnack inequalities for Sierpinski carpets. Bull. A.M.S. 29, 208–212 (1993).

    Article  MathSciNet  MATH  Google Scholar 

  • [BB6] M.T. Barlow and R.F. Bass: Brownian motion and harmonic analysis on Sierpinski carpets. Preprint 1997.

    Google Scholar 

  • [BB7] M.T. Barlow and R.F. Bass: Random walks on graphical Sierpinski carpets. In preparation.

    Google Scholar 

  • [BBS] M.T. Barlow, R.F. Bass, and J.D. Sherwood: Resistance and spectral dimension of Sierpinski carpets. J. Phys. A, 23, L253–L258 (1990).

    Article  MathSciNet  MATH  Google Scholar 

  • [BH] M.T. Barlow and B.M. Hambly: Transition density estimates for Brownian motion on scale irregular Sierpinski gaskets. To appear Ann. IHP.

    Google Scholar 

  • [BHHW] M.T. Barlow, K. Hattori, T. Hattori and H. Watanabe: Weak homogenization of anisotropic diffusion on pre-Sierpinski carpets. To appear Comm Math. Phys.

    Google Scholar 

  • [BK] M.T. Barlow and J. Kigami: Localized eigenfunctions of the Laplacian on p.c.f. self-similar sets. To appear J. Lond. Math. Soc.

    Google Scholar 

  • [BP] M.T. Barlow and E.A. Perkins: Brownian motion on the Sierpinski gasket. Probab. Th. Rel. Fields 79, 543–623 (1988).

    Article  MathSciNet  MATH  Google Scholar 

  • [Bas] R.F. Bass: Diffusions on the Sierpinski carpet. Trends in Probability and related Analysis: Proceedings of SAP'96. World Scientific, Singapore, 1997. To appear.

    MATH  Google Scholar 

  • [BAH] D. Ben-Avraham and S. Havlin: Exact fractals with adjustable fractal and fraction dimensionalities. J. Phys. A. 16, L559–563 (1983).

    Article  MathSciNet  Google Scholar 

  • [Blu] L.M. Blumenthal: Theory and applications of distance geometry. Oxford, 1953.

    Google Scholar 

  • [BG] R.M. Blumenthal and R.K. Getoor: Markov processes and potential theory. Academic Press, New York, 1968.

    MATH  Google Scholar 

  • [CKS] E.A. Carlen, S. Kusuoka and D.W. Stroock: Upper bounds for symmetric Markov transition functions. Ann. Inst. H. Poincare Sup. no. 2, 245–287 (1987).

    MathSciNet  MATH  Google Scholar 

  • [Ca1] D. Carlson: What are Schur complements, anyway? Linear Alg. Appl. 74, 257–275 (1986).

    Article  MathSciNet  MATH  Google Scholar 

  • [CRRST] A.K. Chandra, P. Raghavan, W.L. Ruzzo, R. Smolensky and P. Tiwari: The electrical resistance of a graph captures its commute and cover times. Proceedings of the 21st ACM Symposium on theory of computing, 1989.

    Google Scholar 

  • [Co] T.C. Coulhon: Ultracontractivity and Nash type inequalities, J. Funct. Anal. 141, 510–539 (1996).

    Article  MathSciNet  MATH  Google Scholar 

  • [DSV] K. Dalrymple, R.S. Strichartz and J.P. Vinson: Fractal differential equations on the Sierpinski gasket. Preprint 1997.

    Google Scholar 

  • [Da] E.B. Davies: Heat kernels and spectral theory. Cambridge University Press 1989.

    Google Scholar 

  • [DaSi] E.B. Davies and B. Simon: Ultracontractivity and the heat kernel for Schrödinger operators and Dirichlet Laplacians. J. Funct. Anal. 59, 335–395, (1984).

    Article  MathSciNet  MATH  Google Scholar 

  • [DS] P.G. Doyle and J.L. Snell: Random walks and electrical networks. Washington, Math. Assoc. of America, 1984.

    MATH  Google Scholar 

  • [Dud] R.M. Dudley: Sample functions of the Gaussian process. Ann. Prob. 1, 66–103 (1971).

    Article  MathSciNet  MATH  Google Scholar 

  • [FaS] E.B. Fabes and D.W. Stroock, A new proof of Moser's parabolic Harnack inequality via the old ideas of Nash. Arch. Mech. Rat. Anal. 96, 327–338 (1986).

    Article  MathSciNet  MATH  Google Scholar 

  • [Fa1] K.J. Falconer: Geometry of fractal sets. Cambridge Univ. Press, 1985.

    Google Scholar 

  • [Fa2] K.J. Falconer: Fractal Geometry. Wiley, 1990

    Google Scholar 

  • [Fe] H. Federer: Geometric measure theory. Springer, New York, 1969.

    MATH  Google Scholar 

  • [FHK] P.J. Fitzsimmons, B.M. Hambly, T. Kumagai: Transition density estimates for Brownian motion on affine nested fractals. Comm. Math. Phys. 165, 595–620 (1995).

    Article  MathSciNet  MATH  Google Scholar 

  • [FOT] M. Fukushima, Y. Oshima, and M. Takeda, Dirichlet forms and Symmetric Markov Processes. de Gruyter, Berlin, 1994.

    Book  MATH  Google Scholar 

  • [Fu1] M. Fukushima, Dirichlet forms, diffusion processes, and spectral dimensions for nested fractals. Ideas and methods in stochastic analysis, stochastics and applications, 151–161. Cambridge Univ. Press, Cambridge, 1992.

    MATH  Google Scholar 

  • [FS1] M. Fukushima and T. Shima: On a spectral analysis for the Sierpinski gasket. J. of Potential Analysis 1, 1–35 (1992).

    Article  MathSciNet  MATH  Google Scholar 

  • [FS2] M. Fukushima and T. Shima, On discontinuity and tail behaviours of the integrated density of states for nested pre-fractals. Comm. Math. Phys. 163, 461–471 (1994).

    Article  MathSciNet  MATH  Google Scholar 

  • [FST] M. Fukushima, T. Shima and M. Takeda: Large deviations and related LILs for Brownian motions on nested fractals. Preprint 1997.

    Google Scholar 

  • [GAM1] Y. Gefen, A. Aharony and B.B. Mandelbrot: Phase transitions on fractals. I. Quasilinear lattices. J. Phys. A 16, 1267–1278 (1983).

    Article  MathSciNet  Google Scholar 

  • [GAM2] Y. Gefen, A. Aharony, Y. Shapir and B.B. Mandelbrot: Phase transitions on fractals. II. Sierpinski gaskets. J. Phys. A 17, 435–444 (1984).

    Article  MathSciNet  Google Scholar 

  • [GAM3] Y. Gefen, A. Aharony and B. Mandelbrot: Phase transitions on fractals. III. Infinitely ramifield lattices. J. Phys. A 17, 1277–1289 (1984).

    Article  MathSciNet  Google Scholar 

  • [GK] R.K. Getoor and H. Kesten: Continuity of local times of Markov processes. Comp. Math. 24, 277–303 (1972).

    MathSciNet  MATH  Google Scholar 

  • [Go] S. Goldstein: Random walks and diffusion on fractals. In: Kesten, H. (ed.) Percolation theory and ergodic theory of infinite particle systems (IMA Math. Appl., vol. 8) Springer, New York, 1987, pp. 121–129.

    Chapter  Google Scholar 

  • [Gra] P.J. Grabner: Functional equations and stopping times for the Brownian motion on the Sierpinski gasket. To appear in Mathematica.

    Google Scholar 

  • [GrT] P.J. Grabner and R.F. Tichy: Equidistribution and Brownian motion on the Sierpinski gasket. Preprint 1997.

    Google Scholar 

  • [GrW] P.J. Grabner and W Woess: Functional iterations and periodic oscillations for simple random walk on the Sierpinski gasket. To appear in Stoch. Proc. Appl.

    Google Scholar 

  • [Gre] R.F. Green: A simple model for the construction of Brownian motion on Sierpinski's hexagonal gasket. Technical report, Univ. of Minnesota, Duluth. 1989.

    Google Scholar 

  • [HS] L. deHaan, and U. Stadtmuller: Dominated variation and related concepts and Tauberian theorems for Laplace transforms. J. Math. Anal. Appl. 108, 344–365 (1985).

    Article  MathSciNet  MATH  Google Scholar 

  • [Ham1] B.M. Hambly: Brownian motion on a homogeneous random fractal. Probab. Th. Rel. Fields 94, 1–38, (1992).

    Article  MathSciNet  MATH  Google Scholar 

  • [Ham2] B.M. Hambly: Brownian motion on a random recursive Sierpinski gasket. Ann. Prob. 25, 1059–1102 (1997).

    Article  MathSciNet  MATH  Google Scholar 

  • [HK1] B.M. Hambly, T. Kumagai: Heat kernel estimates and homogenization for asymptotically lower dimensional processes on some nested fractals. Preprint 1996.

    Google Scholar 

  • [HK2] B.M. Hambly, T. Kumagai: Transition density estimates for diffusion processes on p.c.f. self-similar fractals. Preprint 1997.

    Google Scholar 

  • [Har] T.E. Harris: The theory of branching processes. Springer 1963.

    Google Scholar 

  • [Hat1] T. Hattori: Asympotically one-dimensional diffusions on scale-irregular gaskets. Preprint 1994.

    Google Scholar 

  • [HH] K. Hattori, and T. Hattori: Self-avoiding process on the Sierpinski gasket. Probab. Th. Rel. Fields 88, 405–428 (1991).

    Article  MathSciNet  MATH  Google Scholar 

  • [HatK] T. Hattori, S. Kusuoka: The exponent for mean square displacement of self-avoiding random walk on Sierpinski gasket. Probab. Th. Rel. Fields 93, 273–284 (1992).

    Article  MathSciNet  MATH  Google Scholar 

  • [HHK] K. Hattori, T. Hattori, and S. Kusuoka, Self-avoiding paths on the three-dimensional Sierpinski gasket. Publ. Res. Inst. Math. Sci. 29, 455–509 (1993).

    Article  MathSciNet  MATH  Google Scholar 

  • [HHW] K. Hattori, T. Hattori, and H. Watanabe: Reasoning out the empirical rule d<2, Physics Letters A 115, 207–212 (1986).

    Article  Google Scholar 

  • [HHW1] K. Hattori, T. Hattori, and H. Watanabe: Gaussian field theories and the spectral dimensions. Prog. Th. Phys. Supp. No. 92, 108–143, (1987).

    Article  MathSciNet  Google Scholar 

  • [HHW2] K. Hattori, T. Hattori, and H. Watanabe: New approximate renormalisation method on fractals. Phys Rev. A 32, 3730–3733 (1985).

    Article  Google Scholar 

  • [HHW3] K. Hattori, T. Hattori, and S. Kusuoka: Self-avoiding paths on the pre-Sierpinski gasket. Probab. Th. Rel. Fields 84, 1–26 (1990).

    Article  MathSciNet  MATH  Google Scholar 

  • [HHW4] K. Hattori, T. Hattori, and H. Watanabe: Asymptotically one-dimensional diffusions on the Sierpinski gasket and the abc-gaskets. Prob. Th. Rel. Fields 100, 85–116 (1994).

    Article  MathSciNet  MATH  Google Scholar 

  • [HBA] S. Havlin, and D. Ben-Avraham: Diffusion in disordered media, Adv. Phys. 36, 695–798 (1987).

    Article  Google Scholar 

  • [He] M.K. Heck: Homogeneous diffusions on the Sierpinski gasket. Preprint 1996.

    Google Scholar 

  • [Hu] J.E. Hutchinson: Fractals and self-similarity. Indiana J. Math. 30, 713–747 (1981).

    Article  MathSciNet  MATH  Google Scholar 

  • [Jo] O.D. Jones: Transition probabilities for the simple random walk on the Sierpinski graph. Stoch. Proc. Appl. 61, 45–69 (1996).

    Article  MathSciNet  MATH  Google Scholar 

  • [J] A. Jonsson: Brownian motion on fractals and functions spaces. Math. Zeit. 122, 495–504 (1986).

    MATH  Google Scholar 

  • [Ka] M. Kac: On some connections between probability theory and differential and integral equations. Proc. 2nd Berkelely Symposium, 1950, 189–215, (1951).

    Google Scholar 

  • [Ke] H. Kesten: Subdiffusive behaviour of random walk on random cluster. Ann. Inst. H. Poincare B 22, 425–487 (1986).

    MathSciNet  MATH  Google Scholar 

  • [Ki1] J. Kigami: A harmonic calculus on the Sierpinski space. Japan J. Appl. Math. 6, 259–290 (1989).

    Article  MathSciNet  MATH  Google Scholar 

  • [Ki2] J. Kigami: A harmonic calculus for p.c.f. self-similar sets. Trans. A.M.S. 335, 721–755 (1993).

    MathSciNet  MATH  Google Scholar 

  • [Ki3] J. Kigami: Harmonic metric and Dirichlet form on the Sierpinski gasket. In Asymptotic Problems in Probability Theory, ed. K.D. Elworthy, N. Ikeda, 201–218, Longman Scientific, Harlow UK, 1990.

    Google Scholar 

  • [Ki4] J. Kigami: A calculus on some self-similar sets. To appear in Proceedings on the 1st IFIP Conference “Fractal 90”, Elsevier.

    Google Scholar 

  • [Ki5] J. Kigami: Effective resistance for harmonic structures on P.C.F. self-similar sets. Math. Proc. Camb. Phil. Soc. 115, 291–303 (1994).

    Article  MathSciNet  MATH  Google Scholar 

  • [Ki6] J. Kigami: Hausdorff dimension of self-similar sets and shortest path metric. J. Math. Soc. Japan 47, 381–404 (1995).

    Article  MathSciNet  MATH  Google Scholar 

  • [Ki7] J. Kigami: Harmonic calculus on limits of networks and its application to dendrites. J. Functional Anal. 128, 48–86 (1995).

    Article  MathSciNet  MATH  Google Scholar 

  • [KL] J. Kigami, M. Lapidus: Weyl's spectral problem for the spectral distribution of Laplacians on P.C.F. self-similar fractals. Comm. Math. Phys. 158, 93–125 (1993).

    Article  MathSciNet  MATH  Google Scholar 

  • [Kn] F.B. Knight: Essentials of Brownian motion and diffusion. AMS Math. Surveys, Vol. 18. AMS 1981.

    Google Scholar 

  • [Koz] S.M. Kozlov: Harmonization and homogenization on fractals. Comm. Math. Phys. 153, 339–357 (1993).

    Article  MathSciNet  MATH  Google Scholar 

  • [Kr1] W.B. Krebs: A diffusion defined on a fractal state space. Stoch. Proc. Appl. 37, 199–212 (1991).

    Article  MathSciNet  MATH  Google Scholar 

  • [Kr2] W.B. Krebs: Hitting time bounds for Brownian motion on a fractal. Proc. A.M.S. 118, 223–232 (1993).

    Article  MathSciNet  MATH  Google Scholar 

  • [Kr3] W.B. Krebs: Brownian motion on the continuum tree. Prob. Th. Rel. Fields 101, 421–433 (1995). Preprint 1993.

    Article  MathSciNet  MATH  Google Scholar 

  • [Kum1] T. Kumagai: Construction and some properties of a class of non-symmetric diffusion processes on the Sierpinski gasket. In Asymptotic Problems in Probability Theory, ed. K.D. Elworthy, N. Ikeda, 219–247, Longman Scientific, Harlow UK, 1990.

    Google Scholar 

  • [Kum2] T. Kumagai: Estimates of the transition densities for Brownian motion on nested fractals. Prob. Th. Rel. Fields 96, 205–224 (1993).

    Article  MathSciNet  MATH  Google Scholar 

  • [Kum3] T. Kumagai: Regularity, closedness, and spectral dimension of the Dirichlet forms on p.c.f. self-similar sets. J. Math. Kyoto Univ. 33, 765–786 (1993).

    MathSciNet  MATH  Google Scholar 

  • [Kum4] T. Kumagai: Rotation invariance and characterization of a class of self-similar diffusion processes on the Sierpinski gasket. Algorithms, fractals and dynamics, 131–142. Plenum, New York, 1995.

    MATH  Google Scholar 

  • [Kum5] T. Kumagai: Short time asymptotic behaviour and large deviation for Brownian motion on some affine nested fractals. Publ. R.I.M.S. Kyoto Univ. 33, 223–240 (1997).

    Article  MathSciNet  MATH  Google Scholar 

  • [KK1] T. Kumagai, and S. Kusuoka, Homogenization on nested fractals. Probab. Th. Rel. Fields 104, 375–398 (1996).

    Article  MathSciNet  MATH  Google Scholar 

  • [Kus1] S. Kusuoka: A diffusion process on a fractal. In: Ito, K., N. Ikeda, N. (ed.) Symposium on Probabilistic Methods in Mathematical Physics, Taniguchi, Katata. Academic Press, Amsterdam, 1987, pp. 251–274

    Google Scholar 

  • [Kus2] S. Kusuoka: Dirichlet forms on fractals and products of random matrices. Publ. RIMS Kyoto Univ., 25, 659–680 (1989).

    Article  MathSciNet  MATH  Google Scholar 

  • [Kus3] S. Kusuoka: Diffusion processes on nested fractals. In: Statistical mechanics and fractals, Lect. Notes in Math. 1567, Springer, 1993.

    Google Scholar 

  • [KZ1] S. Kusuoka, and X.Y. Zhou: Dirichlet form on fractals: Poincaré constant and resistance. Prob. Th. Rel. Fields 93, 169–196 (1992).

    Article  MathSciNet  MATH  Google Scholar 

  • [KZ2] S. Kusuoka and X.Y. Zhou: Waves on fractal-like manifolds and effective energy propagation. Preprint 1993.

    Google Scholar 

  • [L1] T. Lindstrøm: Brownian motion on nested fractals. Mem. A.M.S. 420, 1990.

    Google Scholar 

  • [L2] T. Lindstrøm: Brownian motion penetrating the Sierpinski gasket. In Asymptotic Problems in Probability Theory, ed. K.D. Elworthy, N. Ikeda, 248–278, Longman Scientific, Harlow UK.

    Google Scholar 

  • [Mae] F-Y Maeda: Dirichlet Integrals on Harmonic Spaces. Springer L.N.M. 803 1980.

    Google Scholar 

  • [Man] B.B. Mandelbrot: The fractal geometry of nature. W.H. Freeman, San Fransisco, 1982.

    MATH  Google Scholar 

  • [MR] M.B. Marcus, J. Rosen: Sample path functions of the local times of strongly symmetric Markov processes via Gaussian processes. Ann. Prob. 20, 1603–1684 (1992).

    Article  MathSciNet  MATH  Google Scholar 

  • [MW] R.D. Mauldin, S.C. Williams: Hausdorff dimension in graph directed constructions. Trans. A.M.S. 309, 811–829 (1988).

    Article  MathSciNet  MATH  Google Scholar 

  • [Me1] V. Metz: Potentialtheorie auf dem Sierpinski gasket. Math. Ann. 289, 207–237 (1991).

    Article  MathSciNet  MATH  Google Scholar 

  • [Me2] V. Metz, Renormalization of finitely ramified fractals. Proc. Roy. Soc. Edinburgh Ser A 125, 1085–1104 (1995).

    Article  MathSciNet  MATH  Google Scholar 

  • [Me3] V. Metz: How many diffusions exist on the Vicsek snowflake? Acta Appl. Math. 32, 224–241 (1993).

    Article  MathSciNet  MATH  Google Scholar 

  • [Me4] V. Metz: Hilbert's projective metric on cones of Dirichlet forms. J. Funct. Anal. 127, 438–455, (1995).

    Article  MathSciNet  MATH  Google Scholar 

  • [Me5] V. Metz, Renormalization on fractals. Proc. International Conf. Potential Theory 94, 413–422. de Gruyter, Berlin, 1996.

    MATH  Google Scholar 

  • [Me6] V. Metz: Maeda's energy norm explains effective resistance. Preprint.

    Google Scholar 

  • [Me7] V. Metz, Renormalization contracts on nested fractals. C.R. Acad. Sci. Paris 332, 1037–1042 (1996).

    MathSciNet  MATH  Google Scholar 

  • [MS] V. Metz, and K.-T. Sturm, Gaussian and non-Gaussian estimates for heat kernels on the Sierpinski gasket. In: Dirichlet forms and stochastic processes, 283–289, de Gruyter, Berlin, 1995.

    Google Scholar 

  • [Mor] P.A.P. Moran: Additive functions of intervals and Hausdorff measure. Proc. Camb. Phil. Soc. 42, 15–23 (1946).

    Article  MathSciNet  MATH  Google Scholar 

  • [Mos] U. Mosco: Composite media and asymptotic Dirichlet forms. J. Funct. Anal. 123, 368–421 (1994).

    Article  MathSciNet  MATH  Google Scholar 

  • [Nus] R.D. Nussbaum: Hilbert's projective metric and iterated non-linear maps. Mem. A.M.S. 75, 1988.

    Google Scholar 

  • [OSS] M. Okada, T. Sekiguchi and Y. Shiota: Heat kernels on infinte graph networks and deformed Sierpinski gaskets. Japan J. App. Math. 7, 527–554 (1990).

    Article  MathSciNet  MATH  Google Scholar 

  • [O1] H. Osada: Isoperimetric dimension and estimates of heat kernels of pre-Sierpinski carpets. Probab. Th. Rel. Fields 86, 469–490 (1990).

    Article  MathSciNet  MATH  Google Scholar 

  • [O2] H. Osada: Cell fractals and equations of hitting probabilities. Preprint 1992.

    Google Scholar 

  • [OS] B. O'Shaughnessy, I. Procaccia: Analytical solutions for diffusion on fractal objects. Phys. Rev. Lett. 54, 455–458, (1985).

    Article  Google Scholar 

  • [Pe] R. Peirone: Homogenization of functionals on fractals. Preprint, 1996.

    Google Scholar 

  • [PP] K. Pietruska-Paluba: The Lifchitz singularity for the density of states on the Sierpinski gasket. Probab. Th. Rel. Fields 89, 1–34 (1991).

    Article  MathSciNet  MATH  Google Scholar 

  • [RT] R. Rammal and G. Toulouse: Random walks on fractal structures and percolation clusters. J. Physique Lettres 44, L13–L22 (1983).

    Article  Google Scholar 

  • [R] R. Rammal: Spectrum of harmonic excitations on fractals. J. de Physique 45, 191–206 (1984).

    Article  MathSciNet  Google Scholar 

  • [Rog] L.C.G. Rogers, Multiple points of Markov processes in a complete metric space. Sém. de Probabilités XXIII, 186–197. Springer, Berlin, 1989.

    MATH  Google Scholar 

  • [Sab1] C. Sabot: Existence and uniqueness of diffusions on finitely ramified self-similar fractals. Preprint 1996.

    Google Scholar 

  • [Sab2] C. Sabot: Espaces de Dirichlet reliés par un nombre fini de points et application aux diffusions sur les fractales. Preprint 1996.

    Google Scholar 

  • [Sha] M. Sharpe: General Theory of Markov Processes. Academic Press. New York, 1988.

    MATH  Google Scholar 

  • [Sh1] T. Shima: On eigenvalue problems for the random walk on the Sierpinski pre-gaskets. Japan J. Appl. Ind. Math., 8, 127–141 (1991).

    Article  MathSciNet  MATH  Google Scholar 

  • [Sh2] T. Shima: The eigenvalue problem for the Laplacian on the Sierpinski gasket. In Asymptotic Problems in Probability Theory, ed. K.D. Elworthy, N. Ikeda, 279–288, Longman Scientific, Harlow UK.

    Google Scholar 

  • [Sh3] T. Shima: On Lifschitz tails for the density of states on nested fractals. Osaka J. Math. 29, 749–770 (1992).

    MathSciNet  MATH  Google Scholar 

  • [Sh4] T. Shima: On eigenvalue problems for Laplacians on P.C.F. self-similar sets. Japan J. Indust. Appl. Math. 13, 1–23 (1996).

    Article  MathSciNet  MATH  Google Scholar 

  • [Sie1] W. Sierpinski: Sur une courbe dont tout point est un point de ramification. C.R. Acad. Sci. Paris 160, 302–305 (1915).

    MATH  Google Scholar 

  • [Sie2] W. Sierpinski: Sur une courbe cantorienne qui contient une image biunivoque et continue de toute courbe donnée. C.R. Acad. Sci. Paris 162, 629–632 (1916).

    MATH  Google Scholar 

  • [Stu1] K.T. Sturm: Diffusion processes and heat kernels on metric spaces. To appear Ann. Probab.

    Google Scholar 

  • [Stu2] K.T. Sturm: Analysis on local Dirichlet spaces I. Recurrence, conservativeness and L p-Liouville properties. J. reine angew. Math. 456, 173–196 (1994).

    MathSciNet  MATH  Google Scholar 

  • [Stu3] K.T. Sturm: Analysis on local Dirichlet spaces II. Upper Gaussian estimates for the fundamental solutions of parabolic equations. Osaka J. Math. 32, 275–312 (1995).

    MathSciNet  MATH  Google Scholar 

  • [Stu4] K.T. Sturm: Analysis on local Dirichlet spaces III. The parabolic Harnack inequality. J. Math. Pures Appl. 75, 273–297 (1996).

    MathSciNet  MATH  Google Scholar 

  • [Tal] M. Talagrand: Regularity of Gaussian processes Acta Math. 159, 99–149 (1987).

    Article  MathSciNet  MATH  Google Scholar 

  • [Tel] A. Telcs: Random walks on graphs, electric networks and fractals. Prob. Th. Rel. Fields 82, 435–451 (1989).

    Article  MathSciNet  MATH  Google Scholar 

  • [Tet] P. Tetali: Random walks and the effective resistance of networks. J. Theor. Prob. 4, 101–109 (1991).

    Article  MathSciNet  MATH  Google Scholar 

  • [Tri] C. Tricot: Two definitions of fractional dimension. Math. Proc. Camb. Phil. Soc. 91, 54–74, (1982).

    Article  MathSciNet  MATH  Google Scholar 

  • [V1] N. Th. Varopoulos: Isoperimetric inequalities and Markov chains. J. Funct. Anal. 63, 215–239 (1985).

    Article  MathSciNet  MATH  Google Scholar 

  • [Wat1] H. Watanabe: Spectral dimension of a wire network. J. Phys. A. 18, 2807–2823 (1985).

    Article  MathSciNet  MATH  Google Scholar 

  • [Y1] J.-A. Yan: A formula for densities of transition functions. Sem. Prob. XXII, 92–100. Lect Notes Math. 1321.

    Google Scholar 

Download references

Authors
  1. Martin T. Barlow
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Pierre Bernard

Rights and permissions

Reprints and permissions

Copyright information

© 1998 Springer-Verlag

About this chapter

Cite this chapter

Barlow, M.T. (1998). Diffusions on fractals. In: Bernard, P. (eds) Lectures on Probability Theory and Statistics. Lecture Notes in Mathematics, vol 1690. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0092537

Download citation

  • DOI: https://doi.org/10.1007/BFb0092537

  • Published:

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-64620-4

  • Online ISBN: 978-3-540-69228-7

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation