×
Barlow, Martin T.; Perkins, Edwin A.

Brownian motion on the Sierpinski gasket. (English) Zbl 0635.60090

Probab. Theory Relat. Fields 79, No. 4, 543-623 (1988).
We construct a “Brownian motion” taking values in the Sierpinski gasket, a fractal subset of \({\mathbb{R}}^ 2\), and study its properties. This is a diffusion process characterized by local isotropy and homogeneity properties. We show, for example, that the process has a continuous symmetric transition density, \(p_ t(x,y)\), with respect to an appropriate Hausdorff measure and obtain estimates on \(p_ t(x,y)\).
Reviewer: M.T.Barlow

Cited in 9 Reviews
Cited in 274 Documents

MSC:

60J65 Brownian motion

Keywords:

Sierpinski gasket; diffusion process characterized by local isotropy and homogeneity properties; continuous symmetric transition density; Hausdorff measure
Cite Review PDF
Full Text: DOI

References:

[1] Asmussen, S., Hering, H.: Branching processes. Boston: Birkhauser 1983 · Zbl 0516.60095
[2] Banavar, J.R., Willemsen, J.F.: Probability density for diffusion on fractals. Phys. Rev. B 30, 6778-6779 (1984) · doi:10.1103/PhysRevB.30.6778
[3] Barlow, M.T.: Continuity of local times for Lévy processes. Z. Wahrscheinlichkeitstheor. Verw. Geb. 69, 23-35 (1985) · Zbl 0561.60076 · doi:10.1007/BF00532583
[4] Barlow, M.T., Bass, R.F.: Construction of Brownian motion on the Sierpinski carpet. Preprint (1987) · Zbl 0691.60070
[5] Bingham, N.H.: On the limit of a supercritical branching process. J. Appl. Probab. (in press) · Zbl 0669.60078
[6] Blumenthal, R.M., Getoor, R.K.: Local times for Markov processes. Z. Wahrscheinlichkeitstheor. Verw. Geb. 3, 50-74 (1963) · Zbl 0126.33701 · doi:10.1007/BF00531683
[7] Blumenthal, R.M., Getoor, R.K.: Markov processes and potential theory. New York: Academic 1968 · Zbl 0169.49204
[8] Breiman, L.: Probability. Reading, Massachusetts: Addison-Wesley 1968
[9] Carlen, E.A., Kusuoka, S., Stroock, D.W.: Upper bounds for symmetric Markov transition functions. Ann. Inst. Henri Poincaré Nouv. Ser., Sect. B 23, 245-287 (1987) · Zbl 0634.60066
[10] Chacon, R.V., Le Jan, Y., Perkins, E., Taylor, S.J.: Generalised arc length for Brownian motion and Lévy processes. Z. Wahrscheinlichkeitstheor. Verw. Geb. 57, 197-211 (1981) · Zbl 0469.60037 · doi:10.1007/BF00535489
[11] Davis, M.H.A., Varaiya, P.: The multiplicity of an increasing family of ?-fields. Ann. Probab. 2, 958-963 (1974) · Zbl 0292.60071 · doi:10.1214/aop/1176996562
[12] Dubuc, S.: Problèmes relatifs a l’itération de fonctions suggerés par les processus en cascade. Ann. Inst. Fourier 21, 171-251 (1971) · Zbl 0196.12801
[13] Dubuc, S.: Etude théorique et numérique de la fonction de Karlin-McGregor. J. Anal. Math. 42, 15-37 (1982) · Zbl 0545.65084 · doi:10.1007/BF02786869
[14] Eggleston, H.G.: On closest packing by equilateral triangles. Math. Proc. Camb. Philos. Soc. 49, 26-30 (1953) · Zbl 0050.05801 · doi:10.1017/S0305004100027997
[15] Fabes, E.B., Stroock, D.W.: A new proof of Moser’s parabolic Harnack inequality using the old ideas of Nash. Arch. Rational Mech. Anal. 96, 328-338 (1986) · Zbl 0652.35052 · doi:10.1007/BF00251802
[16] Feller, W.: An introduction to probability theory and its applications, vol. II, 2nd edn. New York: Wiley 1971 · Zbl 0219.60003
[17] Fristedt, B.E., Pruitt, W.E.: Lower functions for increasing random walks and subordinators. Z. Wahrscheinlichkeitstheor. Verw. Geb. 18, 167-182 (1971) · Zbl 0197.44204 · doi:10.1007/BF00563135
[18] Fukushima, M.: Dirichlet forms and Markov processes. Amsterdam: North-Holland/Kodansha 1980 · Zbl 0422.31007
[19] Garsia, A.M.: Continuity properties of multi-dimensional Gaussian processes. 6th Berkeley Symposium on Math. In: Statistical probability, vol. 2, pp. 369-374. Berkeley: University of California Press 1970
[20] Getoor, R-K-. Kesten, H.: Continuity of local times of Markov processes. Comp. Math. 24, 277-303 (1972) · Zbl 0293.60069
[21] Goldstein, S.: Random walks and diffusions on fractals. In: Kesten, H. (ed.) Percolation theory and ergodic theory of infinite particle systems. IMA vol. Math Appl. 8, 121-128. New York Berlin Heidelberg: Springer 1987
[22] Guyer, R.A.: Diffusive motion on a fractal; G nm(t). Phys. Rev. A 32, 2324-2335 (1985) · doi:10.1103/PhysRevA.32.2324
[23] de Haan, L., Stadtmuller, U.: Dominated variation and related concepts and Tauberian theorems for Laplace transforms. J. Math. Anal. Appl. 108, 344-365 (1985) · Zbl 0581.44003 · doi:10.1016/0022-247X(85)90030-7
[24] Harris, T.E.: Branching processes. Ann. Math. Stat. 19, 474-494 (1948) · Zbl 0041.45603 · doi:10.1214/aoms/1177730146
[25] Harris, T.E.: The theory of branching processes. Berlin Heidelberg: Springer 1963 · Zbl 0117.13002
[26] Itô, K., McKean Jr., H.P.: Diffusion processes and their sample paths. Berlin Heidelberg: Springer 1965
[27] Kesten, H.: Subdiffusive behaviour of random walk on a random cluster. Ann. Inst. H. Poincaré Nouv. Ser., Sect. B 22, 425-487 (1986) · Zbl 0632.60106
[28] Knight, F.B.: Essentials of Brownian motion and diffusion. AMS Math. Surveys vol. 18. Providence, Rhode Island: AMS 1981 · Zbl 0458.60002
[29] Kusuoka, S.: A diffusion process on a fractal. In: Itô, K., Ikeda, N. (eds.) Probabilistic methods in mathematical physics. Proceedings Taniguchi Symposium, Katata 1985. Amsterdam: Kino Kuniya-North Holland 1987. pp. 251-274
[30] Meyer, P.A.: Un cours sur les intégrales stochastiques. In: Meyer, P.A. (ed.) Seminaire de probabilités X. (Lect. Notes Math., vol. 511, pp. 175-183) Berlin Heidelberg New York: Springer 1976
[31] O’Shaughnessy, B., Procaccia, I.: Diffusion on fractals. Phys. Rev. A 32, 3073-3083 (1985) · doi:10.1103/PhysRevA.32.3073
[32] Rammal, R., Toulouse, G.: Random walks on fractal structures and percolation clusters. J- Physique Lettres 44, L13-L22 (1983)
[33] Riesz, F., Sz.-Nagy, B. Lecons d’analyse fonctionelle. Budapest: Academy of Science, Hungary 1952 · Zbl 0122.11205
[34] Stroock, D.W., Varadhan, S.R.S.: Multidimensional diffusion processes. Berlin Heidelberg New York: Springer 1979 · Zbl 0426.60069
[35] Salisbury, T.S.: An increasing diffusion. Proceedings, seminar on stochastic processes, 1984, pp. 173-194. Boston: Birkhäuser 1986
[36] Sierpinski, W.: Sur une courbe dont tout point est un point de ramification. C.R. Acad. Sci. Paris 160, 302-305 (1915) · JFM 45.0628.02
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.