Homogenization on nested fractals

Probability Theory and Related Fields

Volumn 104, Issue 3, 1996, Pages 375-398

  Kumagai, T ^a   Kusuoka, S ^b  

^a NAGOYA UNIVERSITY   (Japan)

^b UNIVERSITY OF TOKYO   (Japan)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 0030558574     PISSN: 01788051     EISSN: None     Source Type: Journal    
DOI: 10.1007/BF01213686     Document Type: Article

References (27)

1. Barlow, M.T.: Random walks, electrical resistance, and nested fractals. In: Elworthy, K.D., Ikeda, N. (eds.) Asymptotic problems in probability theory: stochastic models and diffusions on fractals, pp. 131-157. Montreal: Pitman 1993
2. Barlow, M.T., Bass, R.F.: The construction of Brownian motion on the Sierpinski carpet. Ann. Inst. H. Poincaré 25, 225-257 (1989)
3. Barlow, M.T., Kigami, J.: Localized eigenfunctions of the Laplacian on p.c.f. self-similar sets. Preprint (1994)
4. Barlow, M.T., Perkins, E.A.: Brownian motion on the Sierpinski gasket. Probab. Theory Relat. Fields 79, 543-624 (1988)
5. Bensoussan, A., Lions, J.L., Papanicolaou, G.: Asymptotic methods for periodic structures. Amsterdam: North-Holland 1978
6. Carlen, E.A., Kusuoka, S., Stroock, D.W.: Upper bounds for symmetric Markov transition functions. Ann. L'IHP, Sup au No. 2, 245-287 (1987)
7. Etheier, S.N., Kurtz, T.G.: Markov processes: characterization and convergence. New York: Wiley 1986
8. Fitzsimmons, P.J., Hambly, B.M., Kumagai, T.: Transition density estimates for Brownian motion on affine nested fractals. Commun. Math. Phys. 165, 595-620 (1994)
9. Fukushima, M.: Dirichlet forms, diffusion processes and spectral dimensions for nested fractals. In: Albeverio, S. et al. (eds.) Ideas and methods in mathematical analysis, stochastics, and applications. In Memory of R. Høegh-Krohn, vol. 1, pp. 151-161. Cambridge: Cambridge University Press: Cambridge 1992
10. Fukushima, M., Oshima, Y., Takeda, M.: Dirichlet forms and symmetric Markov-processes, (de Gruyter Studies in Mathematics 19) Berlin New York: Walter de Gruyter 1994
11. Hutchinson, S.: Fractals and self-similarity. Indiana Univ. Math. J. 30, 713-747 (1981)
12. Kato, T.: Perturbation theory for linear operators (Series: Die Grundlehren der Mathematischen Wissenschaften, Band 132) Berlin Heidelberg New York: Springer 1966
13. Kozlov, S.M.: Harmonization and homogenization on fractals. Commun. Math. Phys. 153, 339-357 (1993)
14. Kumagai, T.: Estimates of the transition densities for Brownian motion on nested fractals. Probab. Theory Relat. Fields 96, 205-224 (1993)
15. Kumagai, T.: Regularity, closedness and spectral dimensions of the Dirichlet forms on P.C.F. self-similar sets. J. Math. Kyoto Univ. 33, 765-786 (1993)
16. Kusuoka, S.: Diffusion processes on nested fractals. In: Dobrushin, R.L., Kusuoka, S. (eds.), Statistical mechanics and fractals (Lect. Notes Math., vol. 1567) Berlin Heidelberg New York: Springer 1993
17. Kusuoka, S., Zhou, X.Y.: Dirichlet forms on fractals: Poincaré constant and resistance. Probab. Theory Relat. Fields 93, 169-196 (1992)
18. Kusuoka, S., Zhou, X.Y.: Waves on fractal-like manifolds and effective energy propagation. Preprint (1993)
19. Lindstrøm, T.: Brownian motion on nested fractals. Memoirs Amer. Math. Soc. 420 (1990)
20. Metz, V.: How many diffusions exist on the Vicsek snowflake? Acta Appl. Math. 32, 227-241 (1993)
21. Metz, V.: Hilbert's projective metric on cones of Dirichlet forms. J. Funct. Anal. 127, 438-455 (1995)
22. Mosco, U.: Composite media and asymptotic Dirichlet forms. J. Funct. Anal. 123, 368-421 (1994)
23. Osada, H.: Homogenization of reflecting barrier Brownian motions. In: Elworthy, K.D., Ikeda, N. (eds.), Asymptotic problems in probability theory: stochastic models and diffusions on fractals, pp. 59-74. Montreal: Pitman 1993
24. Papanicolaou, G., Stroock, D.W., Varadhan, S.: Martingale approach to some limit theorems. 1976 Duke Turbulence Conference, Duke Univ. Math. Series III 1977
25. Papanicolaou, G., Varadhan, S.: Boundary value problems with rapidly oscillating random coefficients. In: Friz, J. et al. (eds.), Random fields (Colloq. Math. Soc. Janos Bolyai 27, pp, 835-873) Amsterdam: North-Holland 1979
26. Papanicolaou, G., Varadhan, S.: Diffusions with random coefficients. In: Kallianpur, G. et al. (eds.), Statistics and probability: essays in honor of C.R. Rao, pp. 547-552. Amsterdam: North-Holland 1982
27. Zhikov, V.V., Kozlov, S.M., Oleinik, O.A., Ngoan, K.T.: Averaging and G-convergence of differential operators. Russian Math. Surveys 34, 69-147 (1979)