Functional iterations and stopping times for Brownian motion on the Sierpiński Gasket

Mathematika

Volumn 44, Issue 2, 1997, Pages 374-400

  Grabner, Peter J ^a  

^a GRAZ UNIVERSITY OF TECHNOLOGY   (Austria)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 0346446093     PISSN: 00255793     EISSN: None     Source Type: Journal    
DOI: 10.1112/s0025579300012699     Document Type: Article

References (38)

1. T. M. Apostol. Introduction to Analytic Number Theory (Springer Verlag, Berlin, New York, 1984).
2. M. T. Barlow and E. A. Perkins. Brownian motion on the Sierpiński gasket. Probab. Th. rel. Fields, 79 (1988), 543-623.
3. A. F. Beardon. Iteration of Rational Functions (Springer Verlag, Berlin, New York, 1991).
4. J. D. Biggins and N. H. Bingham. Near-constancy phenomena in branching processes. Math. Proc. Camb. Phil. Soc., 110 (1991), 545-558.
5. J. D. Biggins and N. H. Bingham. Large deviations in the supercritical branching process. Adv. Appl. Probab., 25 (1993), 757-772.
6. N. H. Bingham. On the limit of a supercritical branching process. J. Appl. Probab. (special volume), 25A (1988), 215-228.
7. N. H. Bingham, J. L. Goldie and J. L. Teugels. Regular variation. Encycl. Math. Appl., vol. 27 (Cambridge University Press, 1987).
8. R. P. Boas. Entire Functions (Academic Press, New York, 1954).
9. R. L. Dobrushin and S. Kusuoka. Statistical Mechanics and Fractals, Lecture Notes in Math., vol. 1567 (Springer, 1993).
10. G. Doetsch. Handbuch der Laplace-Transformation (Birkhäuser Verlag, Basel, 1955).
11. S. Dubuc. La densité de la loi-limite d'un processus en cascade expansif. Z. Wahrscheinlichkeitsth., 19 (1971), 281-290.
12. S. Dubuc. Etude théorique et numérique de la fonction de Karlin-McGregor. J. Analyse Math., 42 (1982), 15-37.
13. S. Dubuc. An Approximation of the gamma function. J. Math. Anal. Appl., 146 (1990), 461-468.
14. R. S. Ellis. Entropy, Large Deviations and Statistical Mechanics (Springer Verlag, Berlin, New York, 1985).
15. K. D. Elworthy and N. Ikeda. Problems in Probability Theory: Stochastic Models and Diffusion on Fractals, vol. 283 (Pitman Research Notes in Math., 1993).
16. K. J. Falconer. The Geometry of Fractal Sets (Cambridge University Press, 1985).
17. P. Flajolet and A. M. Odlyzko. Limit distributions for coefficients of iterates of polynomials with applications to combinatorial enumeration. Math. Proc. Camb. Phil. Soc., 96 (1984), 237-372.
18. M. Fukushima and T. Shima. On a spectral analysis for the Sierpiński gasket. Potential Analysis, 1 (1992), 1-35.
19. I. P. Goulden and D. M. Jackson. Combinatorial Enumeration (J. Wiley and Sons, New York, 1983).
20. P. J. Grabner. Completely q-multiplicative functions: the Mellin transform approach. Acta Arith., 65 (1993), 85-96.
21. P. J. Grabner and R. F. Tichy. Equidistribution and Brownian motion on the Sierpiński gasket. Mh. Math. To appear.
22. P. J. Grabner and W. Woess. Functional Iterations and Periodic Oscillations for the Simple Random Walk on the Sierpiński Graph. Stochastic Processes Appl., 69 (1997), 127-138.
23. T. E. Harris. The Theory of Branching Processes (Springer Verlag, Berlin, New York, 1963).
24. K. Itô and H. P. McKean. Diffusion Processes and Their Sample Paths (Springer Verlag, Berlin, New York, 1965).
25. A. Ivić. The Riemann Zeta Function. (J. Wiley, New York, 1985).
26. S. Karlin and J. McGregor. Embeddability of discrete time branching processes into continuous-time branching processes. Trans. Amer. Math. Soc., 132 (19683, 115-136.
27. S. Karlin and J. McGregor Embedding iterates of analytic functions with two fixed points into continuous groups. Trans. Amer. Math. Soc., 132 (1968), 137-145.
28. J. Kigami. A harmonic calculus on the Sierpiński spaces. Japan J. Appl. Math., 6 (1989), 259-290.
29. B. Y. Levin. Distribution of Zeros of Entire Functions, Transl. Math. Monographs, vol. 5 (AMS Providence, RI, 1964).
30. T. Lindstrøm. Brownian Motion on Nested Fractals, vol. 83, Memoirs of the AMS, (Providence, RI, 1990)
31. H. Mellin. Die Dirichlet'schen Reihen, die zahlentheoretischen Funkionen und die unendlichen Produkte von endlichem Geschlecht. Acta. Math., 28 (1903), 36-64
32. F. Ricsz and B. Sz.-Nagy. Functional Analysis (F. Ungar Publ. Co., New York, 1955).
33. T. Shima. On Eigenvalue problems for the random walks on the Sierpiński pregasket. Japan J. Indust. Appl. Math., 8 (1991), 127-141.
34. W. Sierpiński. Sur une courbe dont tout point est un point de ramification. C. R. Acad. Sci. Paris, 160 (1915), 302-305.
35. G. Tenenbaum. Introduction to Analytic and Probabilistic Number Theory (Cambridge University Press, 1995).
36. J. D. Vaaler. Some extremal functions in Fourier analysis. Bull. Amer. Math. Soc., 12 (1985), 183-216.
37. D. V. Widder. An Introduction to Transform Theory (Academic Press, New York, 1971).
38. K. Yosida. Functional Analysis (Springer Verlag, Berlin, New York, 1971).