Generalized arcsine law and stable law in an infinite measure dynamical system

Journal of Statistical Physics

Volumn 132, Issue 1, 2008, Pages 171-186

  Akimoto, Takuma ^a  

^a WASEDA UNIVERSITY   (Japan)

Author keywords

Generalized arcsine law; Infinite measure; Non equilibrium state; Non stationary chaos; Stable law

Indexed keywords

EID: 44349123337     PISSN: 00224715     EISSN: None     Source Type: Journal    
DOI: 10.1007/s10955-008-9544-9     Document Type: Article

References (30)

1. Manneville, P.: Intermittency, self similarity and 1/f spectrum. J. Phys. 41, 1235-1243 (1980)
2. Ahlers, G., Behringer, R.P.: The Rayleigh-Bénard instability and the evolution of turbulence. Prog. Theor. Phys. Suppl. 64, 186-201 (1978)
3. Pomeau, Y., et al.: Intermittent behavior in the Belousov-Zhabotinsky reaction. J. Phys. Lett. 42, 271-273 (1981)
4. Nirmal, M., et al.: Fluorescence intermittency in single cadmium selenide nanocrystals. Nature 383, 802-804 (1996)
5. Omori, F.: On the aftershocks of earthquakes. J. Coll. Sci., Imp. Univ. Tokyo 7, 111-200 (1894)
6. Bottiglieri, M., Godano, C.: On-off intermittency in earthquake occurrence. Phys. Rev. E 75, 026101 (2007)
7. Aizawa, Y., Kikuchi, Y., Harayama, T., Yamamoto, K., Ota, M., Tanaka, K.: Stagnant motions in Hamiltonian Systems. Prog. Theor. Phys. Suppl. 98, 36-82 (1989)
8. -1 spectral chaos. Prog. Theor. Phys. 72, 659-661 (1984)
9. Brokmann, X., et al.: Statistical aging and nonergodicity in the fluorescence of single nanocrystals. Phys. Rev. Lett. 90, 12061 (2003)
10. Margolin, G., Barkai, E.: Nonergodicity of a time series obeying Lévy statistics. J. Stat. Phys. 122, 137-167 (2006)
11. Darling, D.A., Kac, M.: On occupation times for Markov processes. Trans. Am. Math. Soc. 84, 444-458 (1957)
12. Aaronson, J.: An Introduction to Infinite Ergodic Theory. American Mathematical Society, Providence (1997)
13. Thaler, M.: The Dynkin-Lamperti arc-sine laws for measure preserving transformations. Trans. Am. Math. Soc. 350, 4593-4607 (1998)
14. Thaler, M., Zweimüller, R.: Distributional limit theorems in infinite ergodic theory. Probab. Theory Relat. Fields 135, 15-52 (2006)
15. Thaler, M.: A limit theorem for sojourns near indifferent fixed points of one dimensional maps. Ergod. Theory Dyn. Syst. 22, 1289-1312 (2002)
16. Bel, G., Barkai, E.: Weak ergodicity breaking with deterministic dynamics. Europhys. Lett. 74, 15-21 (2006)
17. Aizawa, Y.: Comments on the non-stationary chaos. Chaos, Soliton Fractals 11, 263-268 (2000)
18. Thaler, M.: Transformations on [0,1] with infinite invariant measures. Isr. J. Math. 46, 67-96 (1983)
19. Cox, D.R.: Renewal Theory. Methuen, New York (1962)
20. Aizawa, Y., Kohyama, T.: In: Kuramoto, Y. (ed.) Kyoto Summer Institute "Chaos and Statistical Mechanics", pp. 109-116. Springer, Berlin (1983)
21. Inoue, T.: Ratio ergodic theorems for maps with indifferent fixed points. Ergod. Theory Dyn. Syst. 17, 625-642 (1997)
22. Akimoto, T., Aizawa, Y.: Scaling exponents of the slow relaxation in non-hyperbolic chaotic dynamics. Nonlinear Phenom. Complex Syst. 6, 178-182 (2006)
23. Akimoto, T., Aizawa, Y.: New aspects of the correlation functions in non-hyperbolic chaotic systems. J. Korean Phys. Soc. 50, 254-260 (2007)
24. Aizawa, Y.: Non-stationary chaos revisited from large deviation theory. Prog. Theor. Phys. Suppl. 99, 149-164 (1989)
25. Lamperti, J.: An occupation time theorem for a class of stochastic processes. Trans. Am. Math. Soc. 88, 380-387 (1958)
26. Tasaki, S., Gaspard, P.: Spectral properties of a piecewise linear intermittent map. J. Stat. Phys. 109, 803-820 (2002)
27. Barkai, E.: Residence time statistics for normal and fractional diffusion in a force field. J. Stat. Phys. 123, 883-907 (2006)
28. Burnov, S., Barkai, E.: Occupation time statistics in the quenched trap model. Phys. Rev. Lett. 98, 250601 (2007)
29. Birkhoff, G.D.: Proof of the ergodic theorem. Proc. Natl. Acad. Sci. USA 17, 656-660 (1931)
30. Feller, W.: In: An Introduction to Probability Theory and Its Applications, 2nd edn., vol. II, p. 448. Wiley, New York (1971)