Subordinating regular diffusion – namely, Brownian motion – to random time flows generated by Lévy noises may result in anomalous diffusion. Motivated by this phenomena, and by the recent interest in the phenomena of blinking in various physical systems, we explore the subordination of regular stochastic pulsation – namely, Poisson process – to random time flows generated by Lévy noises. We show that such subordination may yield, analogous to the case of diffusion, anomalous pulsation. Anomalous pulsation displays the following anomalous behaviors, which are impossible in the case of regular pulsation: (i) simultaneous emission of multiple pulses; (ii) non-linear local pulsation rates; (iii) clustering of pulsation epochs.
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References
Y. J. Jung E. Barkai R. J. Silbey (2002) Chem. Phys 284 181 Occurrence Handle10.1016/S0301-0104(02)00547-5
H. Yang et al. (2003) Science 302 262 Occurrence Handle10.1126/science.1086911 Occurrence Handle14551431
M. Orrit (2003) Science 302 239 Occurrence Handle10.1126/science.1090850 Occurrence Handle14551425
A. Meller (2003) J. Phys. Cond. Matt 15 581 Occurrence Handle10.1088/0953-8984/15/17/202
R. D. Reiss, A Course on Point Processes (Springer-Verlag, 1993).
P. Lévy, Calcul des Probabilités(Gauthier-Villars, 1925).
P. Lévy, Théorie de L’Addition des Variables Aléatoires (Gauthier-Villars, 1954).
P. Lévy, Processus Stochastiques et Mouvement Brownien (Gauthier-Villars, 1965).
V.M. Zolotarev, One-Dimensional Stable Distributions (AMS, 1986).
G. Samrodintsky and M. S. Taqqu, Stable non-Gaussian Random Processes (CRC Press, 1994).
J. Bertoin, Lévy Processes (Cambridge University Press, 1996); J. Bertoin, Subordinators: examples and applications, Lecture Notes in Mathematics 1717 (Springer, 1999).
K. Sato, Lévy Processes and Infinitely Divisible Distributions (Cambridge University Press, 1999).
V. V. Uchaikin and V. M. Zolotarev, Chance and Stability, Stable Distributions and Their Applications (V.S.P. Intl. Science, 1999).
O. E. Barndorff-Nielsen, T. Mikosch, and S. Resnic eds., Lévy Processes (Birkhauser, 2001).
J. P. Bouchaud A. Georges (1990) Phys. Rep 195 12 Occurrence Handle10.1016/0370-1573(90)90099-N
M. F. Shlesinger G. M. Zaslavsky J. Klafter (1993) Nature 363 31 Occurrence Handle10.1038/363031a0
M. F. Shlesinger, G. M. Zaslavsky, and U. Frisch eds., Lévy Flights and Related Topics (Springer, 1995).
J. Klafter M. F. Shlesinger G. Zumofen (1996) Phys. Today 49 2 33
G. M. Viswanathan et al. (1999) Nature 401 911 Occurrence Handle10.1038/44831 Occurrence Handle10553906
G. H. Weiss, Aspects and Applications of the Random Walk (North-Holland, 1994).
B. D. Hughes, Random Walks and Random Environments (Oxford University Press, 1995).
R. Metzler J. Klafter (2000) Phys. Rep 339 1 Occurrence Handle10.1016/S0370-1573(00)00070-3
W. Feller, An Introduction to Probability Theory and Its Applications, Vol. II, 2nd ed (Wiley, 1971).
I. M. Sokolov, Phys. Rev. E 63 011104 (2000); I. M. Sokolov, Phys. Rev. E 66: 041101 (2002).
A. A. Stanislavsky (2004) Theor. Appl. Math. Phys 138 IssueID3 481
I. Eliazar J. Klafter (2004) Physica D 187 30 Occurrence HandleMR2046690
I. Eliazar and J. Klafter, to appear in J. Stat. Phys. (titled: “Lévy, Ornstein-Uhlenbeck, and Subordination: Spectral vs Jump description”).
D. R. Cox J. R. Stat (1955) Soci. B 17 129
J. F. C. Kingman, Poisson Processes (Oxford University Press, 1993).
J. Grandell, Doubly Stochastic Poisson Processes (Springer, 1976).
D. R. Cox (1962) Renewal Theory Methuen London
P. Embrechts and M. Maejima, Selfsimilar Processes (Princeton University Press, 2002).
N. H. Bingham, C. M. Goldie, J. L. Teugels, Regular Variation (Cambridge University Press, 1987).
B. B. Mandelbrot, Multifractals and 1/f Noise (Springer,1999).
H. C. Tijms, Stochastic Models: An Algorithmic Approach (Wiley, 1995).
E. Barlow and F. Proschan, Mathematical Theory of Reliability (Classics in Appplied Mathematics 17, reprint edition) (Society for Industrial & Appplied Mathematics, 1996).
T. M. Cover and J. A. Thomas, Elements of Information Theory (Wiley-Interscience, 1991).
I. Eliazar J. Klafter (2003) J. Stat. Phys 111 IssueID314 739 Occurrence Handle10.1023/A:1022894030773
J. M. Chambers C. L. Mallows B. Stuck (1976) J. Amer. Stat. Assoc 71 340
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Eliazar, I., Klafter, J. Anomalous Pulsation. J Stat Phys 120, 587–626 (2005). https://doi.org/10.1007/s10955-005-5478-7
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DOI: https://doi.org/10.1007/s10955-005-5478-7