Lévy flights and Lévy-Schrödinger semigroups

• Authors: Piotr Garbaczewski
• Languages of publication: EN

Abstracts

EN

We analyze two different confining mechanisms for Lévy flights in the presence of external potentials. One of them is due to a conservative force in the corresponding Langevin equation. Another is implemented by Lévy-Schrödinger semigroups which induce so-called topological Lévy processes (Lévy flights with locally modified jump rates in the master equation). Given a stationary probability function (pdf) associated with the Langevin-based fractional Fokker-Planck equation, we demonstrate that generically there exists a topological Lévy process with the same invariant pdf and in reverse.

Keywords

EN

symmetric stable noise; Lévy flights; Lévy semigroups; Schrödinger boundary data problem; stationary densities; Cauchy noise; confining potentials

Discipline

• 05.40.Jc: Brownian motion
• 02.50.Ey: Stochastic processes
• 05.20.-y: Classical statistical mechanics
• 05.10.Gg: Stochastic analysis methods (Fokker-Planck, Langevin, etc.)

• Publisher: De Gruyter Open
• Journal: Open Physics
• Year: 2010
• Volume: 8
• Issue: 5
• Pages: 699-708

Dates

published

1 - 10 - 2010

online

22 - 7 - 2010

Contributors

author

Piotr Garbaczewski

• Institute of Physics, University of Opole, 45-052, Opole, Poland

References

• [1] M. F. Shlesinger, G. M. Zaslavsky, U. Frisch (Eds.), Lévy Flights and Related Topics in Physics, Lecture Notes in Physics (Springer-Verlag, Berlin, 1995)
• [2] P. Garbaczewski, M. Wolf, A. Weron (Eds.), Chaos: The interplay between deterministic and stochastic behavior, Lecture Notes in Physics (Springer-Verlag, Berlin, 1995)
• [3] D. Applebaum, Lévy processes and stochastic calculus (Cambridge University Press, 2004)
• [4] K.-I. Sato, Lévy processes and infinitely divisible distributions (Cambridge University Press, 1999)
• [5] P. Garbaczewski, J. R. Klauder, R. Olkiewicz, Phys. Rev. E 51, 4114 (1995) http://dx.doi.org/10.1103/PhysRevE.51.4114[Crossref]
• [6] P. Garbaczewski, R. Olkiewicz, J. Math. Phys. 40, 1057 (1999) http://dx.doi.org/10.1063/1.532706[Crossref]
• [7] P. Garbaczewski and R. Olkiewicz, J. Math. Phys. 41, 6843 (2000) http://dx.doi.org/10.1063/1.1290054[Crossref]
• [8] S. Jespersen, R. Metzler, H. C. Fogedby, Phys. Rev. E 59, 2736 (1999) http://dx.doi.org/10.1103/PhysRevE.59.2736[Crossref]
• [9] P. D. Ditlevsen, Phys. Rev. E 60, 172 (1999) http://dx.doi.org/10.1103/PhysRevE.60.172[Crossref]
• [10] V. V. Janovsky et al., Physica A 282, 13 (2000) http://dx.doi.org/10.1016/S0378-4371(99)00565-8[Crossref]
• [11] A. Chechkin et al., Chem. Phys. 284, 233 (2002) http://dx.doi.org/10.1016/S0301-0104(02)00551-7[Crossref]
• [12] A. A. Dubkov, B. Spagnolo, Acta Phys. Pol. A 38, 1745 (2007)
• [13] A. A. Dubkov, B. Spagnolo, V. V. Uchaikin, Int. J. Bifurcat. Chaos 18, 2549 (2008)
• [14] V. V. Belik, D. Brockmann, New. J. Phys. 9, 54 (2007) http://dx.doi.org/10.1088/1367-2630/9/3/054[Crossref]
• [15] N. Cufaro Petroni, M. Pusterla, Physica A 388, 824 (2009) http://dx.doi.org/10.1016/j.physa.2008.11.035[Crossref]
• [16] D. Brockmann, I. Sokolov, Chem. Phys. 284, 409 (2002) http://dx.doi.org/10.1016/S0301-0104(02)00671-7[Crossref]
• [17] D. Brockmann, T. Geisel, Phys. Rev. Lett. 90, 170601 (2003) http://dx.doi.org/10.1103/PhysRevLett.90.170601[Crossref]
• [18] D. Brockmann, T. Geisel, Phys. Rev. Lett. 91, 048303 (2003) http://dx.doi.org/10.1103/PhysRevLett.91.048303[Crossref]
• [19] M. C. Mackey, M. Tyran-Kamińska, J. Stat. Phys. 124, 1443 (2006) http://dx.doi.org/10.1007/s10955-006-9181-0[Crossref]
• [20] J. C. Zambrini, Phys. Rev. A 35, 3631 (1987) http://dx.doi.org/10.1103/PhysRevA.35.3631[Crossref]
• [21] P. Garbaczewski, Acta Phys. Pol. B 40, 1353 (2009)
• [22] P. Garbaczewski, Phys. Rev. E 78, 031101 (2008) http://dx.doi.org/10.1103/PhysRevE.78.031101[Crossref]
• [23] H. Risken, The Fokker-Planck equation (SpringerVerlag, Berlin, 1989)
• [24] I. S. Gradshteyn, I. M. Ryzhik, Table of integrals, series and products (Academic Press, NY, 2007)
• [25] J. N. Kapur, H. K. Kesavan, Entropy Optimization Principles with Applications (Academic Press, Boston, 1992)

Identifiers

DOI

10.2478/s11534-009-0156-z