Skip to main content Accessibility help
×
Home

First exit time of a Lévy flight from a bounded region in ℝ N

  • Yoora Kim (a1), Irem Koprulu (a2) and Ness B. Shroff (a2)

Abstract

In this paper we characterize the mean and the distribution of the first exit time of a Lévy flight from a bounded region in N-dimensional spaces. We characterize tight upper and lower bounds on the tail distribution of the first exit time, and provide the exact asymptotics of the mean first exit time for a given range of step-length distribution parameters.

    • Send article to Kindle

      To send this article to your Kindle, first ensure no-reply@cambridge.org is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about sending to your Kindle. Find out more about sending to your Kindle.

      Note you can select to send to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be sent to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

      Find out more about the Kindle Personal Document Service.

      First exit time of a Lévy flight from a bounded region in ℝ N
      Available formats
      ×

      Send article to Dropbox

      To send this article to your Dropbox account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your <service> account. Find out more about sending content to Dropbox.

      First exit time of a Lévy flight from a bounded region in ℝ N
      Available formats
      ×

      Send article to Google Drive

      To send this article to your Google Drive account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your <service> account. Find out more about sending content to Google Drive.

      First exit time of a Lévy flight from a bounded region in ℝ N
      Available formats
      ×

Copyright

Corresponding author

Postal address: Department of Mathematics, University of Ulsan, 93 Daehak-ro, Nam-gu, Ulsan, South Korea. Email address: yrkim@ulsan.ac.kr
∗∗ Postal address: Department of Electrical and Computer Engineering, The Ohio State University, Columbus, OH 43210, USA. Email address: irem.koprulu@gmail.com
∗∗∗ Postal address: Departments of Electrical and Computer Engineering and Computer Science and Engineering, The Ohio State University, Columbus, OH 43210, USA. Email address: shroff.11@osu.edu

References

Hide All
[1] Andersen, E. Sparre (1953). On sums of symmetrically dependent random variables. Scand. Actuarial J. 1953 123-138.
[2] Buldyrev, S. V. et al. (2001). Average time spent by Lévy flights and walks on an interval with absorbing boundaries. Phys. Rev. E 64 041108.
[3] Buldyrev, S. V. et al. (2001). Properties of Lévy flights on an interval with absorbing boundaries. Physica A 302 148-161.
[4] Chen, Z.-Q., Kim, P. and Song, R. (2010). Heat kernel estimates for the Dirichlet fractional Laplacian. J. Eur. Math. Soc. 12 1307-1329.
[5] Dybiec, B., Gudowska-Nowak, E. and Hänggi, P. (2006). Lévy–Brownian motion on finite intervals: mean first passage time analysis. Phys. Rev. E 73 046104.
[6] Gawronski, W. (1984). On the bell-shape of stable densities. Ann. Prob. 12 230-242.
[7] Getoor, R. K. (1961). First passage times for symmetric stable processes in space. Trans. Amer. Math. Soc. 101 75-90.
[8] Katzav, E. and Adda-Bedia, M. (2008). The spectrum of the fractional Laplacian and first-passage-time statistics. EPL 83 30006.
[9] Pruitt, W. E. (1981). The growth of random walks and Lévy processes. Ann. Prob. 9 948-956.
[10] Vahabi, M., Schulz, J. H. P., Shokri, B. and Metzler, R. (2013). Area coverage of radial Lévy flights with periodic boundary conditions. Phys. Rev. E 87 042136.
[11] Whitt, W. (2002). Stochastic-Process Limits: An Introduction to Stochastic-Process Limits and Their Application to Queues. Springer, New York.
[12] Zoia, A., Rosso, A. and Kardar, M. (2007). Fractional Laplacian in bounded domains. Phys. Rev. E 76 021116.
[13] Zolotarev, V. M. (1986). One-Dimensional Stable Distributions. American Mathematical Society, Providence, RI.

Keywords

MSC classification

First exit time of a Lévy flight from a bounded region in ℝ N

  • Yoora Kim (a1), Irem Koprulu (a2) and Ness B. Shroff (a2)

Metrics

Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed.