Hostname: page-component-cd9895bd7-mkpzs Total loading time: 0 Render date: 2024-12-27T22:53:33.889Z Has data issue: false hasContentIssue false
  • English
  • Français

First exit time of a Lévy flight from a bounded region in ℝN

Part of: Stochastic processes Markov processes

Published online by Cambridge University Press:  30 March 2016

Yoora Kim ,
Irem Koprulu  and
Ness B. Shroff
Yoora Kim*
Affiliation:
University of Ulsan
Irem Koprulu*
Affiliation:
The Ohio State University
Ness B. Shroff*
Affiliation:
The Ohio State University
*
∗ 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
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

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.

Keywords

Lévy flightfirst exit timefirst passage timebounded region

MSC classification

Primary: 60J75: Jump processes
Secondary: 60G50: Sums of independent random variables; random walks
Type
Research Papers
Information
Journal of Applied Probability , Volume 52 , Issue 3 , September 2015 , pp. 649 - 664
Copyright
Copyright © 2015 by the Applied Probability Trust 

References

Andersen, E. Sparre (1953). On sums of symmetrically dependent random variables. Scand. Actuarial J. 1953 123-138.Google Scholar
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.CrossRefGoogle Scholar
Buldyrev, S. V. et al. (2001). Properties of Lévy flights on an interval with absorbing boundaries. Physica A 302 148-161.Google Scholar
Chen, Z.-Q., Kim, P. and Song, R. (2010). Heat kernel estimates for the Dirichlet fractional Laplacian. J. Eur. Math. Soc. 12 1307-1329.Google Scholar
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.Google Scholar
Gawronski, W. (1984). On the bell-shape of stable densities. Ann. Prob. 12 230-242.Google Scholar
Getoor, R. K. (1961). First passage times for symmetric stable processes in space. Trans. Amer. Math. Soc. 101 75-90.Google Scholar
Katzav, E. and Adda-Bedia, M. (2008). The spectrum of the fractional Laplacian and first-passage-time statistics. EPL 83 30006.CrossRefGoogle Scholar
Pruitt, W. E. (1981). The growth of random walks and Lévy processes. Ann. Prob. 9 948-956.Google Scholar
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.Google Scholar
Whitt, W. (2002). Stochastic-Process Limits: An Introduction to Stochastic-Process Limits and Their Application to Queues. Springer, New York.Google Scholar
Zoia, A., Rosso, A. and Kardar, M. (2007). Fractional Laplacian in bounded domains. Phys. Rev. E 76 021116.Google Scholar
Zolotarev, V. M. (1986). One-Dimensional Stable Distributions. American Mathematical Society, Providence, RI.Google Scholar