Electronic Journal of Probability

Principal eigenvalue for Brownian motion on a bounded interval with degenerate instantaneous jumps

Iddo Ben-Ari

Full-text: Open access

Abstract

We consider a model of Brownian motion on a bounded open interval with instantaneous jumps. The jumps occur at a spatially dependent rate given by a positive parameter times a continuous function positive on the interval and vanishing on its boundary. At each jump event the process is redistributed uniformly in the interval. We obtain sharp asymptotic bounds on the principal eigenvalue for the generator of the process as the parameter tends to infinity. Our work answers a question posed by Arcusin and Pinsky.

Article information

Source
Electron. J. Probab., Volume 17 (2012), paper no. 87, 13 pp.

Dates
Accepted: 4 October 2012
First available in Project Euclid: 4 June 2016

Permanent link to this document
https://projecteuclid.org/euclid.ejp/1465062409

Digital Object Identifier
doi:10.1214/EJP.v17-1791

Mathematical Reviews number (MathSciNet)
MR2988402

Zentralblatt MATH identifier
1256.35033

Subjects
Primary: 35P15: Estimation of eigenvalues, upper and lower bounds
Secondary: 60J65: Brownian motion [See also 58J65]

Keywords
principal eigenvalue brownian motion random space-dependent jumps

Rights
This work is licensed under aCreative Commons Attribution 3.0 License.

Citation

Ben-Ari, Iddo. Principal eigenvalue for Brownian motion on a bounded interval with degenerate instantaneous jumps. Electron. J. Probab. 17 (2012), paper no. 87, 13 pp. doi:10.1214/EJP.v17-1791. https://projecteuclid.org/euclid.ejp/1465062409


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References

  • Nitay Arcusin; Ross G. Pinsky, Asymptotic behavior of the principal eigenvalue for a class of non-local elliptic operators related to Brownian motion with spatially dependent random jumps. Commun. Contemp. Math. 13 (2011), no. 6, 1077–1093.
  • Ross G. Pinsky, Asymptotics for exit problem and principal eigenvalue for a class of non-local elliptic operators related to diffusion processes with random jumps and vanishing diffusion, preprint.
  • Ross G. Pinsky, Spectral analysis of a class of nonlocal elliptic operators related to Brownian motion with random jumps. Trans. Amer. Math. Soc. 361 (2009), no. 9, 5041–5060.
  • Daniel Revuz and Marc Yor, Continuous martingales and Brownian motion, third ed., Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 293, Springer-Verlag, Berlin, 1999.