Spectral analysis of a class of nonlocal elliptic operators related to Brownian motion with random jumps
Author:
Ross G. Pinsky
Journal:
Trans. Amer. Math. Soc. 361 (2009), 50415060
MSC (2000):
Primary 35P15, 60F10, 60J65
Published electronically:
April 16, 2009
MathSciNet review:
2506436
Fulltext PDF Free Access
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Abstract: Let be a bounded domain and let denote the space of probability measures on . Consider a Brownian motion in which is killed at the boundary and which, while alive, jumps instantaneously at an exponentially distributed random time with intensity to a new point, according to a distribution . From this new point it repeats the above behavior independently of what has transpired previously. The generator of this process is an extension of the operator , defined by with the Dirichlet boundary condition, where is a nonlocal ``centering'' potential defined by The operator is symmetric only in the case that is normalized Lebesgue measure; thus, only in that case can it be realized as a selfadjoint operator. The corresponding semigroup is compact, and thus the spectrum of consists exclusively of eigenvalues. As is well known, the principal eigenvalue gives the exponential rate of decay in of the probability of not exiting the domain by time . We study the behavior of the eigenvalues, our main focus being on the behavior of the principal eigenvalue for the regimes and . We also consider conditions on that guarantee that the principal eigenvalue is monotone increasing or decreasing in .
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 BenAri, I. and Pinsky, R. G., Ergodic behavior of diffusions with random jumps from the boundary, Stochastic Processes and their Applications, to appear.
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 Davidson, F. A. and Dodds, N. Spectral properties of nonlocal uniformlyelliptic operators, Electron. J. Differential Equations (2006) No. 126, 15 pp. (electronic). MR 2255241 (2007e:35046)
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 Freitas, P., A nonlocal SturmLiouville eigenvalue problem, Proc. Roy. Soc. Edinburgh Sect. A 124 (1994), 169188. MR 1272438 (95b:34036)
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 Freitas, P. and Vishnevskii, M. P., Stability of stationary solutions of nonlocal reactiondiffusion equations in dimensional space, Differential Integral Equations 13 (2000), 265288. MR 1811959 (2001m:35173)
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 Friedman, A. Partial Differential Equations of Parabolic Type, PrenticeHall, Inc., Englewood Cliffs, N.J., (1964). MR 0181836 (31:6062)
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 Grieser, D., Uniform bounds for eigenfunctions of the Laplacian on manifolds with boundary, Comm. Partial Differential Equations 27 (2002), 12831299. MR 1924468 (2003g:58036)
 8.
 Grigorescu, I. and Kang, M., Brownian motion on the figure eight, J. Theoret. Probab. 15, (2002), 817844. MR 1922448 (2003f:60144)
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 Grigorescu, I. and Kang, M., Ergodic properties of multidimensional Brownian motion with rebirth, preprint, url: http://www.math.miami.edu/ igrigore/pp/gn.pdf.
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 Leung, Y., Li, W. and Rakesh, Spectral analysis of Brownian motion with jump boundary, Proc. Amer. Math. Soc., 136 (2008), no. 12, 44274436. MR 2431059
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 Pazy, A., Semigroups of Linear Operators and Applications to Partial Differential Equations, Applied Mathematical Sciences, 44, SpringerVerlag, New York, (1983). MR 710486 (85g:47061)
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 Pinsky, R. G., The dead core for reactiondiffusion equations with convection and its connection with the first exit time of the related Markov diffusion process, Nonlinear Anal. 12, (1988), 451471. MR 940604 (89h:35161)
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 Pinsky, R. G., Positive Harmonic Functions and Diffusion, Cambridge Studies in Advanced Mathematics 45, Cambridge University Press, (1995). MR 1326606 (96m:60179)
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Additional Information
Ross G. Pinsky
Affiliation:
Department of Mathematics, Technion—Israel Institute of Technology, Haifa, 32000, Israel
Email:
pinsky@math.technion.ac.il
DOI:
http://dx.doi.org/10.1090/S0002994709048806
PII:
S 00029947(09)048806
Keywords:
Principal eigenvalue,
spectral analysis,
Brownian motion,
random jumps
Received by editor(s):
June 18, 2007
Received by editor(s) in revised form:
June 3, 2008
Published electronically:
April 16, 2009
Additional Notes:
This research was supported by the M. & M. Bank Mathematics Research Fund.
Article copyright:
© Copyright 2009
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.
