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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), 5041-5060
MSC (2000): Primary 35P15, 60F10, 60J65
Published electronically: April 16, 2009
MathSciNet review: 2506436
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Abstract: Let $ D\subset R^d$ be a bounded domain and let $ \mathcal P(D)$ denote the space of probability measures on $ D$. Consider a Brownian motion in $ D$ which is killed at the boundary and which, while alive, jumps instantaneously at an exponentially distributed random time with intensity $ \gamma>0$ to a new point, according to a distribution $ \mu\in\mathcal P(D)$. 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 $ -L_{\gamma,\mu}$, defined by

$\displaystyle L_{\gamma,\mu}u\equiv -\frac12\Delta u+\gamma V_\mu(u), $

with the Dirichlet boundary condition, where $ V_\mu$ is a nonlocal ``$ \mu$-centering'' potential defined by

$\displaystyle V_\mu(u)=u-\int_Du d\mu. $

The operator $ L_{\gamma,\mu}$ is symmetric only in the case that $ \mu$ 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 $ L_{\gamma,\mu}$ consists exclusively of eigenvalues. As is well known, the principal eigenvalue gives the exponential rate of decay in $ t$ of the probability of not exiting the domain by time $ t$. We study the behavior of the eigenvalues, our main focus being on the behavior of the principal eigenvalue for the regimes $ \gamma\gg1$ and $ \gamma\ll1$. We also consider conditions on $ \mu$ that guarantee that the principal eigenvalue is monotone increasing or decreasing in $ \gamma$.

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Additional Information

Ross G. Pinsky
Affiliation: Department of Mathematics, Technion—Israel Institute of Technology, Haifa, 32000, Israel

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.

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