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

DOI:
https://doi.org/10.1090/S0002-9947-09-04880-6

Published electronically:
April 16, 2009

MathSciNet review:
2506436

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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 \begin{equation*} L_{\gamma ,\mu }u\equiv -\frac 12\Delta u+\gamma V_\mu (u), \end{equation*} with the Dirichlet boundary condition, where $V_\mu$ is a nonlocal â€ś$\mu$-centeringâ€ť potential defined by \begin{equation*} V_\mu (u)=u-\int _Du d\mu . \end{equation*} 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 $\gamma \gg 1$ and $\gamma \ll 1$. 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, Technionâ€”Israel Institute of Technology, Haifa, 32000, Israel

Email:
pinsky@math.technion.ac.il

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.