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 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

**1.**Ben-Ari, I. and Pinsky, R. G., Ergodic behavior of diffusions with random jumps from the boundary,*Stochastic Processes and their Applications*, to appear.**2.**Ben-Ari, I. and Pinsky, R. G., Spectral analysis of a family of second-order elliptic operators with nonlocal boundary condition indexed by a probability measure,*J. Funct. Anal.***251**(2007), 122-140. MR**2353702****3.**Davidson, F. A. and Dodds, N. Spectral properties of non-local uniformly-elliptic operators,*Electron. J. Differential Equations*(2006) No. 126, 15 pp. (electronic). MR**2255241 (2007e:35046)****4.**Freitas, P., A nonlocal Sturm-Liouville eigenvalue problem,*Proc. Roy. Soc. Edinburgh Sect. A***124**(1994), 169-188. MR**1272438 (95b:34036)****5.**Freitas, P. and Vishnevskii, M. P., Stability of stationary solutions of nonlocal reaction-diffusion equations in -dimensional space,*Differential Integral Equations***13**(2000), 265-288. MR**1811959 (2001m:35173)****6.**Friedman, A.*Partial Differential Equations of Parabolic Type*, Prentice-Hall, Inc., Englewood Cliffs, N.J., (1964). MR**0181836 (31:6062)****7.**Grieser, D., Uniform bounds for eigenfunctions of the Laplacian on manifolds with boundary,*Comm. Partial Differential Equations***27**(2002), 1283-1299. MR**1924468 (2003g:58036)****8.**Grigorescu, I. and Kang, M., Brownian motion on the figure eight,*J. Theoret. Probab.***15**, (2002), 817-844. MR**1922448 (2003f:60144)****9.**Grigorescu, I. and Kang, M., Ergodic properties of multidimensional Brownian motion with rebirth, preprint, url: http://www.math.miami.edu/ igrigore/pp/gn.pdf.**10.**Karatzas, I. and Shreve, S.,*Brownian Motion and Stochastic Calculus*, Springer-Verlag, New York, 1988. MR**917065 (89c:60096)****11.**Leung, Y., Li, W. and Rakesh, Spectral analysis of Brownian motion with jump boundary,*Proc. Amer. Math. Soc.*,**136**(2008), no. 12, 4427-4436. MR**2431059****12.**Pazy, A.,*Semigroups of Linear Operators and Applications to Partial Differential Equations*, Applied Mathematical Sciences,**44**, Springer-Verlag, New York, (1983). MR**710486 (85g:47061)****13.**Pinsky, R. G., The dead core for reaction-diffusion equations with convection and its connection with the first exit time of the related Markov diffusion process,*Nonlinear Anal.***12**, (1988), 451-471. MR**940604 (89h:35161)****14.**Pinsky, R. G.,*Positive Harmonic Functions and Diffusion*, Cambridge Studies in Advanced Mathematics**45**, Cambridge University Press, (1995). MR**1326606 (96m:60179)****15.**Reed, M. and Simon, B.,*Methods of Modern Mathematical Physics, I, Functional Analysis*, Academic Press, New York, (1972). MR**0493419 (58:12429a)**

Retrieve articles in *Transactions of the American Mathematical Society*
with MSC (2000):
35P15,
60F10,
60J65

Retrieve articles in all journals with MSC (2000): 35P15, 60F10, 60J65

Additional Information

**Ross G. Pinsky**

Affiliation:
Department of Mathematics, Technion—Israel Institute of Technology, Haifa, 32000, Israel

Email:
pinsky@math.technion.ac.il

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

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.