|
A capture problem in Brownian motion and eigenvalues of spherical domains
Author(s):
Jesse
Ratzkin;
Andrejs
Treibergs
Journal:
Trans. Amer. Math. Soc.
361
(2009),
391-405.
MSC (2000):
Primary 60J65;
Secondary 35P15
Posted:
August 19, 2008
MathSciNet review:
2439411
Retrieve article in:
PDF
Abstract |
References |
Similar articles |
Additional information
Abstract:
We resolve a question of Bramson and Griffeath by showing that the expected capture time of four predators pursuing a single prey, all moving by standard Brownian motion on a line, is finite. Our main tool is an eigenvalue estimate for a particular spherical domain, which we obtain by a coning construction and domain perturbation.
References:
-
- [BG]
- M. Bramson and D. Griffeath, Capture problems for coupled random walks. in Random Walks, Brownian Motion and Interacting Particle Systems, ed. R. Durrett and H. Kesten, Birkhäuser, 1991. MR 1146445 (93e:60192)
- [C]
- I. Chavel, Eigenvalues in Riemannian Geometry. Academic Press, 1984. MR 768584 (86g:58140)
- [DB]
- R. D. DeBlassie, Exit times from cones in
 of Brownian motion. Prob. Theory and Rel. Fields. 74:1-29, 1987. MR 863716 (88d:60205) - [F]
- A. R. Forsythe, Theory of Functions of a Complex Variable, 3rd ed., Dover, 1965, 698-700. (Originally published by Cambridge University Press, Cambridge, 1918.) MR 0178116 (31:2374)
- [L]
- N. Lebedev, Special Functions and Their Applications. Dover, 1972. (Originally published by Prentice-Hall, Inc., 1965.) MR 0350075 (50:2568)
- [LS]
- W. Li and Q.-M. Shao, Capture time of Brownian pursuits. Prob. Theory and Rel. Fields. 121:30-48, 2001. MR 1857107 (2002h:60173)
- [N]
- Z. Nehari, Conformal Mapping. Dover, 1975. (Originally published by McGraw-Hill Book Co., Inc., New York, 1952.) MR 0045823 (13:640h)
- [PS]
- G. Pólya and G. Szegő, Isoperimetric Inequalities in Mathematical Physics. Princeton University Press, 1951, 29-30. MR 0043486 (13:270d)
- [R]
- Lord Rayleigh (J. W. Strutt), The Theory of Sound (2nd. ed.), Dover, 1945, 336-342. (Originally published by Mcmillan Co., 1894-1896.) MR 0016009 (7:500e)
- [S]
- E. Sperner, Zur Symmetrisierung von Funktionen auf Sphären. Math. Z. 134:317-327, 1973. MR 0340558 (49:5310)
- [St]
- F. Stenger, Numerical Methods Based on Sinc and Analytic Functions. Springer-Verlag, 1993. MR 1226236 (94k:65003)
Similar Articles:
Retrieve articles in Transactions of the American Mathematical
Society
with
MSC (2000):
60J65,
35P15
Retrieve articles in all Journals with
MSC (2000):
60J65,
35P15
Additional Information:
Jesse
Ratzkin
Affiliation:
Department of Mathematics, University of Connecticut, 196 Auditorium Road, Storrs, Connecticut 06269
Address at time of publication:
Department of Mathematics, University of Georgia, Boyd Hall, Athens, Georgia 30602
Email:
ratzkin@math.uconn.edu, jratzkin@math.uga.edu
Andrejs
Treibergs
Affiliation:
Department of Mathematics, University of Utah, 155 South 1400 East, JWB 233, Salt Lake City, Utah 84112
DOI:
10.1090/S0002-9947-08-04505-4
PII:
S 0002-9947(08)04505-4
Received by editor(s):
June 9, 2005
Received by editor(s) in revised form:
February 21, 2007
Posted:
August 19, 2008
Copyright of article:
Copyright
2008,
American Mathematical Society
The copyright for this article reverts to public domain after 28 years from publication.
|
