Thick points for intersections of planar sample paths
Authors:
Amir Dembo, Yuval Peres, Jay Rosen and Ofer Zeitouni
Journal:
Trans. Amer. Math. Soc. 354 (2002), 49695003
MSC (2000):
Primary 60J55; Secondary 60J65, 28A80, 60G50
Published electronically:
August 1, 2002
MathSciNet review:
1926845
Fulltext PDF Free Access
Abstract 
References 
Similar Articles 
Additional Information
Abstract: Let denote the number of visits to of the simple planar random walk , up to step . Let be another simple planar random walk independent of . We show that for any , there are points for which . This is the discrete counterpart of our main result, that for any , the Hausdorff dimension of the set of thick intersection points for which , is almost surely . Here is the projected intersection local time measure of the disc of radius centered at for two independent planar Brownian motions run until time . The proofs rely on a ``multiscale refinement'' of the second moment method. In addition, we also consider analogous problems where we replace one of the Brownian motions by a transient stable process, or replace the disc of radius centered at by for general sets .
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R. Bass and D. Khoshnevisan, Intersection local times and Tanaka formulas, Ann. Inst. Henri Poincaré Probab. Statist. 29 (1993), 419451. MR 95c:60073
 2.
A. Dembo, Y. Peres, J. Rosen and O. Zeitouni, Thick points for spatial Brownian motion: multifractal analysis of occupation measure, Ann. Probab. 28 (2000), 135. MR 2001g:60194
 3.
A. Dembo, Y. Peres, J. Rosen and O. Zeitouni, Thin points for Brownian Motion, Annales de L'Institut Henri Poincaré, 36 (2000), 749774. CMP 2001:05
 4.
A. Dembo, Y. Peres, J. Rosen and O. Zeitouni, Thick points for transient symmetric stable processes, Electron. J. of Probab. 4 (1999) Paper no. 10, 113. MR 2000f:60117
 5.
A. Dembo, Y. Peres, J. Rosen and O. Zeitouni, Thick points for planar Brownian motion and the ErdosTaylor conjecture on random walk, Acta Math. 186 (2001), 239270.
 6.
U. Einmahl, Extensions of results of Komlós, Major, and Tusnády to the multivariate case. J. Multivariate Anal. 28 (1989), 2068. MR 90g:60032
 7.
P. Fitzsimmons and J. Pitman, Kac's moment formula and the FeynmanKac formula for additive functionals of a Markov process, Stochastic Processes and Applications. 79 (1999), 117134. MR 2000a:60136
 8.
K. Ito and H. McKean, Diffusion processes and their sample paths, SpringerVerlag, NewYork, (1974). MR 49:9963
 9.
J.P. Kahane, Some random series of functions: Second Edition, Cambridge University Press, (1985). MR 87m:60119
 10.
R. Kaufman, Une propriété metriqué du mouvement brownien, C. R. Acad. Sci. Paris 268 (1969), 727728. MR 39:2219
 11.
J. F. Le Gall, The exact Hausdorff measure of Brownian multiple points, Seminar on Stochastic Processes 1986, E. Cinlar, K.L. Chung, R.K. Getoor, editors, Birkhäuser (1987), 107137. MR 89a:60188
 12.
W. König and P. Mörters, Brownian intersection local times: upper tail asymptotics and thick points, Preprint (2001). To appear, Ann. Probab. (2002).
 13.
G. F. Lawler, Intersections of Random Walks, Birkhäuser, Boston, 1991. MR 92f:60122
 14.
J.F. Le Gall, Some properties of planar Brownian motion, École d'été de probabilités de St. Flour XX, 1990 (Berlin). Lecture Notes Math, 1527, SpringerVerlag, Berlin, 1992. MR 94g:60156
 15.
J.F. Le Gall, The exact Hausdorff measure of Brownian multiple points I and II, Seminar on Stochastic Processes, 1986, 107137 and 1988, 193197, Birkhäuser, Boston. MR 89a:60188, MR 90f:60139
 16.
P. Mattila, Geometry of sets and measures in Euclidean spaces, Cambridge University Press, (1995). MR 96h:28006
 17.
E. A. Perkins and S. J. Taylor, Uniform measure results for the image of subsets under Brownian motion, Probab. Theory Related Fields 76 (1987), 257289. MR 88m:60122
 18.
P. Révész, Random Walk in Random and NonRandom Environments, World Scientific, Teaneck, 1990. MR 92c:60096
 1.
 R. Bass and D. Khoshnevisan, Intersection local times and Tanaka formulas, Ann. Inst. Henri Poincaré Probab. Statist. 29 (1993), 419451. MR 95c:60073
 2.
 A. Dembo, Y. Peres, J. Rosen and O. Zeitouni, Thick points for spatial Brownian motion: multifractal analysis of occupation measure, Ann. Probab. 28 (2000), 135. MR 2001g:60194
 3.
 A. Dembo, Y. Peres, J. Rosen and O. Zeitouni, Thin points for Brownian Motion, Annales de L'Institut Henri Poincaré, 36 (2000), 749774. CMP 2001:05
 4.
 A. Dembo, Y. Peres, J. Rosen and O. Zeitouni, Thick points for transient symmetric stable processes, Electron. J. of Probab. 4 (1999) Paper no. 10, 113. MR 2000f:60117
 5.
 A. Dembo, Y. Peres, J. Rosen and O. Zeitouni, Thick points for planar Brownian motion and the ErdosTaylor conjecture on random walk, Acta Math. 186 (2001), 239270.
 6.
 U. Einmahl, Extensions of results of Komlós, Major, and Tusnády to the multivariate case. J. Multivariate Anal. 28 (1989), 2068. MR 90g:60032
 7.
 P. Fitzsimmons and J. Pitman, Kac's moment formula and the FeynmanKac formula for additive functionals of a Markov process, Stochastic Processes and Applications. 79 (1999), 117134. MR 2000a:60136
 8.
 K. Ito and H. McKean, Diffusion processes and their sample paths, SpringerVerlag, NewYork, (1974). MR 49:9963
 9.
 J.P. Kahane, Some random series of functions: Second Edition, Cambridge University Press, (1985). MR 87m:60119
 10.
 R. Kaufman, Une propriété metriqué du mouvement brownien, C. R. Acad. Sci. Paris 268 (1969), 727728. MR 39:2219
 11.
 J. F. Le Gall, The exact Hausdorff measure of Brownian multiple points, Seminar on Stochastic Processes 1986, E. Cinlar, K.L. Chung, R.K. Getoor, editors, Birkhäuser (1987), 107137. MR 89a:60188
 12.
 W. König and P. Mörters, Brownian intersection local times: upper tail asymptotics and thick points, Preprint (2001). To appear, Ann. Probab. (2002).
 13.
 G. F. Lawler, Intersections of Random Walks, Birkhäuser, Boston, 1991. MR 92f:60122
 14.
 J.F. Le Gall, Some properties of planar Brownian motion, École d'été de probabilités de St. Flour XX, 1990 (Berlin). Lecture Notes Math, 1527, SpringerVerlag, Berlin, 1992. MR 94g:60156
 15.
 J.F. Le Gall, The exact Hausdorff measure of Brownian multiple points I and II, Seminar on Stochastic Processes, 1986, 107137 and 1988, 193197, Birkhäuser, Boston. MR 89a:60188, MR 90f:60139
 16.
 P. Mattila, Geometry of sets and measures in Euclidean spaces, Cambridge University Press, (1995). MR 96h:28006
 17.
 E. A. Perkins and S. J. Taylor, Uniform measure results for the image of subsets under Brownian motion, Probab. Theory Related Fields 76 (1987), 257289. MR 88m:60122
 18.
 P. Révész, Random Walk in Random and NonRandom Environments, World Scientific, Teaneck, 1990. MR 92c:60096
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Additional Information
Amir Dembo
Affiliation:
Departments of Mathematics and Statistics, Stanford University, Stanford, California 94305
Email:
amir@math.stanford.edu
Yuval Peres
Affiliation:
Department of Statistics, University of California Berkeley, Berkeley, California 94720 and Institute of Mathematics, Hebrew University, Jerusalem, Israel
Email:
peres@stat.berkeley.edu
Jay Rosen
Affiliation:
Department of Mathematics, College of Staten Island, CUNY, Staten Island, New York 10314
Email:
jrosen3@earthlink.net
Ofer Zeitouni
Affiliation:
Department of Electrical Engineering, Technion, Haifa 32000, Israel
Email:
zeitouni@ee.technion.ac.il
DOI:
http://dx.doi.org/10.1090/S0002994702030805
PII:
S 00029947(02)030805
Keywords:
Thick points,
intersection local time,
multifractal analysis,
stable process
Received by editor(s):
May 9, 2001
Received by editor(s) in revised form:
April 16, 2002
Published electronically:
August 1, 2002
Additional Notes:
The first author’s research was partially supported by NSF grant #DMS0072331
The second author’s research was partially supported by NSF grant #DMS9803597
The third author’s research was supported, in part, by grants from the NSF and from PSCCUNY
The research of all authors was supported, in part, by a USIsrael BSF grant
Article copyright:
© Copyright 2002
American Mathematical Society
