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), 4969-5003

MSC (2000):
Primary 60J55; Secondary 60J65, 28A80, 60G50

DOI:
https://doi.org/10.1090/S0002-9947-02-03080-5

Published electronically:
August 1, 2002

MathSciNet review:
1926845

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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 ``multi-scale 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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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:
https://doi.org/10.1090/S0002-9947-02-03080-5

Keywords:
Thick points,
intersection local time,
multi-fractal 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 #DMS-0072331

The second author’s research was partially supported by NSF grant #DMS-9803597

The third author’s research was supported, in part, by grants from the NSF and from PSC-CUNY

The research of all authors was supported, in part, by a US-Israel BSF grant

Article copyright:
© Copyright 2002
American Mathematical Society