On the volume of the intersection of two Wiener sausages

Permanent URL to this publication: http://dx.doi.org/10.5167/uzh-21826

van den Berg, M; Bolthausen, E; den Hollander, F (2004). On the volume of the intersection of two Wiener sausages. Annals of Mathematics. Second Series, 159(2):741-782.

Preview

PDF
1925Kb

Abstract

For $a>0$ , let $W_i^a(t)$ be the $a$ -neighbourhoods of the $i$ th copy of a standard Brownian motion in $\Bbb R^d$ starting at 0, until time $t$ . The authors prove large deviations results about $|V_2^a(ct)|=|W_1^a(ct)\cap W_2^a(ct)|$ , for $d\geq2$ , and suggest extensions applicable to $|V_k^a(ct)|$ , the volume of the intersection of $k$ sausages.

In particular, for $d\geq3$ , $$${\log{\rm Pr}[|V_2^a(ct)|\geq t]\over t^{(d-2)/d}}\rightarrow$ -I_d^ ${\kappa_a}$ (c) $\quad\text{\ as\ }$ t $\rightarrow\infty$$$ (here $\kappa_a$ is the Newtonian capacity of the ball of radius $a$ ). A similar result holds for $d=2$ with $t^{(d-2)/d}$ replaced by $\log t$ and ${\rm Pr}[|V_2^a(ct)|\geq t]$ replaced by ${\rm Pr}[|V_2^a(ct)|\geq t/\log t]$ . The sizes of the large deviations come from the asymptotic value of the expected volume of a single Wiener sausage. A variational representation is derived for $I_d^{\kappa_a}(c)$ , and the authors also investigate the dependence of $I_d^{\kappa_a}(c)$ on $c$ for different values of $d$ .

The work is motivated by the desire to address a number of open problems arising in the discrete setting from the study of the tail of the distribution of the intersection of the ranges of two independent random walks in $\Bbb Z^d$ (in such cases no exact rate constant is known).

The results in the paper draw on ideas and techniques developed by the authors to handle large deviations for the volume of a single Wiener sausage.

Item Type:	Journal Article, refereed, original work
Communities & Collections:	07 Faculty of Science > Institute of Mathematics
DDC:	510 Mathematics
Uncontrolled Keywords:	Wiener sausages, intersection volume, large deviations, variational problems, Sobolev inequalities.
Language:	English
Date:	2004
Deposited On:	21 Apr 2010 15:15
Last Modified:	23 Nov 2012 15:49
Publisher:	Mathematical Sciences Publishers
ISSN:	0003-486X
Official URL:	http://annals.math.princeton.edu/annals/2004/159-2/p06.xhtml

Users (please log in): suggest update or correction for this item

Repository Staff Only: item control page