Permanent URL to this publication: http://dx.doi.org/10.5167/uzh21826
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):741782.

PDF
1MB 
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 $d\geq3$, $${\log{\rm Pr}[V_2^a(ct)\geq t]\over t^{(d2)/d}}\rightarrowI_d^{\kappa_a}(c)\quad\text{\ as\ }t\rightarrow\infty$$ (here $\kappa_a$ is the Newtonian capacity of the ball of radius $a$). A similar result holds for $d=2$ with $t^{(d2)/d}$ 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.
Citations  Downloads 
Additional indexing
Item Type:  Journal Article, refereed, original work 

Communities & Collections:  07 Faculty of Science > Institute of Mathematics 
Dewey Decimal Classification:  510 Mathematics 
Uncontrolled Keywords:  Wiener sausages, intersection volume, large deviations, variational problems, Sobolev inequalities. 
Language:  English 
Date:  2004 
Deposited On:  21 Apr 2010 13:15 
Last Modified:  28 Nov 2013 00:34 
Publisher:  Mathematical Sciences Publishers 
ISSN:  0003486X 
Official URL:  http://annals.math.princeton.edu/annals/2004/1592/p06.xhtml 
Users (please log in): suggest update or correction for this item
Repository Staff Only: item control page