On the volume of the intersection of two Wiener sausages

Michiel van den Berg¹ ; Erwin Bolthausen² ; Frank den Hollander³

¹ Department of Mathematics, University of Bristol, Bristol BS8 1TW, United Kingdom
² Institut für Mathematik, Universität Zürich, 8057 Zürich, Switzerland
³ EURANDOM, P. O. Box 513, 5600 MB Eindhoven, The Netherlands

Annals of mathematics, Tome 159 (2004) no. 2, pp. 741-782.

Résumé

For $a>0$, let $W_1^a(t)$ and $W_2^a(t)$ be the $a$-neighbourhoods of two independent standard Brownian motions in $\mathrm{R}^d$ starting at 0 and observed until time $t$. We prove that, for $d \geq 3$ and $c>0$, \[ \lim_{t \to \infty} \frac{1}{t^{(d-2)/d}} \log P\Big(|W_1^a(ct) \cap W_2^a(ct)| \geq t\Big) = – I_d^{\kappa_a}(c) \] and derive a variational representation for the rate constant $I_d^{\kappa_a}(c)$. Here, $\kappa_a$ is the Newtonian capacity of the ball with radius $a$. We show that the optimal strategy to realise the above large deviation is for $W_1^a(ct)$ and $W_2^a(ct)$ to “form a Swiss cheese”: the two Wiener sausages cover part of the space, leaving random holes whose sizes are of order 1 and whose density varies on scale $t^{1/d}$ according to a certain optimal profile.

Détail

DOI : 10.4007/annals.2004.159.741