On the volume of the intersection of two Wiener sausages

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.