On the almost intersections of transient Brownian motions
Let \(\{B_1^d(t), t \geq 0\}\) and \(\{B_2^d(t), t \geq 0\}\) be two independent Brownian motions in \({\mathbb R}^d\) starting from 0 and \(x\), respectively. The probability measure is denoted by \(P_{0, x}\). Let \[ w_i^d(a, b) = \{y \in {\mathbb R}^d: B_i^d(t) = y \text{ for some }\, t \in (a, b)\} \] be the images of \((a, b)\) under \(B_i^d(t)\), respectively. The authors investigate the asymptotic behavior as \(a \to 0+\) of the probability \(P_{0, x} \big\{\text{dist}\big(w_1^d(t_1, t_2), w_2^d(0, t_3)\big)\leq a \big\}\) of ``almost intersection'' for \(d \geq 4\), and the probability \(P_{0, x}\{\text{dist}(w_1^d(t_1, t_2), w_2^d(t_3, t_4))\geq a\}\) of ``almost non-intersection'' when \(d = 4\). For example, they prove that for any \(t_1, t_2, t_3 \in [0, T]\), \[ \begin{multlined} P_{0, a^{-1}x}\left\{\text{dist}(w_1^4(t_1, t_2), w_2^4(0, t_3)) \leq a \right\}\\ \sim 4 | \log a| ^{-1} E_{0, x} \left(\int_{t_1 a^{-2}}^{t_2a^{-2}}\int_0^{t_3a^{-2}}I_{\{| B_1^4(u) - B_2^4(v)| \leq 1\}}\, du dv\right),\end{multlined} \] if the right-hand side goes to 0 as \(a \to 0+\). This leads to an improvement of a result of \textit{M. Aizenman} [Commun. Math. Phys. 97, 91--110 (1985; Zbl 0573.60076)] on intersection of Wiener sausages in \({\mathbb R}^4\).
scientific article; zbMATH DE number 2153907