On the volume of the intersection of two independent Wiener sausages
Let \(W_{K_1}(t),\dots,W_{K_k}\) be Wiener sausages for independent Brownian motions \(B_1,\dots,B_k\) and corresponding non-polar, compact sets \(K_1,\dots,K_k\), then \[ N_{k_1,\dots,K_k}(t)=E_0^1\otimes\cdots\otimes E_0^k \left[\bigg| \bigcap_{i=1}^kW_{K_i}(t)\bigg|\right]. \] Let us denote for \(m>2\) the equilibrium measure for a non-polar, compact set \(K\) by \(\mu_K\), the Newtonian capacity of \(K\) by \(C(K)=\int\mu_K(dx)\), and its Lebesgue measure by \(|K|\). The expected volume of intersection of two independent Wiener sausages in \(\mathbb R^m\), \(m\geq3\), up to time \(t\), and associated to non-polar, compact sets \(K_1\) and \(K_2\) respectively, is obtained in the limit of large \(t\). One of obtained result is following. Let \(m=3\), and let \(K_1,K_2\) be non-polar, compact sets in \(\mathbb R^3\). Then for \(t\to\infty\), \[ \begin{multlined} N_{K_1,K_2}(t)= (1-2^{-1/2}) \pi^{-3/2}C(K_1)C(K_2)t^{1/2}- (8\pi)^{-1} \iint\mu_{K_1}(dy_1)\mu_{K_2}(dy_2) |y_1-y_2|\\+ 2^{-5}\pi^{-2}C(K_1) C(K_2)(C(K_1)+C(K_2))+ O(t^{-1/2}\log t). \end{multlined} \]
scientific article; zbMATH DE number 5837786