On the expected volume of intersection of independent Wiener sausages and the asymptotic behaviour of some related integrals
Let \((S(s),P_x)\) be a Brownian motion on \(R^m\) \((m\geq 3)\) with generator \(\Delta\) and \(T_K\) the first hitting time of a compact non-polar set \(K\). Put \(u(x;t)=P_x(T_K\leq t)\) and \(N_{k,m}=\int_{R^m} u(x;t)^k dx\). \(N_{k,m}\) is related to the Wiener sausages for independent Brownian motions \(B_1,B_2,\dots,B_k\) by the relation \(N_{k,m}(t)=E^1_0\otimes \cdots \otimes E^k_0[| \bigcap_{i=1}^k W_i(t)| ]\), where \(W_i(t)=\{B_i(s)+z: 0\leq s\leq t, z \in K\}\). The main purpose of this paper is to get the leading asymptotic behavior of \(N_{k,m}(t)\) as \(t \to \infty\). The results are as follows: \[ N_{k,m}(t)=c_{k,m}\left(C(K)\right)^k t^{(m+(2-m)k)/2}\left(1+o(1)\right)\text{ for }k<m/(m-2), \] \[ N_{m/(m-2),m}(t)=c_m \left(C(K)\right)^{m/(m-2)}\log t \left(1+o(1)\right) \] and \[ \lim_{t\to \infty} N_{k,m}(t)=c'_{k,m} \int_{R^m} dx \left(\int | x-y| ^{2-m} \mu_K(dy)\right)^k, \] where \(C(K)\) and \(\mu_K\) are the capacity and the equilibrium measure of \(K\), respectively. For the proof, the explicit expression of the last exit time distribution is used. In the case of \(m=2\), a similar result is given by \textit{J.-F. Le Gall} [in: Probabilités, Ec. d'Eté XX. Lect. Notes Math. 1527, 111--235 (1992; Zbl 0779.60068)].
scientific article; zbMATH DE number 2172988