Intersection local times of independent Brownian motions as generalized white noise functionals
\textit{M. De Faria, T. Hida, L. Streit} and \textit{H. Watanabe} [Acta Appl. Math. 46, 351--362 (1997; Zbl 0876.60068)] considered the self-intersection local time of a \(d\)-dimensional Brownian motion. Similarly, the intersection local time of two independent \(d\)-dimensional Brownian motions \(B^{(1)}\) and \(B^{(2)}\) can be defined for \(d<4\) using Donsker's \(\delta\)-function, by \(L\equiv \int d^2t\delta(B^{(1)}(t_1)-B^{(2)}(t_2))\). Further procedures will make \(L\) a well-defined generalized function of Brownian motion for all \(d\geq 1\). However, the chaos expansion of \(L\) is done in terms of `normal product' of white noise and it still corresponds to that in terms of the multiple Wiener integrals when one considers the Wiener process as the fundamental random variable. For all \(d\geq 1\) and \(\varepsilon>0\), the kernel functions of the intersection time \(L_{\varepsilon}\) are given explicitly. For the projection onto chaos of order \(n\geq N\) with \(2N>d-4\), the \(L_p\) properties of the kernel functions \(G_{m,k}\) appearing in the expansion of the truncated local time \(L^{(N)}\) are considered. It is shown in particular that for \(d<4\) they are square-integrable on \(R^{m+k}\) whenever \(m\neq 0\), \(k\neq 0\).
scientific article; zbMATH DE number 1742913