Multiple intersection local time of planar Brownian motion as a particular Hida distribution
Let \((L^2)=L^2(({\mathcal S}^2)^*,{\mathcal B}^2,P)\) be the space of white noise, where \({\mathcal S}^2\) be the Schwartz space of two variables and \(P\) be the white noise measure derived from the planar Brownian motion \({\mathbf W}=(W_1,W_2)\). Any \({\mathbf g}\in(L^2)\) can be expanded into multiple Wiener-Itô integrals as \({\mathbf g}\sim(g_0,g_1,g_2,\dots)\) with the relation of norms \(|{\mathbf g}|^2=\sum n!|g_n|^2\). For \(c>1\), define the space of testing functionals \[ {\mathcal G}_c=\{{\mathbf g}\sim(g_0,g_1,g_2,\dots)\in (L^2);\;\sum c^{2n}n!|g_n|^2<\infty\}, \] and its dual space \({\mathcal G}^*_c\). The triplet \({\mathcal G}\equiv\bigcap_c{\mathcal G}_c\subset(L^2)\subset{\mathcal G}^*\equiv\bigcup_c{\mathcal G}^*_c\) represents a framework of white noise calculus they use. The authors consider the following renormalized \((k+1)\)-fold intersection local time of \({\mathbf W}\), \[ \overline\delta^k=\int_{\Delta_k} \prod^k_{i=1}\Biggl[\delta_0({\mathbf W}_{t_{i+1}}-{\mathbf W}_{t_i})-{1\over 2\pi(t_{i+1}-t_i)}\Biggr] dt_1\cdots dt_{k+1}, \] where \(\delta_0\) is Dirac's delta function. They prove that \(\delta^k\in{\mathcal G}^*_c\) for any \(c>k\), and obtain the explicit form of its Wiener-Itô decomposition.
scientific article; zbMATH DE number 933306