On generalized local time for the process of brownian motion

We prove that the functionals \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\delta _\Gamma (B_t ) and \frac{{\partial ^k }}{{\partial x_1^k ...\partial x_d^{k_d } }}\delta _\Gamma (B_t ), k_1 + ... + k_d = k > 1,$$ \end{document} of a d-dimensional Brownian process are Hida distributions, i.e., generalized Wiener functionals. Here, δΓ(·) is a generalization of the δ-function constructed on a bounded closed smooth surface Γ⊂Rd, k≥1 and acting on finite continuous functions φ(·) in Rd according to the rule \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$(\delta _\Gamma ,\varphi ) : = \int\limits_\Gamma {\varphi (x} )\lambda (dx),$$ \end{document} where ι(·) is a surface measure on Γ.