On generalized local time for the process of brownian motion

Abstract

We prove that the functionals \(\delta _\Gamma (B_t ) and \frac{{\partial ^k }}{{\partial x_1^k ...\partial x_d^{k_d } }}\delta _\Gamma (B_t ), k_1 + ... + k_d = k > 1,\) 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 \((\delta _\Gamma ,\varphi ) : = \int\limits_\Gamma {\varphi (x} )\lambda (dx),\) where ι(·) is a surface measure on Γ.