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⊂R d, k≥1 and acting on finite continuous functions φ(·) in R d according to the rule \((\delta _\Gamma ,\varphi ) : = \int\limits_\Gamma {\varphi (x} )\lambda (dx),\) where ι(·) is a surface measure on Γ.
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Bakun, V.V. On generalized local time for the process of brownian motion. Ukr Math J 52, 173–182 (2000). https://doi.org/10.1007/BF02529632
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DOI: https://doi.org/10.1007/BF02529632