Smoothness of Brownian local times and related functionals

Abstract

In this paper we show that the local time of the Brownian motion belongs to the Sobolev space $\mathbb{D}^{\alpha {\text{,}}p}$ for any p≥2 and 0<α<1/p. In order to prove this result we first discuss the smoothness and integrability properties of the composition of the Dirac function δ_α with a Wiener integral W(h), and we show that this composition belongs to $\mathbb{D}^{ - \alpha {\text{,}}p}$, for any α>0 and p>1 such that α+1/p>1.