On Some Properties of the Fractional Derivative of the Brownian Local Time

Abstract

We study the properties of the fractional derivative \(D_\alpha l(t,x)\) of order \(\alpha<1/2\) of the Brownian local time \(l(t,x)\) with respect to the variable \(x\). This derivative is understood as the convolution of the local time with the generalized function \(|x|^{-1-\alpha}\). We show that \(D_\alpha l(t,x)\) appears naturally in Itô’s formula for the process \(|w(t)|^{1-\alpha}\). Using the martingale technique, we also study the limit behavior of \(D_\alpha l(t,x)\) as \(t\to\infty\).