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

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 $.