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 \(t\to\infty\) .
