Fractional Brownian motions as "higher-order" fractional derivatives of Brownian local times

Endre Csáki, Zhan Shi and Marc Yor

Summary: Fractional derivatives ${\cal D}^\gamma$ of Brownian local times are well defined for all $\gamma<3/2$. We show that, in the weak convergence sense, these fractional derivatives admit themselves derivatives which feature all fractional Brownian motions. Strong approximation results are also developed as counterparts of limit theorems for Brownian additive functionals which feature the fractional derivatives of Brownian local times.

Keywords: Local time, additive functional, principal value, Brownian sheet, fractional Brownian motion, Hilbert transform, fractional derivative.

AMS 1991 subject classification: 60J55, 60J65, 60F05.