Brownian motion can be characterized as a generalized random process and, as such, has a generalized derivative whose covariance functional is the delta function. In a similar fashion, fractional Brownian motion can be interpreted as a generalized random process and shown to possess a generalized derivative. The resulting process is a generalized Gaussian process with mean functional zero and covariance functional that can be interpreted as a fractional integral or fractional derivative of the delta-function.
Zinde-Walsh, Victoria and Phillips, Peter C.B., "Fractional Brownian Motion as a Differentiable Generalized Gaussian Process" (2003). Cowles Foundation Discussion Papers. 1656.