Fokker-Planck equation of the fractional Brownian motion

The fractional Brownian motion X-beta (t) is the solution of the Sturm-Liouville fractional differential equation of order beta, (with beta a positive real number), enforced by a zero mean normal white noise. The main aim of this paper is to derive the fractional Fokker-Planck equation (FFP) related to the above fractional differential equation. It is shown that FFP is ruled by the fractional derivative of order 2H, with Hurst index H = beta-1/2. This means that the diffusive term in the FFP equation is found. Further studies are necessary for the complete FFP equation in the more general case in which the equation is enforced not only by the white noise, but also by a nonlinear transformation of the response itself.

Automated summary

The fractional Brownian motion X-beta (t) is a solution of the Sturm-Liouville fractional differential equation enforced by a zero mean normal white noise. The main aim of the paper is to derive the fractional Fokker-Planck equation related to the fractional differential equation. It is shown that the FFP is governed by a fractional derivative of order 2H with Hurst index H = beta-1/2. Further studies are needed for a complete understanding of the FFP equation in more general cases involving nonlinear transformations of the response.