Lévy stable distribution and space-fractional Fokker–Planck type equation

The space-fractional Fokker–Planck type equation ∂p∂t+γ∂p∂x=-D(-Δ)α/2p(0<α⩽2) subject to the initial condition p(x,0)=δ(x)p(x,0)=δ(x) is solved in terms of Fox H functions. The solution as γ=0γ=0 expresses the Lévy stable distribution with the index αα. From the properties of Fox H functions, the series representation and asymptotic behavior for the solution are also obtained. Lévy stable distribution as 0<α<20<α<2 describes anomalous superdiffusion and its diffusion velocity is characterized by xd∝(Dt)1/αxd∝(Dt)1/α.