Schneider’s grey Brownian motions are two parameter stochastic processes starting at zero with variance and characteristic function . The marginal density of the gBM is given by
(*)
where is the MWright function with Laplace transform . Now the marginal density satisfies a timefractional diffusion equation. For more information see muramainardigeneralizedgreybm greybrownianmotions. The general method of solving nonlinear FokkerPlanck equation via a solution of the Markovian FokkerPlanck was known to Sokolov sokolovsolvingnonmarkovianfokkerplanck2002. If is a solution of and we are interested in solving then we take the Green’s function of the Markovian solution and find where
In general the inversion of this Laplace transform is difficult. Srokowski eliminates by applying a Fourier transform.
SROKOWSKI 2006 LEVY FRACTIONAL DIFFUSION SOLUTION

5:44 PM (8 minutes ago)



Fp( t, xi ) = 1 – (a0 * gamma^alpha * Gamma(1+alpha))/(Gamma(2*alpha+gamma*alpha1)) * xi^mu * t^{2*alphagamma*alpha}
Fp(t,xi) = 1 – a0*gamma*Gamma(2)/Gamma(1+gamma) * xi^mu * t^(2gamma)
as the solution of the Levy distributed case which is the direct implementation of a nonMarkovian FokkerPlanck equation with Levy stable density and time fractionation. Now the caveat is that one has to worry about slow versus fast diffusions, which is a case that Mura et. al. handle and for which the gray Brownian motion densities handle. Regardless, this Srokowski solution for power law memory kernel is fast and easy to fit and estimate its parameters on actual data with infrastructure I’ve built already —
Preview attachment fokkerplanck.py
Preview attachment nonmarkovianlevydiffusionsrokowski.pdf
Leave a Reply