Information Geometric Investigation of Solutions to the Fractional Fokker–Planck Equation

Abstract

A novel method for measuring distances between statistical states as represented by probability distribution functions (PDF) has been proposed, namely the information length. The information length enables the computation of the total number of statistically different states that a system evolves through in time. Anomalous transport can presumably be modeled fractional velocity derivatives and Langevin dynamics in a Fractional Fokker–Planck (FFP) approach. The numerical solutions or PDFs are found for varying degree of fractionality ( α ) of the stable Lévy distribution as solutions to the FFP equation. Specifically, the information length of time-dependent PDFs for a given fractional index α is computed.

Numerical solution for Fokker-Planck equation using a two-level scheme

“…A fast algorithm for the numerical solution of the FP equation is presented by [17,17] and a finite difference scheme, in one-dimension, using a staggered grid to solve the Fokker-Planck equations with drift-admitting jumps is presented in [10]. In the year 2020, the research to find the numerical solution to the stochastic models and henece the FP equation is still on; e.g., in [8], a discretization scheme is developed to solve the one-dimensional nonlinear Fokker-Planck-Kolmogorov equation that preserves the nonnegativity of the solution and conserves the mass; a solution to the FokkerPlanck Equation with piecewise-constant drift is proposed in [11], a numerical method, named as information length, for measuring distances between statistical states as represented by PDF has been proposed in [1]. Also, there has been work on Fractional Fokker-Planck Equation as well, e.g., a space-time Petrov-Galerkin spectral method for time fractional FP equation with nonsmooth solution has been studied in [25] and a numerical solution of the Cauchy problem for the fractional FP equation in connection with Sinc convolution methods is proposed in [2].…”