A Fast and Accurate Solver for the Fractional Fokker-Planck Equation with Dirac-Delta Initial Conditions

TL;DR

This paper introduces a novel, high-precision numerical solver for the fractional Fokker-Planck equation with Dirac-delta initial conditions, enabling efficient high-dimensional computations in physics and related fields.

Contribution

It presents the first high-accuracy, efficient numerical method for the free-space FFPE with Dirac-delta initial data, using integral representations and fast algorithms.

Findings

The method achieves high precision for the FFPE with Dirac-delta initial conditions.

It efficiently handles high-dimensional problems using fast algorithms.

The approach is effective for initial conditions given by sums of Gaussians.

Abstract

The classical Fokker-Planck equation (FPE) is a key tool in physics for describing systems influenced by drag forces and Gaussian noise, with applications spanning multiple fields. We consider the fractional Fokker-Planck equation (FFPE), which models the time evolution of probability densities for systems driven by L\'evy processes, relevant in scenarios where Gaussian assumptions fail. The paper presents an efficient and accurate numerical approach for the free-space FFPE with constant coefficients and Dirac-delta initial conditions. This method utilizes the integral representation of the solutions and enables the efficient handling of very high-dimensional problems using fast algorithms. Our work is the first to present a high-precision numerical solver for the free-space FFPE with Dirac-delta initial conditions. In addition to Dirac-delta initial data, we demonstrate the effectiveness of our method for initial conditions given by sums of Gaussians. This opens the door for future research on more complex scenarios, including those with variable coefficients and other types of initial conditions.