A Fast and Accurate Solver for the Fractional Fokker-Planck Equation with Dirac-Delta Initial Conditions
This work provides a foundational numerical tool for researchers studying systems driven by Lévy processes, where Gaussian assumptions fail.
The paper introduces the first high-precision numerical solver for the free-space fractional Fokker-Planck equation with Dirac-delta initial conditions, enabling efficient handling of high-dimensional problems via fast algorithms.
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 Lévy 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.