Operational solutions for the generalized Fokker-Planck and generalized diffusion-wave equations

TL;DR

This paper develops operator-based solutions for generalized Fokker-Planck and diffusion-wave equations with memory effects, linking them to probability densities and stable Lévy distributions, extending classical stochastic process methods.

Contribution

It introduces an operator method for solving generalized equations with memory kernels, connecting solutions to probability densities and stable Lévy distributions, and analyzing evolution operator properties.

Findings

Solutions are expressed via evolution operators with probability density functions.

Memory functions in Laplace space are Stieltjes functions, ensuring probabilistic interpretation.

Power-law memory functions relate to stable Lévy distributions.

Abstract

The evolution operator method is used to solve the generalized Fokker-Planck equations and the generalized diffusion-wave equations in the (1+1) dimensional space in which $x\in\mathbb{R}$ and $t\in\mathbb{R}_+$. These equations contain either the first- or the second-time derivatives smeared by memory functions, each of which forms an integral kernel (denoted by $f(\xi, t)$, $\xi\in\mathbb{R}_+$) of suitable evolution operators. If memory functions in the Laplace space are Stieltjes functions, then $f(\xi, t)$ satisfy normalization, non-negativity, and infinite divisibility to be considered a probability density function. The evolution operators also contain exponential-like operators whose action on initial condition $p_0(x) > 0$ leads to the parent process distribution functions. This makes the results fully analogous to those obtained within the standard subordination approach. The above conclusion is satisfied by the solution of the generalized Fokker-Planck equation. In the case of the generalized diffusion-wave equation, to get this property, we should employ a special class, namely "diffusion-like" initial conditions. The key models of the operator method involve power-law memory functions. It leads to the characterization of $f(\xi, t)$ by applying one-sided stable L\'{e}vy distributions. The article also examines the properties of evolution operators in terms of evolution and self-reproduction.