Fractional Fokker-Planck-Kolmogorov equations associated with stochastic differential equations in a bounded domain

TL;DR

This paper develops a fractional generalization of the Fokker-Planck equation for stochastic differential equations driven by complex Levy processes in bounded domains, establishing existence and uniqueness of solutions.

Contribution

It introduces a fractional Fokker-Planck equation with distributed order operators linked to Levy-driven SDEs in bounded domains, extending classical models.

Findings

Derived fractional Fokker-Planck equations with distributed order operators.

Proved existence and uniqueness of solutions for the initial-boundary value problems.

Extended classical Fokker-Planck equations to fractional, Levy-driven contexts.

Abstract

This paper is devoted to the fractional generalization of the Fokker-Planck equation associated with a stochastic differential equation in a bounded domain. The driving process of the stochastic differential equation is a L\'evy process subordinated to the inverse of L\'evy's mixed stable subordinators. The Fokker-Planck equation is given through the general Waldenfels operator, while the boundary condition is given through the general Wentcel's boundary condition. As a fractional operator a distributed order differential operator with a Borel mixing measure is considered. In the paper fractional generalizations of the Fokker-Planck equation are derived and the existence of a unique solution of the corresponding initial-boundary value problems is proved.