Implicit convolution Fokker-Planck equations: extended Feller convolution

Abstract

Fokker-Planck equations are partial differential equations in the transition function of the Markov process. In the evolution equation approach, we re-write partial differential equations as ordinary differential equations in Banach spaces. In particular, an implicit evolution equation is used to re-write the Fokker-Planck equation for a pair of discontinuous Markov processes. In this paper we consider the continuous analogue in the form of two homogeneous Markov processes intertwined by the extended Chapman-Kolmogorov equation. Abstract harmonic analysis techniques are used to extend the Feller convolution. Then the associated Fokker-Planck equations are re-written as an implicit evolution equation expressed in terms of the extended Feller convolution