Fokker-Planck equation for Feynman-Kac transform of anomalous processes

In this paper, we develop a novel and rigorous approach to the Fokker-Planck equation, or Kolmogorov forward equation, for the Feynman-Kac transform of non-Markov anomalous processes. The equation describes the evolution of the density of the anomalous process Y-t = X-Et under the influence of potentials, where X is a strong Markov process on a Lusin space chi that is in weak duality with another strong Markov process (X) over cap on chi and {E-t, t >= 0} is the inverse of a driftless subordinator S that is independent of X and has infinite Levy measure. We derive a probabilistic representation of the density of the anomalous process under the Feynman-Kac transform by the dual Feynman-Kac transform in terms of the weak dual process (X) over cap (t) and the inverse subordinator {E-t ; t >= 0}. We then establish the regularity of the density function, and show that it is the unique mild solution as well as the unique weak solution of a non-local Fokker-Planck equation that involves the dual generator of X and the potential measure of the subordinator S. During the course of the study, we are naturally led to extend the notation of Riemann-Liouville integral to measures that are locally finite on [0, infinity). (C) 2022 Elsevier B.Y. All rights reserved.