Singular Boundaries in the Forward Chapman-Kolmogorov Differential Equation

The forward Chapman-Kolmogorov differential equation is used to model the time evolution of the Probability Density Function of fluctuations. This equation may be restricted to either Master, Fokker-Planck or Liouville equations. A derivation of the Liouville equation with possible singular boundary conditions has already been presented in a previous publication (Valiño and Hierro in Phys. Rev. E 67:046310, 2003). In this paper, that derivation is extended to the full Chapman-Kolmogorov differential equation.