Generalizations and extensions of the Fokker- Planck-Kolmogorov equations

In this paper, the classical Fokker-Planck-Kolmogorov equations are generalized to hold for conditional probability density functions of arbitrary random processes. Conditions are derived under which the generalized equations are of finite order both for one-dimensional and for vector random processes. An extension of the generalized equations which overcomes degeneracy occurring in the steady-state case is also presented.