On evolution of marginal distributions for Markov processes

We provide a study of the time evolution of the marginals of Markov processes. If this evolution can be controlled by an associated function, the process is said to satisfy the FoED property. This property is used to derive a theorem for an Itô–McKean diffusion, which states that the distribution of the process, under mild conditions, is determined by its one-dimensional marginals. Moreover, the FoED property together with the notion of bridge operators determines the backward dependence structure of a Markov process, providing an alternative to the usual forward dependence structure determined by the semigroup operators. This paves the way for a new approach to dealing with the finite-dimensional distributions of Markov processes. This approach is illustrated by new convolution formulas for Lévy processes. Additional examples illustrating the approach are also presented.