On Function of Evolution of Distribution for Time Homogeneous Markov Processes

A study of time homogeneous, real valued Markov processes with a special property and a non-atomic initial distribution is provided. The new notion of a function of evolution of distribution which determines the dependency between one dimensional distributions of a process is introduced. This, along with the notion of bridge operators which determine the backward structure, as opposed to the forward structure determined by the usual semi-group operators, paves a way to the new approach for dealing with nite-dimensional distributions of Markov processes. This, in particular, produces explicit formulas which eec-tively simplify the computations of nite-dimensional distributions, giving an alternative to the standard approach based on computations using the chain rule of transition densities. Various examples illustrating the new approach are presented.