On Function of Evolution of Distribution for Time Homogeneous Markov Processes

TL;DR

This paper introduces a novel approach to analyzing time homogeneous Markov processes by defining a function of evolution of distribution and bridge operators, simplifying the computation of finite-dimensional distributions.

Contribution

It presents a new concept of a function of evolution of distribution and bridge operators, offering an alternative to traditional semi-group methods for Markov process analysis.

Findings

Explicit formulas for finite-dimensional distributions are derived.

The new approach simplifies computations compared to standard methods.

Examples demonstrate the effectiveness of the proposed framework.

Abstract

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 finite-dimensional distributions of Markov processes. This, in particular, produces explicit formulas which effectively simplify the computations of finite-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.