# On semi-Markov processes and their Kolmogorov's integro-differential equations

## TL;DR

This paper derives integro-differential Kolmogorov equations for semi-Markov processes, generalizing Markov process theory and including fractional cases, with applications to weak limits.

## Contribution

It introduces a new integro-differential form of Kolmogorov's equations for semi-Markov processes, encompassing fractional and limit cases.

## Key findings

- Derived integro-differential equations for semi-Markov processes
- Established equivalence with differential forms of the equations
- Analyzed weak limits and their equations

## Abstract

Semi-Markov processes are a generalization of Markov processes since the exponential distribution of time intervals is replaced with an arbitrary distribution. This paper provides an integro-differential form of the Kolmogorov's backward equations for a large class of homogeneous semi-Markov processes, having the form of an abstract Volterra integro-differential equation. An equivalent evolutionary (differential) form of the equations is also provided. Fractional equations in the time variable are a particular case of our analysis. Weak limits of semi-Markov processes are also considered and their corresponding integro-differential Kolmogorov's equations are identified.