Law of Large Numbers for Semi-Markov inhomogeneous Random Evolutions on Banach spaces

TL;DR

This paper develops a Law of Large Numbers for inhomogeneous Random Evolutions driven by Semi-Markov processes on Banach spaces, demonstrating weak convergence to inhomogeneous semigroups and providing applications to Levy processes.

Contribution

It introduces a novel framework for inhomogeneous Random Evolutions on Banach spaces driven by Semi-Markov processes, establishing their weak convergence and martingale characterization.

Findings

Proved weak convergence of inhomogeneous Random Evolutions to inhomogeneous semigroups.

Established a martingale characterization for these evolutions.

Applied results to inhomogeneous Levy Random Evolutions.

Abstract

Using backward propagators, we construct inhomogeneous Random Evolutions on Banach spaces driven by (uniformly ergodic) Semi-Markov processes. After studying some of their properties (measurability, continuity, integral representation), we establish a Law of Large Numbers for such inhomogeneous Random Evolutions, and more precisely their weak convergence - in the Skorohod space $D$ - to an inhomogeneous semigroup. A martingale characterization of these inhomogeneous Random Evolutions is also obtained. Finally, we present applications to inhomogeneous L\'{e}vy Random Evolutions.