# On Laws of Large Numbers for Systems with Mean-Field Interactions and Markovian Switching

## TL;DR

This paper establishes laws of large numbers for mean-field systems with Markovian switching, where the empirical measure limit is a random measure influenced by the switching process's history.

## Contribution

It introduces a novel approach to characterize the limit as a conditional distribution, addressing challenges posed by the coupling and randomness in switching diffusions.

## Key findings

- Law of large numbers for mean-field switching diffusions established.
- Limit of empirical measures is a random measure dependent on switching history.
- Characterization of the limit via stochastic McKean-Vlasov equations with Markovian switching.

## Abstract

Focusing on stochastic systems arising in mean-field models, the systems under consideration belong to the class of switching diffusions, in which continuous dynamics and discrete events coexist and interact. The discrete events are modeled by a continuous-time Markov chain. Different from the usual switching diffusions, the systems include mean-field interactions. Our effort is devoted to obtaining laws of large numbers for the underlying systems. One of the distinct features of the paper is the limit of the empirical measures is not deterministic but a random measure depending on the history of the Markovian switching process. A main difficulty is that the standard martingale approach cannot be used to characterize the limit because of the coupling due to the random switching process. In this paper, in contrast to the classical approach, the limit is characterized as the conditional distribution (given the history of the switching process) of the solution to a stochastic McKean-Vlasov differential equation with Markovian switching.