STOCHASTIC MODELING AND CONTROL BANACH CENTER PUBLICATIONS, VOLUME 122 INSTITUTE OF MATHEMATICS POLISH ACADEMY OF SCIENCES WARSZAWA 2020 SWITCHING DIFFUSIONS WITH MEAN-FIELD INTERACTIONS: LIMIT RESULTS, MAXIMUM PRINCIPLE, AND NON-MARKOV SYSTEMS GEORGE YIN Department of Mathematics, University of Connecticut Storrs, CT 06269 ORCID: 0000-0002-2951-0704 E-mail: gyin@uconn.edu SON LUU NGUYEN University of Puerto Rico, Rio Piedras campus San Juan, PR 00936, USA ORCID: 0000-0002-6082-6686 E-mail: sonluu.nguyen@upr.edu Abstract. This paper is devoted to switching diffusions with mean-field interactions. We first review some of the recent results. Then we examine a case of systems not driven by Brownian motion but stationary mixing processes. We obtain the limit of the systems by weak convergence analysis together with our limit results on a law of large numbers for switching diffusion processes with mean-field terms. 1. Introduction. In this work, we study switching diffusions with mean-field interac- tions. The motivation stems from two lines of work. One of them is to treat hybrid diffusions that include both continuous dynamics and discrete events, in which the dis- crete events are modeled as a continuous-time Markov chain taking values in a finite state space M = {1,...,m 0 }. The other line is the consideration of systems with mean-field interactions. Both lines are of considerable current interests. What we are looking here is at the intersection of the two lines. Before proceeding further, let us briefly recall some of the works in each of these 2010 Mathematics Subject Classification: 60J25; 60J27; 60J60; 93E20. Key words and phrases: mean-field model, Markovian switching diffusion, law of large number, McKean–Vlasov equation, maximum principle, non-Markov model, weak convergence. The paper is in final form and no version of it will be published elsewhere. DOI: 10.4064/bc122-14 [233] c  Instytut Matematyczny PAN, 2020