Stochastic Mean Field Games within Complex Systems and Interacting Particle Frameworks

This dissertation aims to analyze several advanced topics in stochastic analysis. We begin by examining diffusion approximations for transport equations with dissipative drifts, addressing both time-dependent and time-independent cases. We establish results on the existence and uniqueness of solutions for perturbed stochastic differential equations (SDEs) and demonstrate the convergence of these solutions to those of the unperturbed equations. These findings are then applied to the Cauchy problem and transport equations. Next, we explore mean field games (MFGs) with state dynamics described by stochastic differential equations affected by both idiosyncratic and common noise, and subject to the constraint that the terminal state variable belongs to a nonempty convex closed set. We derive the stochastic maximum principle (SMP) and introduce a suitable forward-backward stochastic differential equation (FBSDE) of conditional McKean-Vlasov type. Additionally, we apply our findings to MFGs of optimal investment. Finally, we investigate classical continuous systems characterized by singular velocity distributions, where the corresponding Radon measures are defined over the entire space with infinite mass. These singular distributions are used to model particle velocities in systems where traditional velocity distributions do not apply. As a result, the particle positions in such systems no longer conform to conventional configurations in physical space. This necessitates the development of novel analytical tools to understand the underlying models. To address this, we introduce a new conceptual framework that redefines particle configurations in phase space, where each particle is represented by its spatial position and a velocity vector. The key idea is the construction of the Plato space, which is designed to represent idealized particle configurations where the total velocity remains bounded within any compact subset of phase space. This space serves as a crucial bridge to the space of vector-valued discrete Radon measures, where each measure captures the velocity distribution over the entire system. Given the inherent complexity of analyzing infinite-dimensional spaces, we tackle the problem by reformulating it onto a finite-dimensional configuration space. This is achieved by decomposing the infinite space into smaller, more manageable components. A central tool in this reformulation is the K-transform, which is pivotal in enabling harmonic analysis on the space. The K-transform allows us to represent the system in terms of components that are more amenable to analysis, thus simplifying the study of the system’s dynamics. Furthermore, we extend previous results in the study of correlation functions by developing correlation measures tailored for these vector-valued Radon measures. Moreover, we define Gibbs measures on the cone of vector-valued measures and prove existence of such measures. We define Gibbs measures as solutions to the Dobrushin-Lanford-Ruelle equation, focusing on the subset of tempered Gibbs measures. To prove the existence of the Gibbs measure, we show that the subset of tempered Gibbs measures is non-empty and relatively compact.