From Stochastic N Particle Systems to Deterministic Differential Equations - with Applications to Econophysics and Averaging Dynamics

The mechanisms behind the emergence of collective behaviors arising from physics, biology, economics and many other related fields have drawn a lot of attention among the applied math community in the last few decades. Broadly speaking, collective behaviors in natural, life and social sciences are all modelled by interacting particle systems, in which a bulk of N particles are engaging in some simple binary pairwise interactions. In this dissertation, some prototypical interacting particle systems having applications in econophysics and statistical averaging dynamics are investigated. It is also emphasized that there is an increasing tendency among the applied math community to apply tools or concepts for studying many particle systems to the (rigorous) investigation of artificial (deep) neural networks.