Using kinetic theory to study econophysics

In this talk, we discuss money exchange models where individuals randomly exchange dollars with each others  according to (simple) rules. The goal is to link the individual behavior and the wealth distribution of an entire population. Mathematically, starting from a simple money exchange model, we study the limit of the dynamics in two regimes: i) letting the number of players goes to infinity (i.e. N→+∞), ii) studying the long time behavior (i.e. t→+∞). Our strategy is to first show that the dynamics becomes deterministic as N→+∞  proving the so-called propagation of chaos (i.e. a refined version of the "law of large number"). Then, we use energy method to study the long time behavior of the derived deterministic dynamics. In many cases, we are able to show that the dynamics converge to an equilibrium distribution. But there is also an interesting scenario where the dynamics only converge weakly since all the money is "lost" as t→∞. This is a joint work with Fei Cao.