On the Propagation of Chaos for Recombination Models - Ong, Lu Kevin (Science Diliman: A Journal of Pure and Applied Sciences, 2021)
We consider binary interactions in an N-particle system. In particular, we use probability distributions known as recombination models to describe these interactions. Chaos propagates when the stochastic independence of two random particles in a particle system persists in time, as the number of particles tends to infinity. The concept of propagation of chaos was first introduced by Kac in connection with the Boltzmann equation, while modeling binary collisions in a gas. We obtain a development of Kac’s program in the framework of recombination models. Specifically, our aim is to prove the relevant propagation of chaos phenomenon for our particle system. We first show that the solution for the master equation of our time-continuous process converges. Then, we use this solution together with the concepts of marginal measure and chaos to prove our desired result. Our main theorem for this study says that if a sequence of measures on our defined particle system is chaotic, then the resulting sequence of measures that had undergone the recombination process is also chaotic. This implies that the study of one particle after recombination gives information on the behavior of a group of particles in our particle system.
Keywords: particle system, recombination, master equation, propagation of chaos