We present a convenient joint generalization of mixing and the local version of the central limit theorem (MLLT) for probability preserving dynamical systems. We verify that MLLT holds for several examples of hyperbolic systems by reviewing old results for maps and presenting new results for flows. Then we discuss applications such as proving various mixing properties of infinite measure preserving systems. Based on joint work with Dmitry Dolgopyat.[-]

We consider stochastic models of scalable biological reaction networks in the form of continuous time pure jump Markov processes. The study of the mean field behavior of such Markov processes is a classical topic, with fundamental results going back to Kurtz, Athreya, Ney, Pemantle, etc. However, there are still questions that are not completely settled even in the case of linear reaction rates. We study two such questions. First is to characterize all possible rescaled limits for linear reaction networks. We show that there are three possibilities: a deterministic limit point, a random limit point and a random limit torus. Second is to study the mean field behavior upon the depletion of one of the materials. This is a joint work with Lai-Sang Young.[-]