Title: Finite time distribution and slow mixing for stochastic reaction network models.
Abstract: Continuous-time Markov chains on infinite positive integer grids with polynomial rates are often used in modeling chemical reaction networks, and it is known that stochastically modeled reaction networks that are complex balanced admit a stationary distribution that is a product of Poisson distributions. In this talk we will consider two related questions about finite time distributions of chemical reaction networks. First supposing that the initial distribution of a stochastically modeled reaction network is a product of Poisson, under what conditions will the distribution remain a product of Poisson for all time? By drawing inspiration from Crispin Gardiner’s “Poisson representation”, we provide a necessary and sufficient condition for such a product-form distribution to hold for all time. Second, what is the rate of convergence of the distribution of the process to the stationary distribution? We will introduce and focus on boundary-induced slow mixing chain as the slow mixing is caused due to the trapping effect taking place around the boundary of the state space. Moreover, to establish sharp mixing time estimation, we establish the upper bound of the first hitting times that are closely linked to the mixing times.
