Seminar: Functional Inequalities for Markov Semigroups of Finite Markov Chains

2021. 8/27 FridayTime: Aug 27, 10:00-11:00 (Taipei Time)
Speaker: Jhih­-Hong Lyu (Ph.D. student in NTU Math)
Title: Functional Inequalities for Markov Semigroups of Finite Markov Chains
The classical theory of Markov chains studied fixed chains. The traditional goal was to estimate the rate of convergence to stationary state
as t goes to infinity. In the past several decades, as interest in chains with large state spaces has increased, a different asymptotic analysis
has emerged. Certain target distance to the stationary distribution is prescribed, and the number of steps required to reach this target distance
is called the mixing time of the chain. The new goal is to understand how the mixing time grows as the size of the statespace increases.

On the other hand, the evolution of a Markov chain on finite state space can be captured by one-parameter semigroup. For semigroups, we can
use functional inequalities to bound the mixing time. One common approach is to define general notion of Ricci curvature bounds for discrete
spaces, and use the geometric flow to set up the desired functional inequalities. In this talk, we will present a paper written in 2012 by
Matthias Erbar and Jan Maas. A new notion of Ricci curvature that applies to Markov chains on discrete spaces is introduced. The results
in this paper can be regarded as the discrete analogues of the fundamental results by Bakry-Emery and Otto-Villani.
Mr. Jhih-Hong Lyu is a Ph.D. student in Math Department, National Taiwan University, under the supervision of Dr. Tai-Chia Lin.