CC2601: From Random Walks to Heat Kernels: An Introduction to Markov Chains
Speaker: Aritrabha Majumdar (M.Math 1st year)
Abstract: Markov chains provide a fundamental framework for describing stochastic systems whose future evolution depends on the present state but not on the past. This talk develops an elementary introduction to both discrete-time and continuous-time Markov chains, with particular emphasis on the connections between the two settings. We begin with transition matrices, the Chapman–Kolmogorov equations, simple random walks, stationary distributions, and convergence to equilibrium in finite state spaces. We then pass to continuous time by introducing exponential holding times and jump rates, leading naturally to the infinitesimal generator and the Kolmogorov equations. Finally, we interpret the transition kernel as a matrix exponential, connecting Markov-chain evolution with semigroup theory and the heat equation. The goal is to develop a unified picture of Markov chains in which transition matrices, generators, equilibrium distributions, and heat kernels arise as different descriptions of the same underlying stochastic evolution.
Pre-requisites: Idea of conditional probability, expectation, variance, discrete and continuous random variable (and their pmf and pdf), linear algebra, notion of matrix exponential, notion of convergence in metric spaces.
Video: (Recorded, link will be given shortly after uploading to YouTube)
Date and Time: Saturday (22nd August), 12 noon to 1:30 p.m.