Singularity of Stationary Measures in Markov Chains

Explore the intricacies of stationary measures in Markov chains through this lecture delivered at the Erwin Schrödinger International Institute for Mathematics and Physics. Delve into the complexities of compound chains formed by convex combinations or products of Markov operators. Examine the question of whether equivalent stationary measures for original operators guarantee an equivalent stationary measure for the compound operator. Discover a negative answer to this query through examples utilizing boundary processes associated with random walks on the modular group PSL(2,Z). Investigate how these examples demonstrate the singularity of harmonic measures for convex combinations or convolutions, despite the equivalence of original harmonic measures. Learn about the connection between these boundary measures and classical constructions by Minkowski and Denjoy. Based on joint work with Behrang Forghani, this lecture offers a deep dive into advanced concepts in probability theory and dynamical systems.