Exploring Fixed Points in Markov Chains: A Mathematical Perspective

Markov chains play a crucial role in modeling stochastic processes across mathematics, computer science and the applied sciences. This study investigates the relationship between fixed-point theory and Markov chains. The objectives are threefold: to analyze how stationary distributions in Markov chains can be interpreted as fixed points, to illustrate the connection between stochasticiterative methods and Page Rank type algorithms and fixed-point theorems and to provide concrete examples that demonstrate these links. The methodology adopted involves a literature-based review of fixed point theory and Markov chains, followedby mathematical formulation and worked examples. The results show that stationary distributions of Markov chains satisfy fixed-point equations and can be computed iteratively under contraction-type conditions. A numerical case study demonstrates convergence of an iterative scheme to the stationary distribution. These findings reinforce the role of fixed-point theory in stochastic analysis and provide tools for practical applications such as PageRank and Monte Carlo simulations