Markov chain

A Markov chain is a stochastic process in which the probability of transitioning to any particular state depends only on the current state, not on the sequence of events that preceded it. This memoryless property — the Markov property — makes Markov chains analytically tractable while still capable of modeling complex temporal dynamics. They are used across domains: from modeling random walks and PageRank algorithms to describing chemical reactions, population genetics, and queueing systems. The stationary distribution of a Markov chain represents its long-term statistical equilibrium, analogous to the attractor of a deterministic dynamical system. The theory connects to ergodic theory through the study of mixing and recurrence properties.