Markov Chains on Metric Spaces: A Short Course
Markov Chains on Metric Spaces: A Short Course
English | 2022 | ISBN: 3031118219 | 230 Pages | PDF EPUB (True) | 10.3 MB
English | 2022 | ISBN: 3031118219 | 230 Pages | PDF EPUB (True) | 10.3 MB
The focus is on the ergodic properties of such chains, i.e., on their long-term statistical behaviour. Among the main topics are existence and uniqueness of invariant probability measures, irreducibility, recurrence, regularizing properties for Markov kernels, and convergence to equilibrium. These concepts are investigated with tools such as Lyapunov functions, petite and small sets, Doeblin and accessible points, coupling, as well as key notions from classical ergodic theory. The theory is illustrated through several recurring classes of examples, e.g., random contractions, randomly switched vector fields, and stochastic differential equations, the latter providing a bridge to continuous-time Markov processes.