The lecture notes collect in an organic and systematic way the teaching materials used in the Probability courses I have taught, covering a broad range of topics: from fundamental concepts and basic tools to more advanced subjects, such as stochastic calculus and its applications. The aim is to provide a complete and coherent reference, useful both for consolidating the knowledge acquired in class and for independent study, by offering examples, rigorous definitions, and theoretical insights that facilitate the understanding of probabilistic principles and their many applications.
Version history:
- 2026-07-21: Current version. Added an appendix reviewing basic topology.
- 2026-07-11: a brand new chapter on McKean-Vlasov equations and nonlinear Markov processes, together with several minor corrections throughout the notes.
- 2026-05-28: a brand new chapter on Markov chains. New examples and minor corrections.
- 2025-12-19: as of this date, only the English edition will be updated. This text has been translated from the original Italian edition by the author with the assistance of AI.