Langevin equation, Markovian process, multiplicative noise, Fokker-Planck equation, Schwinger-Keldysh formalism and much more... During this seminar, almost organised as a lecture, you will embark on a journey into the world of stochastic processes, with focus on their applications in the early Universe in the context of so-called stochastic inflation. We will begin by discussing how stochasticity emerges from the coarse-graining procedure, necessary to describe the effective dynamics of the largest cosmological scales that are affected by the quantum nature of the small-scale fluctuations. Special features of stochastic calculus and stochastic differential equations will be highlighted and applications to both situations of cosmological interest as well as to formal QFT results in curved  
spacetimes will be explained. After this very long introduction, I will summarise the three years of work that it took us to consistently extend the stochastic formalism to multifield models of inflation and solve the so-called inflationary   stochastic anomalies, intrinsically related to the discretisation ambiguity of stochastic differential equations and the quantum nature of cosmological perturbations.