The course is divided into three parts.

First, we start with the example of the Poisson process to introduce several notions concerning continuous-time processes, including the characterization of its law with its finite-dimensional marginals, or the construction of a Markov process via a transition semigroup. We also introduce the notion of conditional distribution and review the weak and strong Markov properties.

Second, we study continuous-time Markov chains. We construct and characterize these processes (possibly up to the time of explosion), and make the connection between the transition semigroup and the matrix of intensities. We provide in particular the Kolmogorov equations and study the invariant measures and the long-time behaviour of the process.

Third, we study Brownian motion and several related processes. We start with the construction of a continuous version of the process and the study of its invariance properties and the regularity properties of its trajectories. We then dig more into the Markov and martingale properties associated with Brownian motion, and see several links with random walks, including Donsker invariance properties. Finally, we look at harmonic functions and the Dirichlet problem via Brownian motion, and if time permits continue with a glimpse of potential theory.