Syllabus
1. The Markov property, transition functions and operators associated with them, construction of a process with a given transition function, shift operators and homogenous prpocesses.
2. Feller processes in locally compact spaces, their C0 semigroups, resolvents and generators, the Hille-Yosida theorem, properties of sample paths, strong Markov processes.
3. Jump processes, processes with independent increments, Lévy processes, the Lévy- Khinchin formula.
4. Diffusion processes: local characteristics, construction via stochastic differential equations, the Kolmogorov equation.
5. Elementary ergodic theory: invariant measures, transient and recurrent processes, basic results on existence of an invariant measure, (Krylov-Bogolyubov, Sunyach), strong Feller processes, uniqueness and stability of invariant measures.
Annotation
The very basic results of the continuous time Markov processes theory will be treated.