Markov Processes (Mukherjee)

Tuesday, 16:00-18:00, M6
Thursday, 16:00-18:00, M6

This course in the course overview
The tutorials in the course overview

Markov Processes are one of the most important stochastic processes.  Intuitively, it is characterized by the property that, conditional on the knowing the history of the process up until a given time, the distribution of the future depends only on the current position of the process. In terms of content, this lecture can be divided into two parts: 

In the first part, we will start out with the basic definitions and properties of Markov processes and provide their characterization via the so-called "martingale problem". These properties will also be demonstrated by a concrete tool for defining Markov processes in discrete spaces, namely Poisson point processes. We will provide techniques that allow to define general Markov processes with continuous state space. In particular, the Hille-Yosida theorem and its consequences will be discussed in detail. 

In the second part, we will be concerned with convergence of Markov processes. In particular, we will provide techniques that  allow one to determine at which speed (if at all) the law of a Markov process approaches its stationary distribution. Of particular interest will be cases where this speed is (sub-) exponential. In particular, we will discuss Lyapunov function techniques, and then move on to an elementary introduction to Malliavin calculus and to a proof of Hörmander’s famous "sums of squares" regularity theorem.