An Introduction to Markov Semigroups

Abstract

This paper contains the notes of a short course on Markov semigroups. The main aim was to give an introduction to some important properties as: ergodicity, irreducibility, strong Feller property, invariant measures, relevant to some important Markov semigroups arising in infinite dimensional analysis and in stochastic dynamical systems. We have considered in particular the heat semigroup in infinite dimensions, the Ornstein-Uhlenbeck semigroup, the transition semigroup of a one dimensional dynamical system perturbed by noise.

The lectures were designed for an audience having a basic knowledge of functional analysis and measure theory but not familiar with probability. An effort has been done in order to make the lectures as self-contained as possible. In this spirit, the first part was devoted to collect some basic properties of Gaussian measures in Hilbert spaces including the reproducing kernel and the Cameron-Martin formula, a tool that was systematically employed.

Several concepts and results contained in this course are taken from the the notes [3] and the monographs [4], [5], [6].