An Intuitive Introduction to Operator Semi-Groups

This article aims to give an intuitive introduction to operator semi-groups and their generators from a probabilistic perspective. By ‘intuitive’ it is meant that the article relies mainly on heuristics and analogies to make its points. For example the definition of the generator is motivated from an analogy to the Cauchy functional equation g(t + s) = g(t)g(s) for real-valued functions. No proofs are given throughout the text. They can be found in either Kallenberg [1997], Yosida [1980] or Butzer and Berens [1967]. Other recommended literature is the article Jacob and Schilling [2001], on which section 5 is based.