An Intuitive Introduction to Operator Semi-Groups Martin Keller-Ressel January 17, 2006 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 Kallen- berg [1997], Yosida [1980] or Butzer and Berens [1967]. Other recommended literature is the article Jacob and Schilling [2001], on which section 5 is based. 1 Operator semi-groups and their generator The theory of operator semi-groups originates from the study of the equation T (t + s)= T (t)T (s) T (0) = I (1.1) where T (t) is an operator-valued function taking values in the set of bounded linear operators acting on a suitable function space. This problem was indepen- dently studied by Hille and Yosida around 1948. They introduced the notion of a operator semi-group, which is defined as follows: Let X be a Banach-space. Then a family {T t } t≥0 of bounded linear Operator Semi- group operators ∈ L(X , X ) is called an operator semi-group if it satisfies T t+s = T t T s T 0 = I A semi-group is called equi-bounded if ‖T t ‖≤ M for some constant M and all t ≥ 0 and is called a contraction semi-group if ‖T t ‖≤ 1 for all t ≥ 0. From a stochastic point of view operator semi-groups originate from the Markov process study of Markov processes. Recall the following definition: Given a stochastic base (Ω, F , F,P ) a process X is called a Markov process with respect to F if F t and σ {X s ; s ≥ t} are conditionally independent given X t for all t ≥ 0. 1