Paul Lévy, analyst and probabilist

Paul Lévy is one of the founders of the theory of processes. In 1935 he introduced the idea of martingale, a process representing a fair game of chance. He also completed the work of de Finetti and Kolmogorov by giving the form (usually called the Lévy-Khinchin formula) of the most general infinitely divisible random variable. In my own analysis in 1969 and later in 2001 of this problem, I express the generating function of the process in terms of cocycles of the abelian group R . I show that de Finetti's examples come from coboundaries, and Kolgogorov's come from global cocycles of the Lie algebra of R . The general Lévy process comes from a general cocycle; as such it is a local but not necessarily global cocycle of the algebra.