Paul Lévy  (1886 - 1971)

Paul  Lévy is one of the greatest mathematicians of our century and can be considered, with A N Kolmogorov ,  to be the forefather of the modern theory of stochastic processes.  Lévy was born into a family counting several mathematicians. His grandfather was a professor of mathematics and Paul's father, Lucien Lévy, was an examiner at the Ecole Polytechnique and wrote papers on geometry. Paul attended the Lycée Saint Louis in Paris where he achieved brilliant results, winning prizes not only in mathematics but also in Greek, chemistry and physics. He ranked first in the entrance examination to the Ecole Normale Supérieure and second for entry to the Ecole Polytechnique. He chose to attend the Ecole Polytechnique and in 1905, while still an undergraduate there, he published his first paper on semi-convergent series. In 1919 Lévy was asked to give three lectures at the Ecole Polytechnique on ``... notions of calculus of probabilities and the role of Gaussian law in the theory of errors.'' Taylor writes ``At that time there was no mathematical theory of probability --- only a
collection of small computational problems. Now it is a fully-fledged branch of mathematics using techniques from all branches of modern analysis and making its own contribution of ideas, problems, results and useful machinery to be applied elsewhere. If there is one person who has influenced the
establishment and growth of probability theory more than any other, that person must be Paul Lévy."

This was the beginning of his lifelong interest in probability theory, which lead to the discovery of a wealth of results, many of which have become today standard material for undergraduate and
graduate courses in probability theory. He made major contributions to the study of Gaussian variables and processes, the law of large numbers, the central limit theorem, stable laws,
infinitely divisible laws and pioneered the study of processes with independent and stationary increments, now known as Lévy processes. The book he wrote on this topic, Théorie de l'addition des variables aléatoires, has served as an inspiration to many researchers in probability and physics, where stable processes with independent increments have become known as Lévy flights. He pioneered the study of the properties of Brownian paths, in which he introduced
the notion of local time. These studies culminated in his classic book Processus stochastiques et mouvement Brownien.
 

Paul  Lévy was an extraordinarily productive mathematician: in parallel with and independently from  the Soviet mathematicians Kolmogorov and Khinchin, he discovered the major part of  what is known today as the theory of stochastic processes. Among his contributions where the study of  various properties of Brownian motion and the discovery of necessary and sufficient conditions in limit theorems for sums of independent random variables.  He proved the Central Limit Theorem using characteristic functions, independently from Lindeberg who proved the same theorem using convolution techniques.  He discovered the class of probability distributions known as "stable distributions" and proved the generalized version of the Central Limit Theorem for independent variables with infinite variance. He also introduced the notion of Brownian local time in the context of study of the properties of Brownian motion: today this concept plays a key role in the study of fine properties of diffusion processes. Michel Loeve gives a vivid description of Lévy's contributions: ``Paul Lévy was a painter in the probabilistic world. Like the very great painting geniuses, his palette was his own and his paintings transmuted forever our vision of reality... His three main, somewhat overlapping, periods were: the limit laws period, the great period of additive processes and of martingales painted in pathtime colours, and the Brownian pathfinder period."

Although he was a contemporary of Kolmogorov, Lévy did not adopt the axiomatic approach to probability. Joseph Doob writes of Lévy: ``[Paul Lévy] is not a formalist. It is typical of his approach to mathematics that he defines the random variables of a stochastic process successively rather than postulating a measure space and a family of functions on it with stated properties, that he is not
sympathetic with the delicate formalism that discriminates between the Markov and strong Markov properties, and that he rejects the idea that the axiom of choice is a separate axiom which need not
be accepted. He has always travelled an independent path, partly because he found it painful to follow the ideas of others.''

This attitude was in strong contrast to the mathematicians of his time, especially in France where the Bourbaki movement dominated the academic scene. Adding this to the fact that probability
theory was not regarded as a branch of mathematics by many of his contemporary mathematicians, one can see why his ideas did not receive in France the attention they deserved at the time of their
publication. P.A. Meyer writes:``Malgré son titre de professeur, malgré son élection à l'Institut ... Paul Lévy a été méconnu en France. Son oeuvre y était considérée avec condéscendance, et on
entendait fréquemment dire que `ce n'était pas un mathématicien."
Translation: Although he was a professor and a member of the Institut [i.e., the Academy of Sciences], Paul Lévy was not well recognized in France. His work was not highly considered and one frequently heard that `he was not a mathematician'.

However, Paul Lévy's work was progressively recognized at an international level. The first issue of Annals of Probability, an international journal of probability theory, was dedicated to his memory in 1973, two years after his death.

Some of  Lévy's works considered "esoteric" at the time he discovered them have turned out to be extremely useful in applications: a well-known example is the study of random variables and processes with infinite variance, known as stable processes, of which he initiated the study and which are now  applied in a variety of contexts,  ranging from  condensed matter physics to finance, to model random phenomena with high variability. Another example is the notion of local time of a stochastic process, introduced by P. Lévy in the study of Brownian motion, which has become a standard tool in the study of stochastic processes and is used  as a tool in the statistical estimation of diffusion processes in econometrics.

Paul Lévy's scientific work by Michel Loève. Appeared in : Annals of Probability, Vol 1, No 1 (1973).

Bibliography of Paul Lévy's works.

 

Text:  R. Cont