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.
See also what Laurent Schwartz writes in his book "Un Mathématicien aux prises avec le siècle" about the relations between Paul Lévy and the Bourbaki group, http://books.google.fr/books?id=Eqc0cyFR0AEC&lpg=PA173&ots=qx9f0eMmcd&dq=paul%20levy%20bourbaki&hl=fr&pg=PA173#v=onepage&q=paul%20levy%20bourbaki&f=false