# Paul-André Meyer

### Born: 21 August 1934 in Boulogne-Billancourt, near Paris, France

Died: 30 January 2003 in Strasbourg, France

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**Paul-André Meyer**was not given the name Meyer when he was born but rather he took the family name of Meyerowitz. Nobody in his family had been interested in mathematics; his father was a trader who fled from France shortly after the outbreak of World War II. The war began on 1 September 1939 when German forces entered Poland. On the following day, Britain, France and several other countries, declared war on Germany but, over the following months, France was not involved in any fighting, but spent time trying to build defences to protect the country from an invasion by Germany. The war changed dramatically for France on 10 May 1940 when the German army crossed the Dutch and Belgium borders and, by June, France had surrendered and fighting had ended. The Meyerowitz family fled from France in 1940 and sailed to Argentina, settling in Buenos Aires. Paul-André attended a French school in Buenos Aires but, spending six years there from the age of six to the age of twelve, he became fluent in Spanish. At first he performed poorly in mathematics but was encouraged by his mother. Soon he was ahead of the rest of his class and was fortunate to have an excellent teacher who taught him to compute square roots and cube roots as well as giving him a liking for geometry. He returned to France with his family in 1946, after the war ended, but his love for the Spanish language lasted through his whole life.

Back in France, Paul-André entered the lycée Janson de Sailly in Paris. This school, one of the best-known lycées in central Paris, traditionally educated many of France's intellectuals. At first he was a boarder at the school but by the middle of his first year he was living with his family, attending the school as a day pupil. It was at this school that Paul-André fell in love with advanced mathematics, mainly due to an outstanding mathematics teacher M Heilbronn who was an expert mathematician having obtained a doctorate with a thesis

*Les équations aux dérivées partielles selon Jules Drach*Ⓣ. Paul-André, who spent four years in Heilbronn's classes, obtained his baccalaureate in 1952. He writes in [10] about influences at this time:-

Paul-André spent the following two years preparing to take the entrance examinations for the grandes écoles. In fact, in 1952 his father changed the family name from Meyerowitz to Meyer and from this time on the subject of this biography was known as Paul-André Meyer. He began to study advanced works [10]:-At that time, I spent much of Thursday afternoon at the Palais de la Découverte. Actually, I spent as much at the Guimet Museum, where the library fascinated me. ... The result of all this was that I went to buy the 'Cours de Mathématiques Spéciales' by A Decerf, which had the advantage of being thin. It was not an extraordinary book, but it was clear and I loved it.

He entered the École Normale Supérieure in 1954 still unsure whether to study physics or mathematics but after encountering rather poor physics teaching contrasted with outstanding lectures in mathematics by Henri Cartan he quickly decided to concentrate on mathematics. Surprisingly, he did not excel at first, in fact up to the time of his agrégation in 1957, he performed rather poorly. However, as soon as he began research under Jacques Deny (1916-), who had been a student of Henri Cartan, he showed himself to be an outstanding student. During his university years he converted to Roman Catholicism and in 1957 he married Genevieve who he met while studying at the Sorbonne. Remarkably she had, like Meyer, lived in Buenos Aires and had even studied at the same French school there although they had not met at that time. They had three children between 1959 and 1962 with a fourth child born in 1969. Meyer's research for his doctorate was put into context by Gustave Alfred Arthur Choquet (see [16]):-It was Raphaël Salem who lent me the first edition of the treatise on integration by Stanisław Saks, Henri Cartan who advised me to read the 'Fonctions de Variables Réelles'Ⓣby Bourbaki, and Laurent Schwartz who recommended me his book on distributions. I bought these books, they have served me later, but of course I could not understand them at that time. More importantly, the physicist P Grivet persuaded my parents that my project to only prepare for the competition of the rue d'Ulm was not crazy.

Gilbert Hunt, referred to in this quote, was a professor of mathematics at Princeton University and a leading authority in probability theory and mathematical analysis. Michel Loève (1907-1979), who had been born in Palestinian, was a French-American probabilist and a mathematical statistician who had studied in Paris under Paul Lévy and had been appointed as Professor of Mathematics at Berkeley. Meyer had spent time at Berkeley following Loève's visit to Paris and, while in the United States, he met Joseph Doob. He had already read Doob's classic textPaul Lévy, who was considered abroad as one of the founding fathers of modern probability, was little known in France. France, which had been the cradle of probability with Pascal, was running the risk of seeing the line of her famous theoreticians(Borel, Fréchet, Lévy ...)cut short. The Parisian mathematicians, being aware of this danger, invited Michel Loève, a former disciple of Paul Lévy, to come from Berkeley and spend a year in Paris, in order to sow the good seed by giving a course and a seminar ... By the end of the year, Meyer had assimilated the probabilistic techniques and was ready to begin his own research; this was the time when Gilbert Hunt was publishing, in the United States, two fundamental memoirs which were renewing at the same time potential theory and the theory of Markov processes by establishing a precise link, in a very general framework, between an important class of Markov processes and the class of kernels in potential theory which French probabilists had just been studying. Meyer, who was in close relation with these potentialists, and who was quite knowledgeable about convex functional analysis, was particularly well positioned to explore Hunt's theory. In1961, under the guidance of Jacques Deny, Meyer defended his thesis on multiplicative and additive functionals of Markov processes, which established him at once as a top researcher among probabilists and potentialists.

*Stochastic Processes*and these events all came together to give Meyer's research career an inspiring beginning. In 1962 and 1963 he published important papers on supermartingales (these can be thought of as processes which model unfair games). These papers contain remarkable deep results which are extensions of work done by Doob. He was appointed to the University of Strasbourg in 1964. In 1966 he published the book

*Probabilités et Potentiel*Ⓣ (an English translation entitled

*Probability and Potential*was published in the same year). Heinz Bauer writes in a review:-

Perhaps the best description of what book contains is given in Meyer's Introduction. We choose, as Heinz Bauer does, to quote from Meyer himself:-It is difficult to explain briefly what this book is. It is much easier to explain what it is not and what it does not pretend to be. It is not a textbook, nor a book on probability theory as a whole, and not at all a book on classical potential theory. ... The book consists of three parts[Elements of Probability Calculus; Martingale theory; Analytical tools in Potential theory]which are "connected by a pattern of analogies rather than by explicit logical relations." But ... the author succeeds perfectly in making clear to his reader the interplay between probabilistic and potential theoretic notions and procedures.

In 1967 the first of the seriesThe fundamental work of Doob and Hunt has shown, during the last ten years or so, that a certain form of potential theory(the study of kernels which satisfy the "complete maximum principle")and a certain branch of probability theory(the study of Markov semigroups and processes)in reality constitute a single theory. It is not a purely formal matter. Probabilistic methods have led to a much better understanding of certain fundamental ideas of potential theory(e.g. balayage, thinness, polar sets); they have above all led to a host of new results in potential theory. In turn, probability theory has received comparable mathematical advantages from this association, and a very important psychological benefit: a marked enlargement of its public, and the end of an old isolation of twenty or thirty years. Because of this isolation, a probabilistic background has been lacking in a number of mathematicians to whom probabilistic methods could be of great service. One can thus imagine the usefulness of a work, intended for researchers rather than students, which might put at their disposal simultaneously the elements of probability theory and some of its more advanced aspects. This need is the raison d'être of the present book.

*Séminaire de Probabilités*Ⓣ appeared. This series, published by Springer in their Lecture Notes in Mathematics series, contains remarkable work by Meyer. In the many volumes, published between 1967 and 1993, Meyer systematically discussed the main developments of the moment, mainly concentrating on his own work and work of his students and collaborators. However, he made very significant improvements when writing about the work of others, giving new proofs, simplifying arguments, extending results or simply explaining things more clearly than in the original papers. In many ways one can consider Meyer's work improving on that of others to be as important, and adding as much to the progress of mathematics, as his original research.

In collaboration with Claude Dellacherie, Meyer wrote five volumes between 1975 and 1992 entitled

*Probabilités et Potentiel*Ⓣ. This title was identical to that of Meyer's single-authored text of 1966 and indeed was considered by the authors as a revision and updating of that earlier text. M G Sur writes in a review of the first of these five volumes:-

In 1993 Meyer publishedMeyer's book 'Probabilités et potentiel'Ⓣ(1966)has earned recognition and enjoys great popularity among specialists. However, since its publication a considerable amount of new material has been added to the theory and a new point of view towards some notions and statements in the book has been developed. For this very reason the authors decided to revise the earlier monograph.

*Quantum probability for probabilists*. His approach, writes David Applebaum, is:-

Stéphane Attal writes in [1] (and [2]) about Meyer as a teacher:-... deeply rooted in the language and perspectives of classical probability theory, as one would expect from a set of notes based on lectures aimed at an audience of classical probabilists with little background in operator algebras. Here the aim is to persuade the audience that quantum probability is a fascinating subject which they would be well advised to study. The author's enthusiasm is apparent on every page; he has read an enormous wealth of literature both within and around the subject and these notes consequently constitute an impressive survey of a field which is still relatively new and rapidly evolving.

He was a very modest man who had many interests outside mathematics (see [1] and [2]):-I thought about this seminar held on Tuesday mornings. Every week we found ourselves in front of the board on the fourth floor of the tower of the Institute de Recherche mathématique Avancée. We were sometimes ten people, sometimes three people, and we listened to Meyer. He came each time with a new typed text: he had proved a new result, or he had read and completely rewritten an article by someone else, or he showed us the10^{th}chapter of his course on von Neumann algebras, Lie algebras or quantum field theory. I thought about the interest and enthusiasm he constantly showed for all that I did.

Meyer received many honours for his mathematical contributions including the Peccot Prize, the Maurice Audin Prize and the Ampère Prize. He was elected as a corresponding member of the Academy of Sciences. Following his death the Institut de Recherche Mathématique Avancée at Strasbourg created the Meyer Prize which is awarded annually to an outstanding young probabilist working in the field of stochastic processes. The prize was awarded for the first time in 2004. The first recipient was Thomas Duquesne.Meyer's four main passions outside of science, I believe, were his family, music, languages and literature. Paul-André Meyer was indeed very close to his family, which for him was paramount. He writes: "I have always given my family a certain priority over my work, and I felt rewarded when one day one of my daughters told me: 'When we were little, we were a little ashamed, because we thought you did nothing!'. Paul-André Meyer was a great lover of music and a great musician. He himself played the violin, viola and especially the flute. His whole family grew up with this passion for music(two girls became professional musicians). In the evening, the "little family Meyer" played at home sonatas, trios, and quartets with piano.

**Article by:** *J J O'Connor* and *E F Robertson*

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