Paul Pierre Lévy (15 September 1886 – 15 December 1971)
[
] was a
French mathematician who was active especially in
probability theory
Probability theory or probability calculus is the branch of mathematics concerned with probability. Although there are several different probability interpretations, probability theory treats the concept in a rigorous mathematical manner by expre ...
, introducing fundamental concepts such as
local time,
stable distributions and
characteristic functions.
Lévy processes,
Lévy flights,
Lévy measures,
Lévy's constant, the
Lévy distribution, the
Lévy area, the
Lévy arcsine law, and the
fractal
In mathematics, a fractal is a Shape, geometric shape containing detailed structure at arbitrarily small scales, usually having a fractal dimension strictly exceeding the topological dimension. Many fractals appear similar at various scale ...
Lévy C curve are named after him.
Biography
Lévy was born in
Paris
Paris () is the Capital city, capital and List of communes in France with over 20,000 inhabitants, largest city of France. With an estimated population of 2,048,472 residents in January 2025 in an area of more than , Paris is the List of ci ...
to a
Jew
Jews (, , ), or the Jewish people, are an ethnoreligious group and nation, originating from the Israelites of ancient Israel and Judah. They also traditionally adhere to Judaism. Jewish ethnicity, religion, and community are highly inte ...
ish family which already included several mathematicians.
His father Lucien Lévy was an examiner at the
École Polytechnique
(, ; also known as Polytechnique or l'X ) is a ''grande école'' located in Palaiseau, France. It specializes in science and engineering and is a founding member of the Polytechnic Institute of Paris.
The school was founded in 1794 by mat ...
. Lévy attended the École Polytechnique and published his first paper in 1905, at the age of nineteen, while still an undergraduate, in which he introduced the
Lévy–Steinitz theorem. His teacher and advisor was
Jacques Hadamard. After graduation, he spent a year in military service and then studied for three years at the
École des Mines, where he became a professor in 1913.
During
World War I
World War I or the First World War (28 July 1914 – 11 November 1918), also known as the Great War, was a World war, global conflict between two coalitions: the Allies of World War I, Allies (or Entente) and the Central Powers. Fighting to ...
Lévy conducted mathematical analysis work for the French Artillery. In 1920 he was appointed Professor of Analysis at the École Polytechnique, where his students included
Benoît Mandelbrot
Benoit B. Mandelbrot (20 November 1924 – 14 October 2010) was a Polish-born French-American mathematician and polymath with broad interests in the practical sciences, especially regarding what he labeled as "the art of #Fractals and the ...
and
Georges Matheron. He remained at the École Polytechnique until his retirement in 1959, with a gap during World War II after his 1940 firing because of the
Vichy anti-Jewish legislation.
Lévy made many fundamental contributions to probability theory and the nascent theory of stochastic processes.
He introduced the notion of 'stable distribution' which share the property of stability under addition of independent variables and proved a general version of the Central Limit theorem, recorded in his 1937 book ''Théorie de l'addition des variables aléatoires'', using the notion of characteristic function. He also introduced, independently from
Aleksandr Khinchin, the notion of infinitely divisible law and derived their characterization through the
Lévy–Khintchine representation.
His 1948 monograph on
Brownian motion
Brownian motion is the random motion of particles suspended in a medium (a liquid or a gas). The traditional mathematical formulation of Brownian motion is that of the Wiener process, which is often called Brownian motion, even in mathematical ...
, ''Processus stochastiques et mouvement brownien'', contains a wealth of new concepts and results, including the
Lévy area, the
Lévy arcsine law,
the
local time of a Brownian path, and many other results.
Lévy received a number of honours, including membership at the
French Academy of Sciences
The French Academy of Sciences (, ) is a learned society, founded in 1666 by Louis XIV at the suggestion of Jean-Baptiste Colbert, to encourage and protect the spirit of French Scientific method, scientific research. It was at the forefron ...
and honorary membership at the
London Mathematical Society
The London Mathematical Society (LMS) is one of the United Kingdom's Learned society, learned societies for mathematics (the others being the Royal Statistical Society (RSS), the Institute of Mathematics and its Applications (IMA), the Edinburgh ...
.
His daughter
Marie-Hélène Schwartz and son-in-law
Laurent Schwartz were also notable mathematicians.
[.]
Works
* 1922 – ''Lecons d'analyse Fonctionnelle''
* 1925 – ''Calcul des probabilités''
* 1937 – ''Théorie de l'addition des variables aléatoires''
* 1948 – ''Processus stochastiques et mouvement brownien''
* 1954 – ''Le mouvement brownien''
See also
*
Cramér's decomposition theorem
*
Lévy distribution
*
Lévy metric
*
Lévy's modulus of continuity
*
Lévy–Prokhorov metric
*
Lévy's continuity theorem
*
Lévy's zero-one law
*
Concentration of measure
*
Lévy process
*
Lévy–Khintchine representation
*
Lévy–Itô decomposition
*
Lévy flight
*
local time
*
Isoperimetric inequality on a sphere
*
Lévy's characterisation of Brownian motion
References
External links
* Rama Cont:
Paul Lévy: a biography* Gérard P. Michon:
*
*
{{DEFAULTSORT:Levy, Paul Pierre
Jewish French scientists
École Polytechnique alumni
Mines Paris - PSL alumni
Corps des mines
1886 births
1971 deaths
20th-century French mathematicians
19th-century French Jews
French probability theorists
Members of the French Academy of Sciences
University of Paris alumni