Paul Lévy (mathematician)

French mathematician

For other uses, see Paul Lévy (disambiguation).

Paul Lévy

Paul Pierre Lévy

Born| (1886-09-15)15 September 1886

Paris, France

Died| 15 December 1971(1971-12-15) (aged 85)

Paris, France

Nationality| French
Alma mater| University of Paris
Known for| Additive process
Brownian excursion
Concentration of measure
Martingale (probability theory)
Universal chord theorem
Lévy alpha-stable distribution
Lévy's arcsine law
Lévy C curve
Lévy's constant
Lévy characterisation
Lévy's continuity theorem
Lévy distribution
Lévy flight
Lévy's local time
Lévy measure
Lévy's modulus of continuity theorem
Lévy process
Lévy's zero–one law
Lévy–Khintchine representation
Lévy–Prokhorov metric
Lévy–Steinitz theorem
Lindeberg–Lévy CLT
Wiener–Lévy theorem
Awards| Emile Picard Medal of the French Academy of Sciences (1953)[1]
Scientific career
Fields| Mathematics
Institutions| École Polytechnique
École des Mines
Doctoral advisor| Jacques Hadamard
Vito Volterra
Doctoral students| Wolfgang Doeblin
Michel Loève
Benoît Mandelbrot
Georges Matheron
Influenced| Kiyosi Ito
Benoit Mandelbrot
Marc Yor

Paul Pierre Lévy (15 September 1886 – 15 December 1971)[2] was a French mathematician who was active especially in probability theory, 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 Lévy C curve are named after him.

## Biography[edit]

Lévy was born in Paris to a Jewish family which already included several mathematicians.[3] His father Lucien Lévy was an examiner at the École Polytechnique. 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.[2]

During World War I 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 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.[2]

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 Ya. Khinchine, the notion of infinitely divisible law and derived their characterization through the Lévy–Khintchine representation.

His 1948 monograph on Brownian motion, 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 and honorary membership at the London Mathematical Society.

His daughter Marie-Hélène Schwartz and son-in-law Laurent Schwartz were also notable mathematicians.[4]

## Works[edit]

* 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[edit]

* 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[edit]

1. ^ Médaille Emile Picard
2. ^ a b c O'Connor, John J.; Robertson, Edmund F., "Paul Lévy (mathematician)", MacTutor History of Mathematics Archive, University of St Andrews
3. ^ Barbut, Marc; Locker, Bernard; Mazliak, Laurent (2013). Paul Lévy and Maurice Fréchet: 50 Years of Correspondence in 107 Letters. p. xii. ISBN 978-1-4471-5618-5.
4. ^ Kosmann-Schwarzbach, Yvette (2015), "Women mathematicians in France in the mid-twentieth century", BSHM Bulletin: Journal of the British Society for the History of Mathematics, 30 (3): 227–242, arXiv:1502.07597, doi:10.1080/17498430.2014.976804, S2CID 119148294.

## External links[edit]

* Rama Cont: Paul Lévy: a biography
* Gérard P. Michon: Paul Lévy and Functional Analysis
* Paul Lévy at the Mathematics Genealogy Project
* O'Connor, John J.; Robertson, Edmund F., "Paul Lévy (mathematician)", MacTutor History of Mathematics Archive, University of St Andrews

* v
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Fractals

Characteristics|

* Fractal dimensions
* Assouad
* Box-counting
* Higuchi
* Correlation
* Hausdorff
* Packing
* Topological
* Recursion
* Self-similarity

Iterated function
system|

* Barnsley fern
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* Koch snowflake
* Menger sponge
* Sierpinski carpet
* Sierpinski triangle
* Apollonian gasket
* Fibonacci word
* Space-filling curve
* Blancmange curve
* De Rham curve
* Minkowski
* Dragon curve
* Hilbert curve
* Koch curve
* Lévy C curve
* Moore curve
* Peano curve
* Sierpiński curve
* Z-order curve
* String
* T-square
* n-flake
* Vicsek fractal
* Hexaflake
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Strange attractor|

* Multifractal system

L-system|

* Fractal canopy
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Escape-time
fractals|

* Burning Ship fractal
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* Misiurewicz point
* Multibrot set
* Newton fractal
* Tricorn
* Mandelbox
* Mandelbulb

Rendering techniques|

* Buddhabrot
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Random fractals|

* Brownian motion
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* Lévy flight
* Percolation theory
* Self-avoiding walk

People|

* Michael Barnsley
* Georg Cantor
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* Desmond Paul Henry
* Gaston Julia
* Helge von Koch
* Paul Lévy
* Aleksandr Lyapunov
* Benoit Mandelbrot
* Hamid Naderi Yeganeh
* Lewis Fry Richardson
* Wacław Sierpiński

Other|

* "How Long Is the Coast of Britain?"
* Coastline paradox
* Fractal art
* List of fractals by Hausdorff dimension
* The Fractal Geometry of Nature (1982 book)
* The Beauty of Fractals (1986 book)
* Chaos: Making a New Science (1987 book)
* Kaleidoscope
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