Paul Malliavin (; September 10, 1925 – June 3, 2010) was a French
mathematician
A mathematician is someone who uses an extensive knowledge of mathematics in their work, typically to solve mathematical problems. Mathematicians are concerned with numbers, data, quantity, mathematical structure, structure, space, Mathematica ...
who made important contributions to
harmonic analysis
Harmonic analysis is a branch of mathematics concerned with investigating the connections between a function and its representation in frequency. The frequency representation is found by using the Fourier transform for functions on unbounded do ...
and
stochastic analysis
Stochastic calculus is a branch of mathematics that operates on stochastic processes. It allows a consistent theory of integration to be defined for integrals of stochastic processes with respect to stochastic processes. This field was created an ...
.
He is known for the
Malliavin calculus
In probability theory and related fields, Malliavin calculus is a set of mathematical techniques and ideas that extend the mathematical field of calculus of variations from deterministic functions to stochastic processes. In particular, it allows ...
, an infinite dimensional calculus for functionals on the
Wiener space and his probabilistic proof of
Hörmander's theorem.
He was
Professor
Professor (commonly abbreviated as Prof.) is an Academy, academic rank at university, universities and other tertiary education, post-secondary education and research institutions in most countries. Literally, ''professor'' derives from Latin ...
at the
Pierre and Marie Curie University
Pierre and Marie Curie University ( , UPMC), also known as Paris VI, was a public research university in Paris, France, from 1971 to 2017. The university was located on the Jussieu Campus in the Latin Quarter of the 5th arrondissement of Paris, ...
and a member of 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 ...
from 1979 to 2010.
Personal life
Malliavin was the son of René Malliavin, also known as ''Michel Dacier'', a political writer and journalist, and Madeleine Delavenne, a physician. On 27 April 1965 he married
Marie-Paule Brameret, who was also a mathematician and with whom he published several mathematical papers.
They had two children.
Scientific contributions
Malliavin's early work was in
harmonic analysis
Harmonic analysis is a branch of mathematics concerned with investigating the connections between a function and its representation in frequency. The frequency representation is found by using the Fourier transform for functions on unbounded do ...
, where he derived important results on the spectral synthesis problem, providing definitive answers to fundamental questions in this field, including a complete characterization of 'band-limited' functions whose Fourier transform has compact support, known as the
Beurling-Malliavin theorem.
In
stochastic analysis
Stochastic calculus is a branch of mathematics that operates on stochastic processes. It allows a consistent theory of integration to be defined for integrals of stochastic processes with respect to stochastic processes. This field was created an ...
, Malliavin is known for his work on the stochastic calculus of variation, now known as the
Malliavin calculus
In probability theory and related fields, Malliavin calculus is a set of mathematical techniques and ideas that extend the mathematical field of calculus of variations from deterministic functions to stochastic processes. In particular, it allows ...
, a mathematical theory which has found many applications in
Monte Carlo simulation
Monte Carlo methods, or Monte Carlo experiments, are a broad class of computational algorithms that rely on repeated random sampling to obtain numerical results. The underlying concept is to use randomness to solve problems that might be det ...
and
mathematical finance
Mathematical finance, also known as quantitative finance and financial mathematics, is a field of applied mathematics, concerned with mathematical modeling in the financial field.
In general, there exist two separate branches of finance that req ...
.
As stated by
Stroock and
Yor: "Like Norbert Wiener, Paul Malliavin came to probability theory from harmonic analysis, and, like Wiener, his analytic origins were apparent in everything he did there."
Malliavin introduced a differential operator on
Wiener space, now called the
Malliavin derivative
In mathematics, the Malliavin derivative is a notion of derivative in the Malliavin calculus. Intuitively, it is the notion of derivative appropriate to paths in classical Wiener space, which are "usually" not differentiable in the usual sense.
...
, and derived an
integration by parts
In calculus, and more generally in mathematical analysis, integration by parts or partial integration is a process that finds the integral of a product of functions in terms of the integral of the product of their derivative and antiderivati ...
formula for Wiener functionals. Using this integration by parts formula, Malliavin initiated a probabilistic approach to
Hörmander's theorem for hypo-elliptic operators and gave a condition for the existence of smooth densities for Wiener functionals in terms of their
Malliavin covariance matrix.
Selected publications
*
La quasi-analyticité généralisée sur un intervalle borné', Annales scientifiques de l’École Normale Supérieure 3
e série 72, 1955, pp. 93–110
*
Impossibilité de la synthèse spectrale sur les groupes abéliens non compacts', Publications Mathématiques de l’IHÉS 2, 1959, pp. 61–68
*
Calcul symbolique et sous-algèbres de L1(G)', Bulletin de la Société Mathématique de France 87, 1959, pp. 181–186,
suite', pp. 187–190
* with
Lee A. Rubel:
On small entire functions of exponential type with given zeros', Bulletin de la Société Mathématique de France 89, 1961, pp. 175–206
*
Spectre des fonctions moyenne-périodiques. Totalité d’une suite d’exponentielles sur un segment', Séminaire Lelong. Analyse 3 Exposé No. 11, 1961
*
Un théorème taubérien relié aux estimations de valeurs propres', Séminaire Jean Leray, 1962–1963, pp. 224–231
*
*
Géométrie riemannienne stochastique', Séminaire Jean Leray 2 Exposé No. 1, 1973–1974
*
* ''Geometrie differentielle stochastique'', Presses de l’Universite de Montreal, 1978
* with Hélène Airault, Leslie Kay, Gérard Letac: ''Integration and Probability'', Springer, 1995
* with H. Airault:
Some heat operators on P(Rd)', Annales mathématiques Blaise Pascal 3 no. 1, 1996, pp. 1–11
* ''Stochastic Analysis'', Springer, 1997
*
References
External links
*
*
{{DEFAULTSORT:Malliavin, Paul
1925 births
2010 deaths
20th-century French mathematicians
21st-century French mathematicians
French probability theorists
University of Paris alumni
Malliavin calculus
Members of the French Academy of Sciences
Members of the Royal Swedish Academy of Sciences