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, structure, space, models, and change. History On ...

who made important contributions to

harmonic analysis Harmonic analysis is a branch of mathematics concerned with the representation of Function (mathematics), functions or signals as the Superposition principle, superposition of basic waves, and the study of and generalization of the notions of Fo ...

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 In mathematics, classical Wiener space is the collection of all Continuous_function#Continuous_functions_between_metric_spaces, continuous functions on a given domain of a function, domain (usually a subinterval of the real line), taking values i ...

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 post-secondary education and research institutions in most countries. Literally, ''professor'' derives from Latin as a "person who pr ...

at the

Pierre and Marie Curie University Pierre and Marie Curie University (french: link=no, Université Pierre-et-Marie-Curie, UPMC), also known as Paris 6, was a public university, public research university in Paris, France, from 1971 to 2017. The university was located on the Jussi ...

and a member of the

French Academy of Sciences The French Academy of Sciences (French: ''Académie des sciences'') is a learned society, founded in 1666 by Louis XIV of France, Louis XIV at the suggestion of Jean-Baptiste Colbert, to encourage and protect the spirit of French Scientific me ...

from 1979 to 2010.

Scientific contributions

Malliavin's early work was in

, 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

, Malliavin is known for his work on the stochastic calculus of variation, now known as the

, 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 determini ...

and

mathematical finance Mathematical finance, also known as quantitative finance and financial mathematics, is a field of applied mathematics, concerned with mathematical modeling of financial markets. In general, there exist two separate branches of finance that require ...

. 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

, 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 antiderivative. ...

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(R^d)
', Annales mathématiques Blaise Pascal 3 no. 1, 1996, pp. 1–11 * ''Stochastic Analysis'', Springer, 1997 *

References

External links

* * 1925 births 2010 deaths 20th-century French mathematicians 21st-century French mathematicians Probability theorists University of Paris alumni Members of the French Academy of Sciences Members of the Royal Swedish Academy of Sciences {{France-mathematician-stub