# Paul Malliavin

Paul Malliavin (French:; September 10, 1925 – June 3, 2010) was a French mathematician who made important contributions to harmonic analysis and stochastic analysis. He is known for the Malliavin calculus, an infinite dimensional calculus for functionals on the Wiener space and his probabilistic proof of Hörmander's theorem. He was Professor at the Pierre and Marie Curie University and a member of the French Academy of Sciences 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, 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, Malliavin is known for his work on the stochastic calculus of variation, now known as the Malliavin calculus, a mathematical theory which has found many applications in Monte Carlo simulation and mathematical finance.
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, and derived an integration by parts 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 L$_1$(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
* Malliavin, Paul (1972). Géométrie différentielle intrinsèque. Paris: Hermann. ISBN 978-2-7056-5696-6.
* Géométrie riemannienne stochastique, Séminaire Jean Leray 2 Exposé No. 1, 1973–1974
* Malliavin, Paul (1978). "Stochastic calculus of variations and hypoelliptic operators". Proceedings of the International Symposium on Stochastic Differential Equations (Res. Inst. Math. Sci., Kyoto Univ., Kyoto, 1976). New York: Wiley. pp. 195–263. MR536013
* 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
* Malliavin, Paul; Thalmaier, Anton (2005). Stochastic Calculus of Variations in Mathematical Finance. Springer. ISBN 978-3-540-43431-3.