Comptes Rendus
Probability Theory
Some remarks about the positivity of random variables on a Gaussian probability space
Comptes Rendus. Mathématique, Volume 339 (2004) no. 12, pp. 873-877.

Let (W,H,μ) be an abstract Wiener space and let LLlogL(μ) is a positive random variable. Using the measure transportation of Monge–Kantorovitch, we prove that the operator corresponding to the kernel of the projection of L on the second Wiener chaos is lower bounded by a semi-positive Hilbert–Schmidt operator.

Soit (W,H,μ) un espace de Wiener abstrait et soit LLlogL une variable aléatoire positive. A l'aide de la théorie de transport de mesure de Monge–Kantorovitch, nous montrons que le noyau de la projection de L dans le second chaos de Wiener est un opérateur de spectre inférieurement borné et que l'opérateur correspondant est inférieurement borné par un opérateur Hilbert–Schmidt semi-positif.

Received:
Accepted:
Published online:
DOI: 10.1016/j.crma.2004.10.014
Denis Feyel 1; A. Suleyman Üstünel 2

1 Université d'Evry-Val-d'Essone, département de mathématiques, 91025 Evry cedex, France
2 ENST, département Infres, 46, rue Barrault, 75013 Paris, France
@article{CRMATH_2004__339_12_873_0,
     author = {Denis Feyel and A. Suleyman \"Ust\"unel},
     title = {Some remarks about the positivity of random variables on a {Gaussian} probability space},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {873--877},
     publisher = {Elsevier},
     volume = {339},
     number = {12},
     year = {2004},
     doi = {10.1016/j.crma.2004.10.014},
     language = {en},
}
TY  - JOUR
AU  - Denis Feyel
AU  - A. Suleyman Üstünel
TI  - Some remarks about the positivity of random variables on a Gaussian probability space
JO  - Comptes Rendus. Mathématique
PY  - 2004
SP  - 873
EP  - 877
VL  - 339
IS  - 12
PB  - Elsevier
DO  - 10.1016/j.crma.2004.10.014
LA  - en
ID  - CRMATH_2004__339_12_873_0
ER  - 
%0 Journal Article
%A Denis Feyel
%A A. Suleyman Üstünel
%T Some remarks about the positivity of random variables on a Gaussian probability space
%J Comptes Rendus. Mathématique
%D 2004
%P 873-877
%V 339
%N 12
%I Elsevier
%R 10.1016/j.crma.2004.10.014
%G en
%F CRMATH_2004__339_12_873_0
Denis Feyel; A. Suleyman Üstünel. Some remarks about the positivity of random variables on a Gaussian probability space. Comptes Rendus. Mathématique, Volume 339 (2004) no. 12, pp. 873-877. doi : 10.1016/j.crma.2004.10.014. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2004.10.014/

[1] Y. Brenier Polar factorization and monotone rearrangement of vector valued functions, Commun. Pure Appl. Math., Volume 44 (1991), pp. 375-417

[2] N. Dunford; J.T. Schwartz Linear Operators, 2, Interscience, 1963

[3] D. Feyel, A survey on the Monge transport problem, Preprint, 2004

[4] D. Feyel; A. de La Pradelle Capacités gaussiennes, Ann. Inst. Fourier, Volume 41 (1991) no. 1, pp. 49-76

[5] D. Feyel; A.S. Üstünel The notion of convexity and concavity on Wiener space, J. Funct. Anal., Volume 176 (2000), pp. 400-428

[6] D. Feyel; A.S. Üstünel Transport of measures on Wiener space and the Girsanov theorem, C. R. Acad. Sci. Paris, Ser. I, Volume 334 (2002) no. 1, pp. 1025-1028

[7] D. Feyel; A.S. Üstünel Monge–Kantorovitch measure transportation and Monge–Ampère equation on Wiener space, Probab. Theory Related Fields, Volume 128 (2004), pp. 347-385

[8] D. Feyel; A.S. Üstünel Monge–Kantorovitch measure transportation, Monge–Ampère equation and the Itô calculus, Adv. Stud. Pure Math., vol. 41, Mathematical Society of Japan, 2004, pp. 49-74

[9] D. Feyel; A.S. Üstünel The strong solution of the Monge–Ampère equation on the Wiener space for log-concave densities, C. R. Acad. Sci. Paris, Ser. I, Volume 339 (2004) no. 1, pp. 49-53

[10] K. Itô Multiple Wiener integral, J. Math. Soc. Japan, Volume 3 (1951), pp. 157-164

[11] P. Malliavin Stochastic Analysis, Springer-Verlag, 1997

[12] H.P. McKean Geometry of differential space, Ann. Probab., Volume 1 (1973), pp. 197-206

[13] J. Ruiz de Chavez; P.A. Meyer Positivité sur l'espace de Fock, Séminaire de Probabilités XXIV, Lecture Notes in Math., vol. 1426, Springer, 1990, pp. 461-465

[14] D.W. Stroock Homogeneous chaos revisited, Séminaire de Probabilités XXI, Lecture Notes in Math., vol. 1247, Springer, 1987, pp. 1-8

[15] A.S. Üstünel Introduction to Analysis on Wiener Space, Lecture Notes in Math., vol. 1610, Springer, 1995

[16] A.S. Üstünel; M. Zakai Transformation of Measure on Wiener Space, Springer-Verlag, 1999

[17] N. Wiener The homogeneous chaos, Amer. J. Math., Volume 60 (1930), pp. 897-936

Cited by Sources:

Comments - Policy


Articles of potential interest

The strong solution of the Monge–Ampère equation on the Wiener space for log-concave densities

Denis Feyel; Ali Suleyman Üstünel

C. R. Math (2004)


Measure transport on Wiener space and the Girsanov theorem

Denis Feyel; Ali Süleyman Üstünel

C. R. Math (2002)


The invertibility of adapted perturbations of identity on the Wiener space

A. Suleyman Üstünel; Moshe Zakai

C. R. Math (2006)