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

Denis Feyel; Ali Suleyman Üstünel

doi:10.1016/j.crma.2004.04.013

Probability Theory

The strong solution of the Monge–Ampère equation on the Wiener space for log-concave densities
[La solution forte de l'équation de Monge–Ampère sur l'espace de Wiener pour les densités log-concaves]

Présenté par : Paul Malliavin

Denis Feyel ¹ ; Ali 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

Comptes Rendus. Mathématique, Volume 339 (2004) no. 1, pp. 49-53.

Résumé
Abstract

Soit (W,H,μ) un espace de Wiener abstrait, on suppose que $d ν = L d μ$ est une autre probabilité sur $(W, ℬ (W))$ où $L = \frac{1}{c} \exp - f$ , avec $f \in 𝔻_{2, 1}$ , inférieurement bornée et H-convexe. Soit T=I_W+∇ϕ, $ϕ \in 𝔻_{2, 1}$ , la solution du problème de Monge qui transporte μ sur ν et qui realise la distance de Wasserstein entre μ et ν par rapport à la métrique de Cameron–Martin. Nous montrons qu'en fait $ϕ \in 𝔻_{2, 2}$ . Par conséquent le jacobien gaussien $Λ = \det_{2} (I + \nabla^{2} ϕ) {\exp {ℒ ϕ - 1 / 2 | \nabla ϕ |}_{H}^{2}}$ est bien défini et T est la solution forte de l'equation de Monge–Ampère ΛL∘T=1 p.s.

Let (W,H,μ) be an abstract Wiener space, assume that $d ν = L d μ$ is a second probability measures on $(W, ℬ (W))$ such that $L = \frac{1}{c} \exp - f$ , with $f \in 𝔻_{2, 1}$ lower bounded and H-convex. Let $T = I_{W} + \nabla ϕ, ϕ \in 𝔻_{2, 1}$ , be the solution of the Monge problem transporting μ to ν and realizing the H-Wasserstein distance between μ and ν. We prove that $ϕ \in 𝔻_{2, 2}$ hence the Gaussian Jacobian $Λ = \det_{2} (I + \nabla^{2} ϕ) {\exp {ℒ ϕ - 1 / 2 | \nabla ϕ |}_{H}^{2}}$ is well-defined and T is the strong solution of the Monge–Ampère equation ΛL∘T=1 a.s. on W.

Reçu le : 2004-04-21
Accepté le : 2004-04-27
Publié le : 2004-05-25

DOI : 10.1016/j.crma.2004.04.013

Affiliations des auteurs :

Denis Feyel ¹ ; Ali 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_1_49_0,
     author = {Denis Feyel and Ali Suleyman \"Ust\"unel},
     title = {The strong solution of the {Monge{\textendash}Amp\`ere} equation on the {Wiener} space for log-concave densities},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {49--53},
     publisher = {Elsevier},
     volume = {339},
     number = {1},
     year = {2004},
     doi = {10.1016/j.crma.2004.04.013},
     language = {en},
}

TY  - JOUR
AU  - Denis Feyel
AU  - Ali Suleyman Üstünel
TI  - The strong solution of the Monge–Ampère equation on the Wiener space for log-concave densities
JO  - Comptes Rendus. Mathématique
PY  - 2004
SP  - 49
EP  - 53
VL  - 339
IS  - 1
PB  - Elsevier
DO  - 10.1016/j.crma.2004.04.013
LA  - en
ID  - CRMATH_2004__339_1_49_0
ER  -

%0 Journal Article
%A Denis Feyel
%A Ali Suleyman Üstünel
%T The strong solution of the Monge–Ampère equation on the Wiener space for log-concave densities
%J Comptes Rendus. Mathématique
%D 2004
%P 49-53
%V 339
%N 1
%I Elsevier
%R 10.1016/j.crma.2004.04.013
%G en
%F CRMATH_2004__339_1_49_0

Denis Feyel; Ali Suleyman Üstünel. The strong solution of the Monge–Ampère equation on the Wiener space for log-concave densities. Comptes Rendus. Mathématique, Volume 339 (2004) no. 1, pp. 49-53. doi : 10.1016/j.crma.2004.04.013. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2004.04.013/

Bibliographie
Cité par

[1] E.F. Beckenbach; R. Bellman Inequalities, Ergebn. Math. Grenzgeb., vol. 30, Springer-Verlag, 1983

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

[3] L.A. Caffarelli The regularity of mappings with a convex potential, J. Amer. Math. Soc., Volume 5 (1992), pp. 99-104

[4] L.A. Caffarelli Monotonicity properties of optimal transportation and the FKG and related inequalities, Comm. Math. Phys., Volume 214 (2000), pp. 547-563

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

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

[7] D. Feyel; A.de La Pradelle Capacités gaussiennes, Ann. Inst. Fourier, Volume 41 (1991), pp. 49-76

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

[9] 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

[10] 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

[11] 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, in press

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

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

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

Cité par Sources :

Commentaires - Politique

Ces articles pourraient vous intéresser

Some remarks about the positivity of random variables on a Gaussian probability space

Denis Feyel; A. 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)