Probability Theory
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.

Let (W,H,μ) be an abstract Wiener space, assume that dν=Ldμ is a second probability measures on (W,(W)) such that L=1 c exp -f, with f𝔻 2,1 lower bounded and H-convex. Let T=I W +ϕ,ϕ𝔻 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 ϕ𝔻 2,2 hence the Gaussian Jacobian Λ= det 2 (I+ 2 ϕ) exp {ϕ-1/2|ϕ| H 2 } is well-defined and T is the strong solution of the Monge–Ampère equation ΛLT=1 a.s. on W.

Soit (W,H,μ) un espace de Wiener abstrait, on suppose que dν=Ldμ est une autre probabilité sur (W,(W))L=1 c exp -f, avec f𝔻 2,1 , inférieurement bornée et H-convexe. Soit T=IW+∇ϕ, ϕ𝔻 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 ϕ𝔻 2,2 . Par conséquent le jacobien gaussien Λ= det 2 (I+ 2 ϕ) exp {ϕ-1/2|ϕ| H 2 } est bien défini et T est la solution forte de l'equation de Monge–Ampère ΛLT=1 p.s.

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DOI: 10.1016/j.crma.2004.04.013
Feyel, Denis 1; Üstünel, Ali Suleyman 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
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Feyel, Denis; Üstünel, Ali Suleyman. 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. http://www.numdam.org/articles/10.1016/j.crma.2004.04.013/

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