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

Presented by: 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.

Abstract
Résumé

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.

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.

Received: 2004-04-21
Accepted: 2004-04-27
Published online: 2004-05-25

DOI: 10.1016/j.crma.2004.04.013

Author's affiliations:

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

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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/

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