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

Feyel, Denis; Üstünel, Ali Suleyman

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

Soit (W,H,μ) un espace de Wiener abstrait, on suppose que $\mathrm{d}\nu =\mathrm{L}\mathrm{d}\mu$ est une autre probabilité sur $(\mathrm{W},ℬ(\mathrm{W}\left)\right)$ où $\mathrm{L}=\frac{1}{c}\exp -\mathrm{f}$, avec $\mathrm{f}\in {𝔻}_{2,1}$, inférieurement bornée et H-convexe. Soit T=I_W+∇ϕ, $\varphi \in {𝔻}_{2,1}$, 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 $\varphi \in {𝔻}_{2,2}$. Par conséquent le jacobien gaussien $\Lambda ={\det }_2(\mathrm{I}+{\nabla }^2\varphi ){\exp \{ℒ\varphi -1/2|\nabla \varphi |}_H^2\}$ 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 $\mathrm{d}\nu =\mathrm{L}\mathrm{d}\mu$ is a second probability measures on $(\mathrm{W},ℬ(\mathrm{W}\left)\right)$ such that $\mathrm{L}=\frac{1}{c}\exp -\mathrm{f}$, with $\mathrm{f}\in {𝔻}_{2,1}$ lower bounded and H-convex. Let $\mathrm{T}={\mathrm{I}}_W+\nabla \varphi ,\varphi \in {𝔻}_{2,1}$, be the solution of the Monge problem transporting μ to ν and realizing the H-Wasserstein distance between μ and ν. We prove that $\varphi \in {𝔻}_{2,2}$ hence the Gaussian Jacobian $\Lambda ={\det }_2(\mathrm{I}+{\nabla }^2\varphi ){\exp \{ℒ\varphi -1/2|\nabla \varphi |}_H^2\}$ is well-defined and T is the strong solution of the Monge–Ampère equation ΛL∘T=1 a.s. on W.

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