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

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 = 1 c exp - f , with f ∈ 𝔻 2 , 1 lower bounded and H -convex. Let T = I W + ∇ ϕ , ϕ ∈ 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 ∈ 𝔻 2 , 2 hence the Gaussian Jacobian Λ = det 2 ( I + ∇ 2 ϕ ) exp { ℒ ϕ - 1 / 2 | ∇ ϕ | H 2 } is well-defined and T is the strong solution of the Monge–Ampère equation ΛL ∘ T =1 a.s. on W .