On the Monge–Ampère equation on Wiener space

Therefore, the image of μ under the mapping x 7→ (x, T (x)) gives minimum to the integral on the right-hand side in (1). According to [4], if μ is absolutely continuous, the optimal transportation T is the gradient of a convex function V . As shown in [5], if μ = f dx and ν = g dx are two absolutely continuous probability measures such that μ is equivalent to Lebesgue measure and V is a convex function such that ∇V takes μ to ν, then, letting det(DacV ) be the determinant of the density D2 acV of the absolutely continuous part of D2V (i.e., the determinant in A.D. Alexandroff’s sense), we obtain that the set M of points where the matrix D2 acV is defined and invertible is of full μ-measure and for almost all x ∈M the following Monge–Ampère equation holds: f(x) = g(∇V (x)) detD2 acV (x).