On the Monge–Ampère equation in infinite dimensions

Abstract. We prove that the optimal transportation mapping that takes a Gaussian measure γ on an infinite dimensional space to an equivalent probability measure g · γ satisfies the Monge–Ampère equation provided that log g ∈ L1(γ) and g log g ∈ L1(γ). 1. Introduction and Main Result The Monge–Kantorovich problem and the Monge–Ampère equation have become a very popular object of research in the last decade (see [1], [15], [20], where one can find addi-tional references). In the finite dimensional case, considerable progress has been achieved by Brenier [5] and McCann [13], whose works stimulated a growing flow of publications