## Advanced Studies in Pure Mathematics

- Adv. Stud. Pure Math.
- Stochastic Analysis and Related Topics in Kyoto: In honour of Kiyosi Itô, H. Kunita, S. Watanabe and Y. Takahashi, eds. (Tokyo: Mathematical Society of Japan, 2004), 49 - 74

### Monge-Kantorovitch Measure Transportation, Monge-Ampère Equation and the Itô Calculus

Denis Feyel and Ali Süleyman Üstünel

#### Abstract

Let $(W, \mu, H)$ be an abstract Wiener space assume two $\nu_i, i =1, 2$ probabilities on $(W, \mathcal{B}(W))$. Assume that the Wasserstein distance between $\nu_1$ and $\nu_2$ with respect to the Cameron-Martin norm $$ d_H (\nu_1, \nu_2)=\left\{ \inf_{\beta} \int_{W \times W} |x-y|_{H}^{2} d \beta (x, y) \right\}^{1/2} $$ is finite, where the infimum is taken on the set of probability measures $\beta$ on $W \times W$ whose first and second marginals are $\nu_1$ and $\nu_2$ and that $\nu_1$ has regular disintegration along a sequence of finite dimensional projections. Then there exists a unique (cyclically monotone) map $T = I_W + \xi$, with $\xi : W \to H$, such that $T$ maps $\nu_1$ to $\nu_2$ and the measure $\gamma = (I \times T) \nu_1$ is the unique solution of the Monge-Kantorovitch problem. Besides, if $\nu_2 \ll \mu^{1)}$, then $T$ is stochastically invertible, i.e., there exists $S : W \to W$ such that $S \circ T = I_W$ $\nu_1$ a.s. and $T \circ S = I_W$ $\nu_2$ a.s. If $\nu_1 = \mu_2$, then there exists a 1-convex function $\phi$ in the Gaussian Sobolev space $\mathbb{D}_{2,1}$, such that $\xi = \nabla \phi$. These results imply that the quasi-invariant transformations of the Wiener space with finite Wasserstein distance from $\mu$ can be written as the composition of a transport map $T$ and a rotation, i.e., a measure preserving map. We give also 1-convex sub-solutions using by calculating the Gaussian jacobian. Finally the full solutions of the Monge-Ampère equation on $W$ are given with the help of the Itô calculus.

#### Article information

**Dates**

Received: 6 March 2003

First available in Project Euclid:
3 January 2019

**Permanent link to this document**

https://projecteuclid.org/
euclid.aspm/1546542605

**Digital Object Identifier**

doi:10.2969/aspm/04110049

**Mathematical Reviews number (MathSciNet)**

MR2083703

**Zentralblatt MATH identifier**

1064.60124

#### Citation

Feyel, Denis; Üstünel, Ali Süleyman. Monge-Kantorovitch Measure Transportation, Monge-Ampère Equation and the Itô Calculus. Stochastic Analysis and Related Topics in Kyoto: In honour of Kiyosi Itô, 49--74, Mathematical Society of Japan, Tokyo, Japan, 2004. doi:10.2969/aspm/04110049. https://projecteuclid.org/euclid.aspm/1546542605