The Brownian Transport Map

In a joint work with Yair Shenfeld, we introduce the Brownian transport map that pushes forward the Wiener measure to a target measure in a finite-dimensional Euclidean space. The map is a solution to a natural transportation problem, compatible with the filtration of the Brownian motion in the Wiener space. It turns out that this construction exhibits qualitatively different properties than the optimal transport map (in the sense of Brenier). Using tools from Ito's and Malliavin's calculus, we show that the map is Lipschitz in several cases of interest. Specifically, our results apply when the target measure satisfies one of the following:
- More log-concave than the Gaussian, recovering a result of Caffarelli.
- Bounded convex support with a semi log-concave density, providing an affirmative answer to a question first posed for the Brenier map.
- A mixture of Gaussians, explaining recent results about dimension-free functional inequalities for such measures.
- log-concave and isotropic. In this case, we establish a direct connection between the Poincare constant and the (averaged) Lipschitz constant of the Brownian transport map. Since the Poincare constant is the object of the famous KLS conjecture, we essentially show that the conjecture is equivalent to the existence of a suitable transportation map.