# Brenier theorem on manifold

Let $$W(\mu,\nu)^2_2$$ be the 2-wasserstein distance between probability distributions $$\mu,\nu$$ over $$\mathbb{R}^d$$ such that $$d\mu,d\nu\ll\text{dVol}$$:

$$W(\mu,\nu)^2_2=\inf_{\pi\in\Pi(\mu,\nu)} \iint \|x-y\|^2 d\pi(x,y).$$

Let $$MP(\mu,\nu)=\{T: \mathbb{R}^d\longrightarrow \mathbb{R}^d| T\#\mu=\nu \}.$$

Brenier's theorem states that:

There exists a unique (up to a $$\mu$$-negligible set) minimizer $$T^*$$ to the problem $$d(\mu,\nu)^2=\inf_{T\in MP(\mu,\nu)} \int \|x-T(x)\|^2 d\mu(x)$$ such that $$d(\mu,\nu)^2=W(\mu,\nu)^2_2$$, and $$T^*$$ can be represented $$\mu$$-almost everywhere as $$T^*=\nabla \psi$$ for some convex function $$\psi:\mathbb{R}^d\longrightarrow \mathbb{R}$$.

Question:

Imagine that one introduces a Riemannian metric $$g$$ in $$\mathbb{R}^d$$, so that all Euclidean distances appearing above are now replaced by geodesic distances w.r.t. $$g$$ . Suppose that $$\mu,\nu$$ are two probability measures on $$(\mathbb{R}^d,g)$$.

Is there an analogous statement to the one above? Namely that there exists a unique $$T^*$$ that is the gradient of a convex function, and that $$d(\mu,\nu)^2$$ and $$W(\mu,\nu)^2_2$$ coincide?

• I think you will have a better chance of receiving an answer by moving your question to mathoverflow.net. Commented Jun 8, 2023 at 7:10
• There is an appropriate generalisation of Brenier's theorem in Villani's OT book. Commented Jun 11, 2023 at 14:28