This course develops optimal transport as a unifying framework for analysis, geometry, and applied modeling. It begins with the dynamic viewpoint, where transport is described by time-dependent flows and the Benamou-Brenier formula connects geometry with kinetic energy. From there, it builds a calculus on the Wasserstein space $W_2$, including Otto’s formal Riemannian picture, which turns probability measures into an infinite-dimensional geometric object on which gradients, geodesics, and curvature-like effects can be studied.

The middle chapters focus on how this geometry controls functionals and evolution equations. Displacement convexity leads to sharp functional inequalities and structural stability results, while the JKO scheme and minimizing movements provide a variational method for constructing gradient flows. These ideas are then applied to nonlinear diffusion, aggregation dynamics, and Monge-Ampère methods, showing how optimal transport informs both PDE theory and classical PDE techniques. The course then extends to Ricci curvature through transport and to concentration, isoperimetry, and stability, where geometric and probabilistic consequences of transport convexity become central.

The final chapters emphasize modern applications and computation. Computational and statistical optimal transport connects the theory to algorithms, regularization, and data analysis, while the synthesis chapter draws together the core message of the course: transport is not just a way to compare measures, but a modeling language for describing motion, interaction, curvature, and optimization across analysis and applied mathematics.

# Introduction

This course begins after the Monge-Kantorovich existence and duality theory is already available. The central question is how optimal transport becomes an analytic method: not only a way to compare probability measures, but a geometry in which PDE, convexity, curvature, concentration, and computation can be organised. The emphasis is on applications, but the applications are treated through the structures that make them work.

The guiding object is the Wasserstein space of probability measures with finite second moment. In the first course, $W_2$ was introduced through transport plans and optimal couplings. Here the same distance is studied dynamically: a curve of measures is interpreted as mass moving with a velocity field, and the squared distance is recovered as a least kinetic action.

## What Changes After the Foundations?

The foundational theory answers when optimal plans and maps exist and how duality identifies them. For applications, the more useful question is often different: if a probability measure evolves in time, what differential equation does the evolution satisfy, and what functional is it decreasing? This shifts the viewpoint from static optimisation to calculus on spaces of measures.

[definition: Wasserstein Space] Let $d \coloneqq W_2$ denote the quadratic Wasserstein distance on \begin{align*} \mathcal P_2(\mathbb R^n) = \left\{\mu \text{ a Borel probability measure on } \mathbb R^n : \int_{\mathbb R^n} |x|^2\,d\mu(x) < \infty\right\}. \end{align*} The metric space $(\mathcal P_2(\mathbb R^n), W_2)$ is called quadratic Wasserstein space. [/definition]

This definition fixes the ambient space for most of the course. Its importance is that the distance is strong enough to see second moments, but flexible enough to contain singular measures, empirical measures, smooth densities, and solutions of weak PDE.

[example: Transport Between Point Masses] For $a,b\in\mathbb R^n$, consider $\delta_a,\delta_b\in\mathcal P_2(\mathbb R^n)$. If $\pi$ is a coupling of $\delta_a$ and $\delta_b$, then its first marginal gives $\pi(\{a\}\times\mathbb R^n)=1$, and its second marginal gives $\pi(\mathbb R^n\times\{b\})=1$. Hence the complement of $\{a\}\times\{b\}$ has $\pi$-measure $0$, so $\pi=\delta_{(a,b)}$. Using the definition of the quadratic Wasserstein distance, the squared cost is therefore \begin{align*} W_2^2(\delta_a,\delta_b)=\int_{\mathbb R^n\times\mathbb R^n}|x-y|^2\,d\delta_{(a,b)}(x,y)=|a-b|^2. \end{align*} Since $W_2$ is nonnegative, this gives $W_2(\delta_a,\delta_b)=|a-b|$. Thus the map $a\mapsto\delta_a$ preserves distances from Euclidean space into Wasserstein space, while Wasserstein space also contains genuinely distributed measures, not only point masses. [/example]

The course repeatedly exploits this mixture of finite-dimensional intuition and infinite-dimensional behaviour. Geodesics between point masses look like straight lines, while geodesics between densities encode interpolation of whole distributions.

## Dynamic Transport as a Calculus of Moving Mass

How should a curve $t\mapsto \mu_t$ be differentiated when its values are measures rather than points? The answer used throughout the course is to pair the curve with a velocity field and impose conservation of mass. This gives the continuity equation as the differential language of transport.

[definition: Continuity Equation] Let $(\mu_t)_{t\in[0,1]}$ be a narrowly continuous curve in $\mathcal P_2(\mathbb R^n)$ and let $(v_t)_{t\in[0,1]}$ be a Borel vector field with $v_t\in L^2(\mu_t;\mathbb R^n)$ for a.e. $t$. The pair $(\mu_t,v_t)$ solves the continuity equation if \begin{align*} \partial_t\mu_t + \nabla\cdot(v_t\mu_t)=0 \end{align*} in the sense of distributions on $(0,1)\times\mathbb R^n$. [/definition]

The formula says that the only way mass changes in a region is by flux through its boundary. It is the bridge between transport plans and PDE: a coupling describes where mass starts and ends, while the continuity equation describes how it moves in between. The next issue is whether this dynamical description gives the same distance as the static Kantorovich problem, because the rest of the course relies on replacing couplings by curves without changing the metric.

[quotetheorem:9556]

[citeproof:9556]

This theorem is the first main technical hinge of the course. It turns the Wasserstein distance into an action functional and makes tools from weak PDE available in transport geometry. The finite second moment assumption is not decorative: without it the squared displacement and the kinetic action may both be infinite, so the formula would no longer give a finite metric statement. Narrow continuity is the minimal continuity condition compatible with testing measures against bounded continuous functions, while the requirement $v_t\in L^2(\mu_t;\mathbb R^n)$ is exactly what makes the kinetic energy meaningful. The formula should not be read as saying that an optimal velocity field is unique or smooth; minimisers may exist only in weak form, and the continuity equation is usually a distributional equation rather than a classical PDE.

[example: Constant-Speed Motion of a Density] Let $\mu_0=\rho_0\mathcal L^n$, let $T_\#\mu_0=\mu_1$, and set $T_t(x)=(1-t)x+tT(x)$. By the definition of pushforward, for every bounded test function $\varphi$, \begin{align*} \int_{\mathbb R^n}\varphi(y)\,d\mu_t(y)=\int_{\mathbb R^n}\varphi(T_t(x))\,d\mu_0(x). \end{align*} Since $\partial_tT_t(x)=T(x)-x$, differentiating the identity for smooth compactly supported $\varphi$ gives \begin{align*} \frac{d}{dt}\int_{\mathbb R^n}\varphi(y)\,d\mu_t(y)=\int_{\mathbb R^n}\nabla\varphi(T_t(x))\cdot (T(x)-x)\,d\mu_0(x). \end{align*} Define the velocity along transported particles by $v_t(T_t(x))=T(x)-x$. Then the previous identity becomes \begin{align*} \frac{d}{dt}\int_{\mathbb R^n}\varphi(y)\,d\mu_t(y)=\int_{\mathbb R^n}\nabla\varphi(y)\cdot v_t(y)\,d\mu_t(y), \end{align*} which is exactly the weak form of $\partial_t\mu_t+\nabla\cdot(v_t\mu_t)=0$. The kinetic action of this curve is computed by changing variables through the pushforward $\mu_t=(T_t)_\#\mu_0$: \begin{align*} \int_0^1\int_{\mathbb R^n}|v_t(y)|^2\,d\mu_t(y)\,dt=\int_0^1\int_{\mathbb R^n}|v_t(T_t(x))|^2\,d\mu_0(x)\,dt. \end{align*} Using $v_t(T_t(x))=T(x)-x$ gives \begin{align*} \int_0^1\int_{\mathbb R^n}|v_t(T_t(x))|^2\,d\mu_0(x)\,dt=\int_0^1\int_{\mathbb R^n}|T(x)-x|^2\,d\mu_0(x)\,dt. \end{align*} The integrand is independent of $t$, so \begin{align*} \int_0^1\int_{\mathbb R^n}|T(x)-x|^2\,d\mu_0(x)\,dt=\int_{\mathbb R^n}|T(x)-x|^2\,d\mu_0(x). \end{align*} Thus the displacement interpolation moves each particle with constant velocity from $x$ to $T(x)$, and its kinetic action equals the quadratic transport cost induced by the optimal map $T$. [/example]