(You may see the rest of the lecture notes here).
In this lecture we continue our illustration of the power of optimal transport through examples. See my handwritten notes.
The following is a rough transcription of my handwritten notes, processed with an LLM.
Reminder: No class this Thursday (I will do a make-up)
OH change: Now Wed/Fri + by appointment
Two illustrations of OT.
Ex 2. Parabolic PDE and minimizing movements
The porous medium eqn is an important PDE: for $m\geq 1$ it is given by
\[\partial_t \rho = \Delta \rho^m\]For $m=1$ this gives back the HE.
Fix $\rho_0 \in L^1(\mathbb{R}^d)$, $\rho_0 \geq 0$, fix $h>0$
\[\min_\rho \left\{ h\int \rho^m\;dx + \frac{1}{2}d_2(\rho,\rho_0) \right\}\]$\ldots$ $\rho_0, \rho_1, \rho_2, \ldots$ Define $\rho^{(h)}(x,t)$
Theorem (Otto ~2000)
The sequence $\rho^{(h)} \to \rho$, unique weak sol. to $\partial_t \rho = \Delta \rho^m$, $\rho(x,0+) = \rho_0(x)$
A follow up paper by JKO considers the same for Fokker-Planck:
\[\partial_t \rho = \text{div}(\nabla \rho + \rho\nabla V) \qquad \text{(FP)}\]If $\rho_\infty := \frac{1}{Z}e^{-V}$, then
\[\min_\rho \left\{ h\int \rho\log(\rho/\rho_\infty)\;dx + \frac{1}{2}d_2(\rho,\rho_\infty)^2 \right\}\]Produces a soln for FP …
If $V$ is $\alpha$-convex, then
\[\text{KL}(\rho(t),\rho_\infty) \leq e^{-\alpha t}\,\text{KL}(\rho_0,\rho_\infty)\]…
Relevant to: Kinetic theory, Monte-Carlo …
Ex 3: OT and curvature in a Riemannian manifold
… Ricci curvature!
Displacement interpolation: (McCann)
Given two prob. meas. $\mu_0, \mu_1$
$\exists !\; t \to \mu_t \in \mathcal{P}_2(M)$ s.t.
\[d_2(\mu_t,\mu_0) = t\; d(\mu_0,\mu_1)\](a geodesic)
Lott-Villani / Sturm:
A mfld has $\text{Ric} \geq 0 \iff$
\[H(\rho) = \int_M \rho\log\rho \; d\text{Vol}\]is displacement-convex.
A word from our sponsor: the Monge-Ampère equation
Suppose $T = \nabla u$ is smooth … and
\[\int_{\mathbb{R}^n} \eta(T(x))f(x)\;dx = \int \eta(y)g(y)\;dy \quad \forall\, \eta \ldots\]Then
\[\int \eta(T(x))f(x)\;dx = \int \eta(T(x))\,g(T(x))\det DT\;dx\] \[\Rightarrow \quad \det(DT(x)) = \frac{f(x)}{g(T(x))}\]or,
\[\boxed{\det(D^2u(x)) = \frac{f(x)}{g(\nabla u(x))}}\]Compare with Poisson:
\[\Delta u(x) = f(x) \ldots\]Regularity of optimal transport map $\Leftrightarrow$ regularity of solution to the Monge-Ampère eqn.
The Monge-Kantorovich problem
Now we describe the Monge-Kantorovich problem for two spaces $X, Y$ (say, compact metric spaces), Borel meas., and $c: X\times Y \to \mathbb{R}$ cont.
Consider two Borel prob. meas. $\mu, \nu$ in $X, Y$.
A transport plan is a prob. meas. $\pi$ in $X\times Y$ s.t.
\[\int_{X\times Y} \phi(x)\;d\pi(x,y) = \int_X \phi(x)\;d\mu(x),\] \[\int_{X\times Y} \psi(y)\;d\pi(x,y) = \int_Y \psi(y)\;d\nu(y)\] \[\forall\, \phi \in C_c(X),\; \psi \in C_c(Y)\]Example: Given $T: X \to Y$ and $\mu \in \mathcal{P}(X)$, define $\pi$ by
\[\int \phi(x,y)\;d\pi(x,y) := \int \phi(x,T(x))\;d\mu(x)\]Observe,
\[\int \phi(x)\;d\pi(x,y) = \int \phi(x)\;d\mu(x),\] \[\int \psi(y)\;d\pi(x,y) = \int \psi(T(x))\;d\mu(x),\]So a Borel transport map gives rise to a transport map.