Optimal Transport and Ricci Curvature

6C.10 Fully Nonlinear Equations and Monge–Ampère met optimal transport as a source of Monge–Ampère equations: Brenier's theorem says that the cheapest way to move one distribution of mass to another, at quadratic cost, is the gradient of a convex function, and that function solves $g(\nabla u)\det D^2u = f$. This chapter takes the next step, which turned optimal transport from a branch of the calculus of variations into a branch of geometry. The minimal cost is a distance between probability measures, and the space of probability measures with that distance behaves like a Riemannian manifold. In it, the heat equation is the gradient flow of entropy (Jordan, Kinderlehrer and Otto); along its geodesics, entropy is convex exactly when the underlying space has nonnegative Ricci curvature (McCann; Cordero-Erausquin, McCann and Schmuckenschläger; von Renesse and Sturm); and that convexity can be stated on spaces with no smooth structure at all, which gave Lott, Sturm and Villani a definition of "Ricci curvature bounded below" for metric measure spaces.

The Ricci flow enters at three points. The quantity $\operatorname{Ric} + \nabla^2f$ that Perelman's $\mathcal F$-functional produces as its gradient (12A.2 Ricci Flow as a Gradient Flow) is the Bakry–Émery Ricci tensor of the weighted space $(M, g, e^{-f}dV)$, the curvature that controls displacement convexity of entropy relative to $e^{-f}dV$, so gradient shrinking solitons are exactly the weighted spaces of constant Bakry–Émery curvature. McCann and Topping (2010) showed that the Ricci flow is characterised by a contraction property: under a super Ricci flow, solutions of the conjugate heat equation draw together in Wasserstein distance. And Topping (2009) and Lott (2009) proved the monotonicity of Perelman's reduced volume by optimal transport with Perelman's $\mathcal L$-length as the cost. The real-world uses sketched in 6C.10 Fully Nonlinear Equations and Monge–Ampère (economics, image processing, machine learning) are joined here by meteorology and cosmology.

By the end of this chapter you will be able to:

• state the Kantorovich problem and its duality, prove the easy half of duality and the cyclical monotonicity of optimal plans for finite measures;
• define the Wasserstein distance and compute it for Diracs, translates and one-dimensional Gaussians;
• define displacement interpolation, and prove McCann's theorem that entropy is displacement convex on $\mathbb{R}^n$;
• explain the Otto calculus and derive, formally, the heat and Fokker–Planck equations as Wasserstein gradient flows of entropy, with Fisher information as the dissipation;
• explain why displacement convexity of entropy on a manifold is equivalent to $\operatorname{Ric} \geq 0$, state the $K$-convexity characterisation of $\operatorname{Ric} \geq K$, and define the Lott–Sturm–Villani condition;
• identify the Bakry–Émery tensor $\operatorname{Ric} + \nabla^2f$ in Perelman's work and read shrinking solitons as weighted spaces of constant curvature;
• state McCann–Topping's characterisation of super Ricci flow and describe the $\mathcal L$-optimal transport proof of reduced-volume monotonicity.

11.1Kantorovich's problem and its dual

Let $\mu$ and $\nu$ be probability measures on $\mathbb{R}^n$ and $c(x, y) \geq 0$ a cost. A transport plan is a probability measure $\pi$ on $\mathbb{R}^n\times\mathbb{R}^n$ with marginals $\mu$ and $\nu$: $\pi(A\times\mathbb{R}^n) = \mu(A)$ and $\pi(\mathbb{R}^n\times B) = \nu(B)$. It says how much mass goes from where to where, and, unlike a map, it may split the mass at a point. Kantorovich's problem is to minimise $\int c\,d\pi$ over plans. It is a linear problem in $\pi$ over a convex set, and it always has a minimiser when $c$ is continuous (the set of plans is weakly compact, by Prokhorov's theorem, and the cost is lower semicontinuous); Brenier's theorem (6C.10 Fully Nonlinear Equations and Monge–Ampère) says that for $c = |x - y|^2$ and $\mu$ absolutely continuous the minimiser is concentrated on the graph of a map $\nabla u$.

Duality. Every linear programme has a dual. If $\varphi(x) + \psi(y) \leq c(x, y)$ for all $x$, $y$, then for every plan $\pi$,

because $\pi$ has marginals $\mu$ and $\nu$. So the supremum of the left side over such pairs is at most the infimum of the right side. Kantorovich duality says they are equal (for lower semicontinuous costs bounded below; statement):

Economically, $\varphi$ and $\psi$ are prices: a shipping company that charges $\varphi(x)$ to pick up at $x$ and $\psi(y)$ to deliver at $y$, never more than the cost of doing it yourself, can extract at most the optimal cost, and in equilibrium extracts exactly that. The optimal $\varphi$, $\psi$ are the Kantorovich potentials, and on the support of an optimal plan the constraint is an equality, $\varphi(x) + \psi(y) = c(x, y)$. For $c = \frac12|x - y|^2$, writing $u(x) = \frac12|x|^2 - \varphi(x)$, the constraint becomes $u(x) + u^*(y) \geq x\cdot y$, the Fenchel inequality, with equality exactly when $y \in \partial u(x)$: this is how the convex function of Brenier's theorem arises.

Cyclical monotonicity. For finitely many points the structure of optimal plans can be seen directly.

Ralph Tyrrell Rockafellar proved (1966) that a cyclically monotone set lies in the subdifferential of a convex function. So, already for finite sets, the optimal plan for quadratic cost is supported on the graph of $\partial u$ for a convex $u$, and Brenier's theorem is the continuum version. (For linear assignment problems such as matching $N$ workers to $N$ jobs, the minimiser over permutations is computed by the Hungarian algorithm of Kuhn, 1955, in time polynomial in $N$; for large problems the entropic regularisation and Sinkhorn iterations mentioned in 6C.10 Fully Nonlinear Equations and Monge–Ampère are used.)

11.2The Wasserstein distance

For probability measures with finite second moments, set

It is a metric on this set of measures (the triangle inequality follows by "gluing" two plans along their common marginal; statement), and convergence in $W_2$ is weak convergence together with convergence of second moments. Its character is best seen in examples (Exercise 11.5).

• $W_2(\delta_x, \delta_y) = |x - y|$: the space of points embeds isometrically.
• If $\nu$ is the translate of $\mu$ by a vector $v$, then $W_2(\mu, \nu) = |v|$, however spread out $\mu$ is. Compare the $L^2$ distance between densities, which for two disjoint bumps is the same whether they are a millimetre or a mile apart.
• For one-dimensional Gaussians, $W_2(N(m_0, \sigma_0^2), N(m_1, \sigma_1^2))^2 = (m_0 - m_1)^2 + (\sigma_0 - \sigma_1)^2$.

The Wasserstein distance is a distance on distributions that knows the geometry of the underlying space, which is why it is the right distance for comparing histograms of images, distributions of data, or the shapes of density fields. It also has a dynamic form.

Jean-David Benamou and Yann Brenier (2000) introduced this formula as a computational method; it says that $W_2^2$ is the minimal kinetic energy of a fluid flow carrying $\mu_0$ to $\mu_1$ in unit time, exactly as the squared Riemannian distance is the minimal energy of a curve joining two points. The minimising curve is the displacement interpolation

in which each particle moves at constant speed along a straight line from $x$ to its destination $\nabla u(x)$: a geodesic of the Wasserstein space, with $W_2(\mu_s, \mu_t) = |t - s|W_2(\mu_0, \mu_1)$.

11.3Displacement convexity

Robert McCann (1997) asked which functionals of a density are convex along displacement interpolations, rather than along the linear interpolations $(1 - t)\rho_0 + t\rho_1$ for which convexity is ordinary convexity of the integrand. The answer for entropy is a short computation with the Monge–Ampère equation.

The same argument, using the concavity of $t \mapsto \det(DT_t)^{1/n}$ (the geometric mean of the affine functions $1 - t + t\lambda_i$), shows that $\int\rho^m$ for $m \geq 1 - \frac1n$, with the right sign, is displacement convex; with $m = 1 - \frac1n$ and uniform measures on two sets $A$ and $B$, it yields the Brunn–Minkowski inequality $|(1 - t)A + tB|^{1/n} \geq (1 - t)|A|^{1/n} + t|B|^{1/n}$ (Exercise 11.9), which in turn implies the isoperimetric inequality. A potential energy $\int V\rho$ is displacement convex exactly when $V$ is convex (Exercise 11.7). Displacement convexity is a new kind of convexity, adapted to the geometry of moving mass, and it gives uniqueness of minimisers for problems that are not convex in the usual sense, such as McCann's original application, the uniqueness of equilibrium shapes of interacting gases.

11.4The heat equation as a gradient flow

Felix Otto (2001), building on Jordan, Kinderlehrer and Otto (1998), gave the space of probability densities a formal Riemannian structure in which $W_2$ is the distance. A tangent vector at $\rho$ is an infinitesimal change $s = \partial_t\rho$ of mass; by the continuity equation it is produced by a velocity field, and among all velocity fields producing it, the one of least kinetic energy is a gradient, $v = \nabla\phi$, with

This is the Otto metric. The gradient of a functional $\mathcal E(\rho)$ in this metric is the tangent vector $s$ with $\langle s, s'\rangle_\rho = \frac{d}{d\varepsilon}\mathcal E(\rho + \varepsilon s')$ for all $s'$. For $s' = -\operatorname{div}(\rho\nabla\phi')$, an integration by parts gives $\frac{d}{d\varepsilon}\mathcal E = \int\frac{\delta\mathcal E}{\delta\rho}s' = \int\rho\nabla\frac{\delta\mathcal E}{\delta\rho}\cdot\nabla\phi'$, so the gradient corresponds to $\phi = \frac{\delta\mathcal E}{\delta\rho}$:

For the entropy $H(\rho) = \int\rho\log\rho$, $\frac{\delta H}{\delta\rho} = \log\rho + 1$, and $\rho\nabla\log\rho = \nabla\rho$, so the gradient flow $\partial_t\rho = -\operatorname{grad}H$ is

the heat equation is the Wasserstein gradient flow of entropy (Exercise 11.8). The rate of dissipation is $\|\operatorname{grad}H\|^2 = \int\rho|\nabla\log\rho|^2 = \int\frac{|\nabla\rho|^2}{\rho}$, the Fisher information, which is the entropy production of 6A.12 Entropy, Information and Diffusion seen as the squared length of a gradient. Adding a potential, $\mathcal E = \int\rho\log\rho + \int V\rho$, gives the Fokker–Planck equation $\partial_t\rho = \Delta\rho + \operatorname{div}(\rho\nabla V)$ (6D.4 Fokker–Planck, Time Reversal and Diffusion Models); taking $\mathcal E = \frac{1}{m-1}\int\rho^m$ gives the porous medium equation of 6A.8 Nonlinear Parabolic Equations (Otto, 2001).

These are formal computations; Jordan, Kinderlehrer and Otto made them rigorous through a discrete scheme. Fix a time step $h$ and define

the JKO scheme, a Wasserstein version of the implicit Euler method for a gradient flow (6A.11 Calculus of Variations and Gradient Flows). They proved (1998) that as $h \to 0$ the piecewise constant interpolations converge to the solution of the Fokker–Planck equation. The theory of gradient flows in metric spaces of Luigi Ambrosio, Nicola Gigli and Giuseppe Savaré (2005) puts this on a general footing: on a geodesic metric space, a functional that is $K$-convex along geodesics has a well-defined gradient flow, which contracts distances at the rate $e^{-Kt}$. Displacement convexity of entropy is thus the reason the heat flow contracts Wasserstein distances on $\mathbb{R}^n$.

11.5Ricci curvature as convexity of entropy

On a Riemannian manifold $(M, g)$ with volume measure $dV$, optimal transport for the cost $d(x, y)^2$ still works: Robert McCann (2001) proved that the optimal map is $T(x) = \exp_x(\nabla\phi(x))$ for a function $\phi$ that is "$\frac{d^2}{2}$-concave", the Riemannian version of a convex potential, and the displacement interpolation moves each point along the geodesic $t \mapsto \exp_x(t\nabla\phi(x))$. The computation of Theorem 11.3 goes through, with one change: the Jacobian $J_t(x) = \det DT_t(x)$ is no longer the determinant of an affine family of matrices, but is built from Jacobi fields along the geodesics (9A.7 Jacobi Fields and Curvature versus Topology), and its logarithm satisfies the Riccati-type inequality of 9B.1 Laplacian Comparison:

The first term is the Euclidean part (equality when the Hessian is a multiple of the identity), and the second is the effect of curvature: positive Ricci curvature focuses geodesics, so $\log J_t$ is more concave, and the entropy $H(\rho_t) = H(\rho_0) - \int\rho_0\log J_t$ is more convex. Negative Ricci curvature spreads geodesics, and convexity can fail. This leads to the theorem that made optimal transport a part of geometry.

Dario Cordero-Erausquin, Robert McCann and Michael Schmuckenschläger proved that (1) implies (2) (2001), and Max-Karl von Renesse and Karl-Theodor Sturm proved the converse (2005), by testing on measures concentrated near a point and moving in a given direction, so that the entropy sees exactly $\operatorname{Ric}(v, v)$.

Spaces without smooth structure. Condition (2) uses only a distance, a measure, and geodesics in the Wasserstein space: it makes sense on any metric measure space $(X, d, m)$. Lott and Villani (2009) and Sturm (2006) took it as a definition: $(X, d, m)$ has Ricci curvature bounded below by $K$ (with a dimension parameter $N$, in the refined condition $CD(K, N)$, using $\int\rho^{1-1/N}$ in place of entropy) if entropy is $K$-convex along Wasserstein geodesics. The definition is stable under measured Gromov–Hausdorff convergence, because convexity inequalities pass to limits, so limits of manifolds with $\operatorname{Ric} \geq K$ (9B.4 Convergence of Manifolds) satisfy it; and many theorems of Riemannian geometry, the Bishop–Gromov volume comparison (9B.2 Volume Comparison), Brunn–Minkowski, the Lichnerowicz eigenvalue bound (6B.16 Eigenvalue Problems and Perelman’s λ), log-Sobolev and Talagrand inequalities, follow from it directly. Felix Otto and Cédric Villani (2000) showed that on a space where entropy is $K$-convex with $K > 0$, the log-Sobolev inequality holds with constant $K$, and implies Talagrand's transport inequality $W_2(\nu, m)^2 \leq \frac2KH(\nu\,|\,m)$, a quantitative form of concentration of measure. The later theory of $RCD$ spaces (Ambrosio, Gigli, Savaré, and others) adds a condition that rules out Finsler geometries and recovers much of the structure theory of Cheeger and Colding.

Weighted spaces and the Bakry–Émery tensor. For a weighted manifold $(M, g, e^{-f}dV)$, entropy relative to $e^{-f}dV$ is $\int\rho\log\rho\,dV + \int f\rho\,dV$, the entropy plus a potential energy, and the potential term adds $\nabla^2f(\dot\gamma, \dot\gamma)$ to the second derivative along geodesics (Exercise 11.7). So $K$-convexity of the weighted entropy is equivalent to

the condition $CD(K, \infty)$ for the weighted space, with Bakry–Émery Ricci tensor $\operatorname{Ric}_f = \operatorname{Ric} + \nabla^2f$ (Dominique Bakry and Michel Émery introduced it in 1985 for diffusion operators $\Delta - \nabla f\cdot\nabla$; 6D.5 Diffusions on Manifolds and Curvature). This is precisely the tensor that appears in Perelman's work. The gradient of $\mathcal F(g, f) = \int(R + |\nabla f|^2)e^{-f}dV$ with the measure $e^{-f}dV$ held fixed is $-(\operatorname{Ric} + \nabla^2f)$ (12A.2 Ricci Flow as a Gradient Flow), and a gradient shrinking soliton $\operatorname{Ric} + \nabla^2f = \frac{1}{2\tau}g$ is a weighted manifold with constant Bakry–Émery curvature $\frac{1}{2\tau}$: the weighted analogue of an Einstein manifold. The Gaussian soliton, $\mathbb{R}^n$ with $f = \frac{|x|^2}{4\tau}$, is the model, and its weighted entropy inequality with $K = \frac{1}{2\tau}$ is the Gaussian log-Sobolev inequality of 6A.12 Entropy, Information and Diffusion (Exercise 11.10). Perelman's entropy $\mathcal W$ is, in this language, a log-Sobolev functional for a weighted space whose curvature the Ricci flow is driving toward a soliton.

11.6Optimal transport and the Ricci flow

McCann–Topping. Robert McCann and Peter Topping (2010) found a characterisation of the Ricci flow by optimal transport. Run time backwards, $\tau = T - t$, so that the Ricci flow is $\partial_\tau g = 2\operatorname{Ric}$, and let two probability measures evolve by the conjugate heat equation (9B.7 The Heat Equation on a Manifold), the natural diffusion for measures along the backward flow. They proved that the Wasserstein distance between any two such measures, measured with $g(\tau)$, is nonincreasing in $\tau$, and that this contraction property characterises super Ricci flows, families of metrics with $\partial_\tau g \leq 2\operatorname{Ric}$ (statement). The fixed-metric version is the classical fact that heat flow contracts $W_2$ exactly when $\operatorname{Ric} \geq 0$; along the Ricci flow the metric moves just enough to make the contraction hold with no curvature assumption, which is another sense in which the Ricci flow is "the heat equation for the metric". Their paper also derived monotonicity of Perelman-type entropies along the flow from the same structure.

$\mathcal L$-optimal transport and the reduced volume. Perelman's reduced distance $\ell(q, \tau)$ (12A.5 Reduced Distance and Reduced Volume) is the minimal value of the $\mathcal L$-length $\int_0^\tau\sqrt s\big(R + |\dot\gamma|^2\big)ds$ over space-time paths, a Hamilton–Jacobi value function (6C.9 Hamilton–Jacobi Equations and Viscosity Solutions), and the reduced volume $\tilde V(\tau) = \int(4\pi\tau)^{-n/2}e^{-\ell}\,dV$ is monotone. Peter Topping (2009) used the $\mathcal L$-length as the cost of an optimal transport problem between measures at two different times of the flow, and proved a convexity inequality for entropy along $\mathcal L$-optimal transport that contains, as special cases, the monotonicity of the reduced volume (transporting from a Dirac mass at the base point) and of Perelman's $\mathcal W$-entropy. John Lott (2009) gave a related treatment of the reduced volume by optimal transport. In this picture the reduced volume is monotone for the same reason that entropy is displacement convex under $\operatorname{Ric} \geq 0$: the space-time Ricci curvature of the backward flow, in Perelman's sense, is nonnegative, and $\mathcal L$-geodesics play the role of straight lines.

11.7History

Monge posed his problem in 1781 and Kantorovich his relaxation and duality in 1942. Kuhn's Hungarian algorithm appeared in 1955 and Rockafellar's theorem on cyclically monotone sets in 1966. Brenier's theorem dates from 1987–91 and McCann's displacement convexity from 1997; Jordan, Kinderlehrer and Otto's gradient-flow interpretation of the Fokker–Planck equation appeared in 1998, Otto's calculus for the porous medium equation in 2001, and Otto and Villani's log-Sobolev–Talagrand theorem in 2000. Benamou and Brenier's dynamic formulation appeared in 2000. Bakry and Émery introduced their curvature condition in 1985. McCann's theorem on manifolds appeared in 2001, as did the theorem of Cordero-Erausquin, McCann and Schmuckenschläger; von Renesse and Sturm's converse appeared in 2005. Ambrosio, Gigli and Savaré's book on gradient flows in metric spaces appeared in 2005, Sturm's papers on the geometry of metric measure spaces in 2006, and Lott and Villani's in 2009. Frisch's reconstruction of the early universe was published in 2002. Topping's $\mathcal L$-optimal transport and Lott's paper on the reduced volume appeared in 2009, and McCann and Topping's characterisation of super Ricci flow in 2010.

11.8Exercises

Solution

(a) If $X \sim N(m_0, \sigma_0^2)$ then $T(X) = m_1 + \frac{\sigma_1}{\sigma_0}(X - m_0) \sim N(m_1, \sigma_1^2)$. $T$ is increasing and affine, the derivative of the convex function $m_1x + \frac{\sigma_1}{2\sigma_0}(x - m_0)^2$; by Brenier's theorem (in one dimension, the monotone rearrangement of 6C.10 Fully Nonlinear Equations and Monge–Ampère), it is optimal. (b) $x - T(x) = (m_0 - m_1) + (1 - \frac{\sigma_1}{\sigma_0})(x - m_0)$; squaring and integrating against $\mu_0$, the cross term vanishes (mean zero) and $\int(x - m_0)^2d\mu_0 = \sigma_0^2$, giving $(m_0 - m_1)^2 + (\sigma_0 - \sigma_1)^2$. (c) $T_t(x) = (1 - t)x + tT(x)$ is affine with slope $(1 - t) + t\frac{\sigma_1}{\sigma_0}$, so $\mu_t = N(m_t, \sigma_t^2)$ with $m_t = (1 - t)m_0 + tm_1$ and $\sigma_t = (1 - t)\sigma_0 + t\sigma_1$: means and standard deviations interpolate linearly.

Solution

$\frac{d}{dt}\log\det M_t = \operatorname{tr}(M_t^{-1}\dot M_t)$ with $\dot M_t = A - I$, and $\frac{d}{dt}M_t^{-1} = -M_t^{-1}\dot M_tM_t^{-1}$, so the second derivative is $-\operatorname{tr}(M_t^{-1}(A - I)M_t^{-1}(A - I))$. The matrix $B = M_t^{-1/2}(A - I)M_t^{-1/2}$ is symmetric, and the trace equals $\operatorname{tr}(B^2) \geq 0$. For Gaussians, $H(\mu_t) = -\frac12\log(2\pi e) - \log\sigma_t$ with $\sigma_t$ affine and positive in $t$, and $-\log$ of a positive affine function is convex, as the theorem says.

Solution

(a) $\mathcal V(\mu_t) = \int V(T_tx)\,d\mu_0(x)$, and $\frac{d^2}{dt^2}V(x + t(Tx - x)) = \nabla^2V(T_tx)[Tx - x, Tx - x]$. (b) If $\nabla^2V \geq 0$ the integrand is nonnegative. Conversely, $\mu_t = \delta_{(1-t)x + ty}$ is a Wasserstein geodesic between Diracs, and $\mathcal V(\mu_t) = V((1 - t)x + ty)$; convexity of this for all $x$, $y$ is convexity of $V$. (c) On a manifold, the entropy term contributes $-\frac{d^2}{dt^2}\int\rho_0\log J_t \geq \int\rho_0\operatorname{Ric}(\dot\gamma, \dot\gamma)$ (from the Riccati inequality, dropping the nonnegative square), and the potential term contributes $\int\rho_0\nabla^2f(\dot\gamma, \dot\gamma)$, by (a) along geodesics. The sum is controlled by $(\operatorname{Ric} + \nabla^2f)(\dot\gamma, \dot\gamma)$, integrated against $\rho_0$, with $|\dot\gamma|$ the transport distance; a lower bound $Kg$ gives $K$-convexity with the term $KW_2^2$.

Solution

(a) $\frac{d}{dt}\int\rho\log\rho = \int(\log\rho + 1)\partial_t\rho = -\int(\log\rho + 1)\operatorname{div}(\rho v) = \int\rho\,\nabla\log\rho\cdot v$. (b) By Cauchy–Schwarz in $L^2(\rho)$, $\int\rho\nabla\log\rho\cdot v \geq -\|\nabla\log\rho\|_{L^2(\rho)}\|v\|_{L^2(\rho)}$, with equality for $v$ a negative multiple of $\nabla\log\rho$. With $v = -\nabla\log\rho$, $\rho v = -\nabla\rho$ and the continuity equation is $\partial_t\rho = \operatorname{div}\nabla\rho = \Delta\rho$. (c) Then $\frac{d}{dt}H = -\int\rho|\nabla\log\rho|^2 = -\int\frac{|\nabla\rho|^2}{\rho}$, which is minus the Fisher information, the identity of 6A.12 Entropy, Information and Diffusion.

Solution

$T_t(x) = (1 - t)x + t\nabla u(x) \in (1 - t)A + tB$ for $x \in A$, since $\nabla u(x) \in \overline B$; so $\rho_t$ is supported in $(1 - t)A + tB$ (up to closure). From $\rho_t(T_tx)\det DT_t(x) = \rho_0(x) = \frac{1}{|A|}$ and $\det(DT_t)^{1/n} \geq (1 - t) + t\big(\frac{|B|}{|A|}\big)^{1/n} = \frac{(1 - t)|A|^{1/n} + t|B|^{1/n}}{|A|^{1/n}}$, we get $\rho_t(T_tx) \leq \frac{1}{|A|}\cdot\frac{|A|}{((1 - t)|A|^{1/n} + t|B|^{1/n})^n}$. Since $\rho_t$ is a probability density supported in $(1 - t)A + tB$ and bounded by $M = ((1 - t)|A|^{1/n} + t|B|^{1/n})^{-n}$, $1 \leq M\,|(1 - t)A + tB|$, which is the inequality.

Solution

(a) $\operatorname{Ric} = 0$ and $\nabla^2\frac{|x|^2}{4\tau} = \frac{1}{2\tau}I$. (b) $\int(4\pi\tau)^{-n/2}e^{-|x|^2/4\tau}dx = 1$ (6A.3 The Heat Equation on ℝⁿ), the Gaussian of covariance $2\tau I$. (c) Talagrand: $W_2(\nu, \gamma)^2 \leq \frac{2}{K}H(\nu\,|\,\gamma) = 4\tau H(\nu\,|\,\gamma)$. For $\nu$ the translate by $v$: $W_2 = |v|$, and $H(\nu\,|\,\gamma) = \int\log\frac{d\nu}{d\gamma}d\nu = \int\frac{|x|^2 - |x - v|^2}{4\tau}d\nu(x) = \frac{2\langle\bar x, v\rangle - |v|^2}{4\tau}$ with $\bar x = v$ the mean of $\nu$, so $H = \frac{|v|^2}{4\tau}$ and $4\tau H = |v|^2 = W_2^2$: equality, as expected for the extremal case.