I learned this Wasserstein P–L viewpoint from Raphaël Barboni. In particular, he taught me the useful distinction at the heart of this post: the condition can control convergence of the gradient-flow trajectory itself, not merely decay of its objective value.
A small gradient is reassuring only if it means that the objective is nearly minimized. The Polyak–Łojasiewicz condition supplies precisely this guarantee [6]. It can yield exponential convergence even without convexity, as in Fast gradient flows without convexity.
Now replace a point by a probability distribution, Euclidean distance by quadratic Wasserstein distance, and the objective by relative entropy. The same condition becomes the logarithmic Sobolev inequality. This is more than a dictionary: one short argument shows that the entire distribution converges, not just its entropy. It also explains the implication from logarithmic Sobolev to Talagrand’s transport inequality [1].
Entropy has a gradient: Fisher information
Let \(\pi(x)=Z^{-1}e^{-V(x)}\) be a smooth positive probability density on \(\mathbb R^d\). We use natural logarithms and write
\[\mathcal F(\rho)=\operatorname{KL}(\rho\mid\pi) =\int\rho\log\frac\rho\pi\,dx,\qquad \mathcal I(\rho\mid\pi)=\int\left|\nabla\log\frac\rho\pi\right|^2\rho\,dx.\]For the calculations, assume finite second moments and enough smoothness and decay to integrate by parts. If a density moves with velocity \(v\), the continuity equation is \(\partial_t\rho+\nabla\cdot(\rho v)=0\). Differentiating gives
\[\frac d{dt}\mathcal F(\rho_t) =\int\nabla\log(\rho_t/\pi)\cdot v_t\,\rho_t\,dx.\]The tangent norm is \(\|v\|_\rho^2=\int|v|^2\rho\). Thus the Wasserstein gradient is represented by \(\nabla\log(\rho/\pi)\), and its squared norm is Fisher information. Its negative gradient flow is
\[v_t=-\nabla\log(\rho_t/\pi),\qquad \partial_t\rho_t=\Delta\rho_t+\nabla\cdot(\rho_t\nabla V).\]This is the Fokker–Planck equation. Along a regular solution,
\[-\frac d{dt}\mathcal F(\rho_t) =\mathcal I(\rho_t\mid\pi) =|\dot\rho_t|_{W_2}^2.\]The equality with metric speed uses the gradient velocity, not an arbitrary velocity solving the same continuity equation. Wasserstein gradient-flow theory makes this identity precise beyond smooth densities [2].
The P–L condition in disguise
Proposition 1 (the dictionary). Logarithmic Sobolev with constant \(\lambda>0\) is the inequality
\[\mathcal I(\rho\mid\pi)\ge 2\lambda\operatorname{KL}(\rho\mid\pi).\]Equivalently,
\[\tfrac12\|\operatorname{grad}_{W_2}\mathcal F(\rho)\|_\rho^2 \ge\lambda(\mathcal F(\rho)-\min\mathcal F).\]This is exactly Wasserstein P–L. Consequently,
\[\mathcal F(\rho_t)\le e^{-2\lambda t}\mathcal F(\rho_0).\]Proof: substitute the gradient norm
The minimum of relative entropy is zero, attained only at \(\pi\). Substitute the gradient–Fisher identity above. Energy dissipation then gives \(\mathcal F'\le-2\lambda\mathcal F\), and Grönwall gives the decay estimate. \(\square\)
The constant convention matters: for the standard Gaussian, \(\lambda=1\). In the classical function formulation, putting \(\rho=h^2\pi/\int h^2d\pi\) turns this into \(\operatorname{Ent}_\pi(h^2)\le(2/\lambda)\int|\nabla h|^2d\pi\). This is the normalization of Gross’s Gaussian inequality [3].
Why the measures themselves converge
Exponential decay of an objective does not, by itself, imply convergence of its argument. A valley can contain many minimizers; a curve might keep travelling along it. Here the dissipation identity gives the missing control: the curve has finite remaining length.
Theorem 1 (finite length under Wasserstein P–L). Let \(\mathcal G:\mathcal P_2(\mathbb R^d)\to(-\infty,+\infty]\) be lower semicontinuous, with finite infimum \(\mathcal G_\star\). Suppose a global locally absolutely continuous flow, of finite initial energy, has locally absolutely continuous energy \(E\) and satisfies for almost every time
\[E(t)=\mathcal G(\rho_t)-\mathcal G_\star,\qquad -E'(t)=|\dot\rho_t|_{W_2}^2=|\partial\mathcal G|^2(\rho_t), \qquad |\partial\mathcal G|^2(\rho_t)\ge2\lambda E(t).\]Here \(|\partial\mathcal G|\) is the descending metric slope. Then the flow converges in \(W_2\) to a minimizer \(\rho_\infty\), and
\[W_2(\rho_t,\rho_\infty) \le\sqrt{\frac{2E(t)}\lambda} \le\sqrt{\frac{2E(0)}\lambda}\,e^{-\lambda t}.\]Proof: differentiate the square root of the energy
Write \(s(t)=|\dot\rho_t|_{W_2}\). As long as \(E(t)>0\), dissipation and P–L give
\[-\frac d{dt}\sqrt{E(t)} =\frac{s(t)^2}{2\sqrt{E(t)}} \ge\sqrt{\lambda/2}\,s(t).\]Integrate from \(a\) to \(b\), and use that distance is bounded by the length of a curve:
\[W_2(\rho_a,\rho_b) \le\int_a^b s(t)\,dt \le\sqrt{2/\lambda}\bigl(\sqrt{E(a)}-\sqrt{E(b)}\bigr).\]Also \(E'\le-2\lambda E\), so \(E(t)\to0\). The last bound makes the curve Cauchy as time tends to infinity. Completeness of \(\mathcal P_2\) gives a limit, and lower semicontinuity gives \(\mathcal G(\rho_\infty)\le\liminf\mathcal G(\rho_t)=\mathcal G_\star\). Let \(b\to\infty\) in the length bound. If energy reaches zero at a finite time, the dissipation identity makes the curve constant thereafter, with the same conclusion. \(\square\)
For entropy, the minimizer is uniquely \(\pi\). Applying the theorem at time zero gives
\[W_2(\rho,\pi)^2\le\frac2\lambda\operatorname{KL}(\rho\mid\pi).\]This is Talagrand’s quadratic transportation inequality, with the same constant. The calculation proves it for initial data admitting the stated dissipative flow; standard approximation extends it to all finite-entropy data under the usual hypotheses. The full logarithmic-Sobolev-to-transport implication is due to Otto and Villani [1].
A Gaussian example attaining both rates
Take \(\pi=\mathcal N(0,I)\) and \(\rho_0=\mathcal N(m_0,I)\). The exact flow is \(\rho_t=\mathcal N(e^{-t}m_0,I)\). Hence
\[\mathcal F(\rho_t)=\tfrac12e^{-2t}|m_0|^2,\qquad W_2(\rho_t,\pi)=e^{-t}|m_0|.\]The objective decays at rate two, the distance at rate one. Both constants in Theorem 1 are sharp.
Why the Gaussian logarithmic Sobolev constant is one
Here is a short semigroup proof for smooth positive densities \(f=d\rho/d\pi\) bounded above and away from zero, with sufficient derivative integrability. Let
\[P_tf(x)=\mathbb E\,f(e^{-t}x+\sqrt{1-e^{-2t}}Z),\qquad Z\sim\pi.\]Gaussian invariance gives \(\int P_tg\,d\pi=\int g\,d\pi\). Differentiation gives \(\nabla P_tf=e^{-t}P_t\nabla f\). Weighted Cauchy–Schwarz therefore yields
\[\frac{|\nabla P_tf|^2}{P_tf} \le e^{-2t}P_t\!\left(\frac{|\nabla f|^2}{f}\right).\]After integration, \(\mathcal I(P_tf\,\pi\mid\pi)\le e^{-2t}\mathcal I(f\pi\mid\pi)\). Entropy dissipation and \(P_tf\to1\) give
\[\operatorname{KL}(f\pi\mid\pi) =\int_0^\infty\mathcal I(P_tf\,\pi\mid\pi)\,dt \le\tfrac12\mathcal I(f\pi\mid\pi).\]Boundedness justifies the entropy limit by dominated convergence. Truncation, smoothing, and lower semicontinuity extend the inequality to its natural finite-information domain. \(\square\)
More than entropy, more than convexity
Uniformly convex confinement. If \(\nabla^2V\succeq\lambda I\), the Bakry–Émery criterion gives logarithmic Sobolev with constant \(\lambda\) [4]. This is a sufficient condition, not a characterization: a bounded perturbation of a uniformly convex potential can preserve a logarithmic Sobolev inequality even when pointwise convexity is lost.
A genuinely nonconvex lifted loss. If \(|\nabla h|^2\ge2\lambda(h-h_\star)\), integration proves Wasserstein P–L for \(\mathcal G(\rho)=\int h\,d\rho\) whenever its slope has the expected gradient representation. For instance,
\[h(x,y)=\tfrac12(y-x^2)^2,\qquad |\nabla h|^2=(1+4x^2)(y-x^2)^2\ge2h.\]Its minimizers are distributions supported on a parabola. A particle stays in its initial sublevel, which bounds \(|y-x^2|\); the equations \(x'=2x(y-x^2)\), \(y'=-(y-x^2)\) then rule out finite-time escape. Theorem 1 applies to distributional flows for which its finite-moment and full-dissipation assumptions hold. Different initializations can select different minimizing measures.
Confined internal energy. For \(m>1\), the functional
\[\mathcal G(\rho)=\frac1{m-1}\int\rho^m\,dx+ \frac\lambda2\int|x|^2\rho\,dx\]is \(\lambda\)-displacement convex and satisfies Wasserstein P–L with constant \(\lambda\). Indeed, its geodesic subgradient inequality bounds the energy gap by \(|\partial\mathcal G|r-\lambda r^2/2\le|\partial\mathcal G|^2/(2\lambda)\), where \(r\) is the distance to a minimizer. Convexity supplies one route to P–L; the finite-length argument needs only P–L itself. See Optimal Transport for Machine Learners [5].
Numerical experimentation: a narrow Gaussian explores a mixture
A diffusion can quickly settle inside one well and still take a long time to explore the others. Start with a small, translated Gaussian cloud and let it evolve towards an equally weighted mixture of \(k\) Gaussians. The controls change the number of components and move their centers from overlapping to separated positions.
\[\pi_{k,R}=\frac1k\sum_{j=0}^{k-1}\mathcal N(\mu_j,\sigma^2I_2), \qquad \mu_j=R\left(\cos\theta_j,\sin\theta_j\right), \qquad \theta_j=\pi+\frac{2\pi j}{k}.\]Here \(\sigma=0.65\) is fixed, while the position slider changes \(R\). For \(k=1\), the single center moves along the negative horizontal axis; for larger \(k\), the centers form a regular polygon. At \(R=0\), every choice of \(k\) gives exactly the same Gaussian target.
With \(V=-\log\pi_{k,R}\), the Langevin equation is \(dX_t=\nabla\log\pi_{k,R}(X_t)\,dt+\sqrt2\,dB_t\). Its drift averages the component centers using the local responsibilities:
\[\nabla\log\pi_{k,R}(x) =\frac{\sum_j w_j(x)\mu_j-x}{\sigma^2}, \qquad w_j(x)=\frac{e^{-|x-\mu_j|^2/(2\sigma^2)}} {\sum_\ell e^{-|x-\mu_\ell|^2/(2\sigma^2)}}.\]Every run starts from the same narrow distribution \(\rho_0=\mathcal N(m_0,\tau^2I_2)\), with \(m_0=(-3.6,0)\) and \(\tau=0.3\). Its center stays clearly to the left of every target center throughout the slider range \(0\le R\le2.5\). Noise strength, physical time, spatial axes, and density colors are identical across settings. Unlike the initial distribution itself, its relative entropy to the target changes with \(k\) and \(R\).
The dark dashed target level sets move immediately when either target control changes; the colored initial Gaussian remains visible while the new diffusion is computed. Changing the target restarts the time at zero, so the display never combines a new target with an old trajectory.
For one component, the exact solution stays Gaussian. Its mean and variance are
\[m_t=\mu_0+(m_0-\mu_0)e^{-t/\sigma^2}, \qquad s_t^2=\sigma^2+(\tau^2-\sigma^2)e^{-2t/\sigma^2}.\]This gives an independent check of the numerical evolution. For separated components, low-density passages create a different challenge: a cloud can look equilibrated around its starting mode while still missing much of the target mass.
Proposition 2 (separating the modes weakens log-Sobolev). For fixed \(k\), the optimal log-Sobolev rate \(\lambda_{k,R}\) is nonincreasing in \(R\). It also satisfies
\[\lambda_{k,R}\le \frac{1}{\sigma^2+\lambda_{\max}(\operatorname{Cov}(\mu_J))}, \qquad J\sim\operatorname{Unif}\{0,\ldots,k-1\}.\]For \(k=1\), the exact constant is \(\sigma^{-2}\). The upper bound is \((\sigma^2+R^2)^{-1}\) for \(k=2\), and \((\sigma^2+R^2/2)^{-1}\) for \(k\ge3\). These upper bounds are not formulas for the observed decay. Finite Gaussian mixtures do satisfy a positive log-Sobolev inequality; quantitative bounds for Gaussian convolutions are developed in [8].
Proof: a linear test and an extra Gaussian smoothing
The upper bound. Log-Sobolev implies \(\lambda_{k,R}\operatorname{Var}_{\pi_{k,R}}(g)\le\int|\nabla g|^2d\pi_{k,R}\). Test \(g(x)=a\cdot x\) with a unit vector \(a\) maximizing the variance. Since a mixture sample is \(\mu_J+\sigma Z\), its covariance is \(\operatorname{Cov}(\mu_J)+\sigma^2I_2\). This proves the bound, by smooth truncation if needed.
Monotonicity in the separation. Write \(C_R=1/\lambda_{k,R}\), and fix \(0\le R_1\le R_2\), with \(R_2>0\). If \(r=R_1/R_2\), \(X\sim\pi_{k,R_2}\), and \(Z'\sim\mathcal N(0,I_2)\) is independent, then
\[rX+\sigma\sqrt{1-r^2}\,Z'\sim\pi_{k,R_1}.\]Apply the product log-Sobolev inequality to \(g(rX+\sigma\sqrt{1-r^2}Z')\). The chain rule gives
\[C_{R_1}\le r^2C_{R_2}+(1-r^2)\sigma^2 \le C_{R_2}.\]The last inequality uses \(C_{R_2}\ge\sigma^2\), already implied by the linear test. When both radii are zero there is nothing to prove. \(\square\)
The plot solves the physical-space Fokker–Planck equation deterministically, using target-preserving Scharfetter–Gummel fluxes and positive implicit steps; see [7] for related finite-volume entropy analysis. It uses a large reflecting box. The full-space mathematical statements above and these finite-domain numerical trajectories should not be confused; the plot does not measure the optimal log-Sobolev constant.
References
- Felix Otto and Cédric Villani. Generalization of an Inequality by Talagrand and Links with the Logarithmic Sobolev Inequality. Journal of Functional Analysis 173(2), 361–400, 2000.
- Luigi Ambrosio, Nicola Gigli, and Giuseppe Savaré. Gradient Flows in Metric Spaces and in the Space of Probability Measures. Second edition, Birkhäuser, 2008.
- Leonard Gross. Logarithmic Sobolev Inequalities. American Journal of Mathematics 97(4), 1061–1083, 1975.
- Dominique Bakry and Michel Émery. Diffusions hypercontractives. Séminaire de Probabilités XIX, 177–206, 1985.
- Gabriel Peyré. Optimal Transport for Machine Learners. Book manuscript, 2025–2026.
- Hamed Karimi, Julie Nutini, and Mark Schmidt. Linear Convergence of Gradient and Proximal-Gradient Methods Under the Polyak–Łojasiewicz Condition. ECML PKDD, 2016.
- Claire Chainais-Hillairet and Maxime Herda. Large-time behaviour of a family of finite volume schemes for boundary-driven convection–diffusion equations. IMA Journal of Numerical Analysis 40(4), 2473–2504, 2020.
- Jean-Baptiste Bardet, Nathaël Gozlan, Florent Malrieu, and Pierre-André Zitt. Functional inequalities for Gaussian convolutions of compactly supported measures: explicit bounds and dimension dependence. 2015.
- André Schlichting. Poincaré and log-Sobolev inequalities for mixtures. Entropy 21(1), 89, 2019.