Log-Sobolev inequalities and Herbst's argument

For a nonnegative integrable Y with the required logarithmic integrability, define Ent_μ(Y)=E_μ[Y log Y]−E_μY log(E_μY), with zero log zero interpreted as zero. Say a probability measure μ on Euclidean space satisfies a log-Sobolev inequality with constant C>0 when Ent_μ(g²)≤2C E_μ||∇g||₂² for admissible real test functions. We initially assume the inequality for bounded continuously differentiable g with the finite energy needed below, and its usual extension by approximation when invoked.

Under this convention the standard Gaussian has C=1, while covariance σ²I has C=σ². Sources that place C rather than 2C before the energy use a different numerical constant. The gradient is Euclidean; changing the metric also changes the energy and its constant. The inequality compares a nonlinear entropy with a quadratic local energy, and it is stronger than merely controlling variance. Expanding near a constant recovers a Poincaré-type variance bound, but that weaker bound alone does not reproduce the full exponential-moment conclusion.

First let f be bounded and continuously differentiable with gradient norm at most L, and center it by subtracting its expectation. Define H(λ)=log E_μ exp(λ(f−E_μf)). Boundedness justifies differentiating under the expectation for every finite real λ. Insert g=exp(λ(f−E_μf)/2) into the log-Sobolev inequality. The squared gradient supplies λ²/4, while the left entropy, divided by the exponential expectation, becomes λH'(λ)−H(λ).

The result bounds λH'−H by Cλ²L²/2. For positive λ, divide by λ² to control the derivative of H(λ)/λ. Its limit at zero is zero because f was centered, so integration gives H(λ)≤CL²λ²/2. Apply the same calculation to −f for negative λ. Chernoff then yields a one-sided tail exp(−t²/(2CL²)) and its two-sided factor-two version. The argument is an explicit derivation for the stated test-function class, not a general license to differentiate any infinite exponential moment.

For an integrable globally Lipschitz f, truncate its values and smooth by a Euclidean mollifier. The approximants can retain Lipschitz constant at most L, converge pointwise, and have expectations converging to Ef; the latter follows from integrability and controlled approximation. Assuming the log-Sobolev inequality extends to these bounded smooth tests, the derived mgf bounds are uniform. Fatou's lemma then passes the bound to exp(λ(f−Ef)), establishing finiteness rather than assuming it. This explains how an unbounded linear Gaussian observable can be included rigorously.

Entropy dissipation under a suitable Markov semigroup is another route to proving a log-Sobolev inequality, but a decaying mode by itself does not establish that inequality for every test function. A periodic heat equation and a Gaussian Ornstein–Uhlenbeck semigroup use different invariant measures and generators. They share an energy-dissipation idea without being the same model. The lab therefore illustrates mode decay only; the concentration constants here come from the stated entropy inequality and its verified Gaussian scaling, not from fitting a decay curve.