Logarithmic Sobolev inequality (${\mathrm{Ent}}_{\mu }(g^2)\le c_{\mathrm{LS}}{\mathcal{E}}_{\mu }(g)$)

Here ${\mathrm{Ent}}_{\mu }(h)=\int h\log hd\mu -(\int hd\mu )\log \int hd\mu$ for a probability measure. The coefficient depends on the normalization of the energy. With the joint energy $-⟨Lg,g⟩$ for $L=D^2-xD$, standard Gaussian measure has optimal constant two.

// Target: analysis.bigb

Differentiate $\log \Vert u{\Vert }_q$ with $u=P_{t}f$ and a changing exponent. The derivative is $q^{\prime }\mathrm{Ent}(u^q)/(q^2\int u^q)-\mathcal{E}(u^{q-1},u)/\int u^q$. Apply the logarithmic Sobolev inequality to $u^{q/2}$ and the power inequality for symmetric Markov energies. Choosing $q^{\prime }=4(q-1)/c_{\mathrm{LS}}$ makes the upper bound zero.

// Target: probability-and-statistics.bigb