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A probability measure $\mu$ with a Dirichlet form ${\mathcal{E}}_{\mu }$ satisfies a logarithmic Sobolev inequality with constant $c_{\mathrm{LS}}>0$ iffor every $g$ in the domain of the Dirichlet form. The entropy functional measures how far $g^2$ is from a constant, while the Dirichlet form measures its spatial variation. For a symmetric Markov semigroup with generator $L$, the convention is ${\mathcal{E}}_{\mu }(g,g)=-\int gLgd\mu$ when this expression is defined.

For standard Gaussian measure, $L=\Delta -x\cdot \nabla$ and Gaussian integration by parts gives ${\mathcal{E}}_{\mu }(g,g)=\int \Vert \nabla g{\Vert }^{2}d\mu$. The Gaussian logarithmic Sobolev inequality has optimal constant two in this convention. Rescaling the Dirichlet form rescales the constant inversely, so a numerical constant needs its energy convention. The logarithmic Sobolev inequality implies hypercontractivity and also supports concentration inequalities through the Herbst argument.^[ref]

// Target: analysis.bigb

A positive linear operator on bounded measurable functions that preserves the constant function one. A Markov semigroup is a time-indexed family of such operators with the semigroup law.