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For a reversible transition matrix $P$ with invariant distribution $\pi$,

This chain energy is a discrete example of a Dirichlet form.

A Gaussian free field with Dirichlet boundary condition is fixed to zero on a boundary set and has density proportional to the exponential of minus one half of its discrete Dirichlet energy on the remaining vertices.

Fix a bounded planar domain $D$ and explicitly choose the Dirichlet form normalizationIts Dirichlet Green function of the Laplacian satisfies $-{\Delta }_x{G}_D(x,y)=2\pi {\delta }_y(x)$. Let $(e_n)$ be an orthonormal basis of the corresponding Sobolev space $H_0^1(D)$. The formal expansionis interpreted as a weak kernel identity. Concretely, for smooth compactly supported test functions $\varphi ,\psi$,The identity follows by applying Parseval's identity to the two Riesz representatives for the integration functionals. It is not a claim that arbitrary Sobolev space basis functions have pointwise values everywhere or that the series converges on the singular diagonal.

For a finite measure $\rho$ with finite Green energy, the same Hilbert-space identity gives ${\sum }_n(\int e_{n}d\rho {)}^2=\iint G_D(x,y)\rho (dx)\rho (dy)$. This is the square-summability condition for a Finite-Green-energy measure pairing with a Gaussian free field. An alternative energy normalization rescales both the basis and the Green kernel; specifying the factor prevents an unnoticed $2\pi$ mismatch.^[ref]

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