Symmetric Feller semigroup ($⟨P_{t}f,g{⟩}_{L^2(\mu )}=⟨f,P_{t}g{⟩}_{L^2(\mu )}$)

A Feller semigroup symmetric in the space associated with an invariant measure has a self-adjoint contraction realization in $L^2(\mu )$. Its two-point kernel measure $\mu (dx)P_t(x,dy)$ is symmetric. This reversibility expresses its joint energy as an integral of products of increments.

// Target: analysis.bigb

For functions in the form domain, this symmetric bilinear energy equals $-⟨Lf,g⟩$ when $f$ is in the generator domain. Its finite-time approximation is $(2t{)}^{-1}\int (f(x)-f(y))(g(x)-g(y))\mu (dx)P_t(x,dy)$. The quadratic energy ${\mathcal{E}}_{\mu }(f)={\mathcal{E}}_{\mu }(f,f)$ may take the extended value infinity.

// Target: analysis.bigb

For $a,b\ge 0$ and $q>2$, Cauchy-Schwarz inequality applied to the integral of $(q/2)s^{q/2-1}$ gives $(a^{q/2}-b^{q/2}{)}^2\le q^2(a^{q-1}-b^{q-1})(a-b)/(4(q-1))$. Integrating against the symmetric two-point kernel measure and taking the energy limit proves the displayed inequality.

// Target: analysis.bigb