Feller semigroup ($P_{t}1=1,f\ge 0⟹P_{t}f\ge 0$)

On a compact metric space, a conservative Feller semigroup is a strongly continuous semigroup of positive unital operators on $C(X)$. The Riesz-Markov-Kakutani representation theorem associates a Markov kernel to each operator. On noncompact spaces the standard definition uses $C_0(X)$; conservativity then refers to the corresponding kernels having mass one, even when $1\notin C_0(X)$.

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

For a Markov process generator and suitable real functions, this bilinear expression records the local quadratic energy. Positivity of $\Gamma (f,f)$ follows by differentiating $P_t(f^2)\ge (P_{t}f{)}^2$. For $L=D^2-xD$ it is $f^{\prime }{g}^{\prime }$, giving the Gaussian Dirichlet energy after integration against the invariant measure.

If real $f$ and $f^2$ belong to the generator domain of a Feller semigroup, Jensen inequality gives $P_t(f^2)-(P_{t}f{)}^2\ge 0$. This expression is zero at $t=0$, so its right derivative is nonnegative, proving the inequality. It is equivalent to positivity of the carré du champ.