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The graphical representation of the contact process preserves coordinatewise order. Its law from the all-infected state decreases in stochastic domination with time, and its limit is invariant by the Feller semigroup property. The all-healthy state is absorbing. Any invariant probability law lies between these two laws. Duality of the contact process gives $\overline{\nu }(\eta (x)=1)=\theta (\lambda )$ when the recovery rate is one. Consequently the two extremal laws coincide exactly when the survival probability of the contact process is zero.

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.

A probability measure is invariant for a Markov process with transition semigroup $P_t$ when starting with that law gives the same law at every later time. Equivalently, $\int P_{t}fd\nu =\int fd\nu$ for every bounded measurable function. Invariant laws may be constructed by weak convergence of probability measures of evolved laws when the Feller semigroup property permits passing the semigroup through the limit, as for the extremal invariant measures of the contact process.

A Borel probability measure is invariant for a Markov kernel semigroup if every transition preserves it. For a Feller semigroup on a compact metric space, it suffices to test the displayed identity on continuous functions. Kernel Jensen inequality gives contraction on $L^p(\mu )$, and sup-norm strong continuity followed by density gives strong continuity there for $1\le p<\infty$.

For a Feller semigroup on a compact metric space, time-averaged transition laws have weakly convergent subsequences by compactness of probability measures on a compact metric space. Shifting a time interval of length $n$ by $t$ changes a bounded test-function average by at most $2t\Vert f{\Vert }_{\infty }/n$. A weak subsequential limit is therefore an invariant probability measure for a semigroup. More generally, tightness of the time averages supplies the required compactness.

On a compact metric space $X$, a conservative Feller semigroup is a strongly continuous semigroup of positive linear maps on $C(X)$ satisfying $P_{t}1=1$. Positivity means $f\ge 0⟹P_{t}f\ge 0$. These conditions imply sup-norm contraction. Conversely, for a contraction semigroup with $1\in D(L)$ and $L1=0$, differentiation on the generator domain gives $\frac{d}{dt}{P}_{t}1=P_{t}L1=0$, so $P_{t}1=1$. On the real space $C(X)$, if $0\le f\le 1$, then $\Vert 1-P_{t}f{\Vert }_{\infty }\le \Vert 1-f{\Vert }_{\infty }\le 1$, implying $P_{t}f\ge 0$. Scaling proves positivity for every nonnegative $f$. This is the unital contraction positivity criterion. For complex $C(X)$ the same conclusion follows because each functional $f\mapsto P_{t}f(x)$ has norm one and value one on $1$, hence is a positive linear functional.

For each $t,x$, the Riesz-Markov-Kakutani representation theorem represents that functional by a unique Borel probability measure $p_t(x,dy)$:Continuity of $P_{t}f$ gives weak continuity of $x\mapsto p_t(x,\cdot )$. Approximating indicators of open sets increasingly by continuous functions proves Borel measurability in $x$; a monotone-class argument then gives a Markov kernel. The semigroup property and uniqueness of representing measures yield $p_0(x,\cdot )={\delta }_x$ and the Chapman-Kolmogorov equationThese are the required transition probabilities.

An invariant probability measure for a semigroup satisfies ${\int }_X{P}_{t}fd\mu ={\int }_{X}fd\mu$ for every $t\ge 0$ and $f\in C(X)$. Equivalently its distribution is preserved by the Markov kernel. For $1\le p<\infty$, Jensen inequality gives $|P_{t}f{|}^p\le P_t(|f{|}^p)$ pointwise. Integrating and using invariance provesAlso $\Vert P_{t}f-f{\Vert }_{L^p(\mu )}\le \Vert P_{t}f-f{\Vert }_{\infty }\to 0$. Continuous functions are dense in these $L^p$ spaces for a finite Borel measure on a compact metric space, so the contraction extends to a strongly continuous semigroup on $L^p(\mu )$.

For real $f$ with $f,f^2\in D(L)$, the kernel form of Jensen inequality gives $P_t(f^2)-(P_{t}f{)}^2\ge 0$, with equality at $t=0$. Take its right derivative to obtain the generator square inequalityThis is nonnegativity of the associated carré du champ.

Each time-average functional ${\varphi }_n$ is positive with ${\varphi }_n(1)=1$, hence represents a Borel probability measure ${\mu }_n$. The compactness of probability measures on a compact metric space gives a subsequence converging weakly to a probability measure $\mu$. Explicitly, $C(X)$ is a separable Banach space, so Banach-Alaoglu theorem and a countable dense family give a subsequence converging on every continuous function, rather than merely a net. For fixed $t\ge 0$, the semigroup property gives the Krylov-Bogolyubov time-average argument:Its absolute value is at most $2t\Vert f{\Vert }_{\infty }/n$. Pass to the subsequential limit, observing that $P_{t}f\in C(X)$, to get ${\int }_X{P}_{t}fd\mu ={\int }_{X}fd\mu$. Thus the limiting measure is invariant.

A linear map between ordered vector spaces is positive if it carries the positive cone of the domain into the positive cone of the codomain. For function spaces this means $f\ge 0⟹Tf\ge 0$. A Feller semigroup consists of such positive maps on continuous functions.

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

A real-linear contraction on $C(X)$ that preserves $1$ is positive: for $0\le f\le 1$, $\Vert 1-Tf{\Vert }_{\infty }\le 1$ forces $Tf\ge 0$, then scale. In the complex case, a unital norm-one functional is real on real functions, since $|1+it\varphi (g)|\le \Vert 1+itg{\Vert }_{\infty }=1+O(t^2)$ for both signs of $t$. The same positivity argument applies pointwise to $\varphi (f)=Tf(x)$. This turns a unital contraction semigroup into a Feller semigroup.