= Solution
On a compact <metric space> $X$, a conservative <Feller semigroup> is a <strongly continuous semigroup> of positive linear maps on $C(X)$ satisfying $P_t1=1$. <Positivity> means $f\geq0\Longrightarrow P_tf\geq0$. 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 $\frac{d}{dt}P_t1=P_tL1=0$, so $P_t1=1$. On the real space $C(X)$, if $0\leq f\leq1$, then $\|1-P_tf\|_\infty\leq\|1-f\|_\infty\leq1$, implying $P_tf\geq0$. 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 $f\mapsto P_tf(x)$ 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> $p_t(x,dy)$:
$$
\boxed{P_tf(x)=\int_X f(y)\,p_t(x,dy).}
$$
Continuity of $P_tf$ 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 equation>
$$
p_{s+t}(x,A)=\int_X p_t(y,A)\,p_s(x,dy).
$$
These are the required <transition probabilities>.
An <invariant probability measure for a semigroup> satisfies $\int_XP_tf\,d\mu=\int_Xf\,d\mu$ for every $t\geq0$ and $f\in C(X)$. Equivalently its distribution is preserved by the <Markov kernel>. For $1\leq p<\infty$, <Jensen inequality> gives $|P_tf|^p\leq P_t(|f|^p)$ pointwise. Integrating and using invariance proves
$$
\boxed{\|P_tf\|_{L^p(\mu)}\leq\|f\|_{L^p(\mu)}.}
$$
Also $\|P_tf-f\|_{L^p(\mu)}\leq\|P_tf-f\|_\infty\to0$. <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 $P_t(f^2)-(P_tf)^2\geq0$, with equality at $t=0$. Take its right <derivative> to obtain the <generator square inequality>
$$
\boxed{L(f^2)\geq2fLf.}
$$
This is nonnegativity of the associated <carré du champ>.
Each time-average functional $\phi_n$ is positive with $\phi_n(1)=1$, 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 $t\geq0$, the <semigroup property> gives the <Krylov-Bogolyubov time-average argument>:
$$
\phi_n(P_tf)-\phi_n(f)=\frac1n\left[\int_n^{n+t}\!\int_XP_sf\,d\nu\,ds-\int_0^t\!\int_XP_sf\,d\nu\,ds\right].
$$
Its absolute value is at most $2t\|f\|_\infty/n$. Pass to the subsequential limit, observing that $P_tf\in C(X)$, to get $\int_XP_tf\,d\mu=\int_Xf\,d\mu$. Thus the limiting measure is invariant.
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