Condition for the existence of a Markov process from the properties of a semigroup $\{T_t\}$ In the article Diffusion processes with continuous coefficients I (1969, Stroock and Varadhan) one finds the following arguments in pages 26-27
"$(\cdots)$ for any $ \epsilon >0, \sup_{x \in \mathbb{R}^d} P(t, x, \mathbb{R}^d - B(x,\epsilon)) \to 0$ as $t \downarrow 0$. Hence by theorem 6.3 in Dynkin [1960]( Dynkin, E. B., Foundations of the Theory of Markov Processes, Pergamon Press, New York, 1960.), there is a right continuous Markov process $(\overline{x}(t),\bar{M}(t),  \bar{P}_x)$  such that $\bar{P}_x(\bar{x}(t) \in \Gamma ) = P(t, X , \Gamma)$, $t \geq 0, x \in \mathbb{R}^d$, and
$\Gamma \in \mathcal{B}(\mathbb{R}^d)$"
The questions are:
a) is there a more recent reference on the subject.
b) It is not clear to me how this conditions plays a role in the existence of a Markov chain with preassigned transition probabilities.
Isn't a direct consequence of the existence of $P(t,x, \Gamma)$ the existence of a Markov chain that has $P(\bullet,\bullet,\bullet)$ as its transition probabilities functions?
Is it the case that this condition assures that 
\begin{equation}
\tag{1}
\lim_{t \to 0} P (t,x,\Gamma) = \delta_x(\Gamma)?
\end{equation}
 c)Is there a case where  condition $(1)$ does not hold even though the transitions functions are derived from a semigroup $\{T_t\}_{t\geq 0}$ that satisfies
$$T_tf(x) - f(x) = \int_0^t T_u Lf(x) \, du $$
 A: (a) Dynkin is still a pretty standard reference for this kind of material.  
(b) The existence of a Markov process with transition function $P$ is an immediate consequence of the Kolmogorov extension theorem, yes.  But Kolmogorov might give you a process that is not strong Markov, not right continuous, or bad in other ways.  To get a "nice" process with transition function $P$, you have to work harder (and perhaps assume more about $P$).
The condition given definitely does not imply (1).  Consider for example the transition semigroup of one-dimensional Brownian motion, given by
$$P(t, x, \Gamma) = \int_{\Gamma} \frac{1}{\sqrt{2 \pi t}} e^{-|x-y|^2/(2t)}\,dy.$$
It's pretty easy to verify that this satisfies Stroock and Varadhan's condition.  But it does not satisfy (1): for example, if you take $\Gamma = \{x\}$, we have $P(t, x, \{x\}) = 0$ for all $t > 0$, yet $\delta_x(\{x\})=1$.
This also answers your question (c), since the Brownian semigroup $T_t$ is strongly continuous and has $Lf = -f''$ as its generator. 
