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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

$$T_tf(x) - f(x) = \int_0^t T_u Lf(x) \, du $$

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(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.

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