We are in the setting of a continuous time MC, as defined by Liggett in his book on continuous time markov processes, on a countable state space $S$. All of his MCs are defined on the space of right continuous paths from $[0, \infty)\rightarrow S$ with finitely many jumps in any finite interval. This space is endowed with the smallest $\sigma$ algebra making all the coordinate maps continuous.
To specify an MC, one specifies a filtration in this space, as well as a set of probability measures $P^x$ so that if $X_t$ are the coordinate maps, then one has $P^x(X_0=x)=1$, the Markov property, $X_t$ adapted to the filtration. This can be found in Liggett Ch. 2.1 There is nothing debatable about his other definitions, I only felt the need to mention MC because he defines it on a particular space and I want to stick to that throughout. I do wonder if his definition of MC is at least standard in the sense that although other people may not pick the same space, they require essentially the same path properties? I have yet to grasp a real reason for right continuity, even though it has been used a number of times, I'm not sure exactly which parts of the correspondence between MC, transition function, and $Q$ matrix would fail without the path regularity.
In Ch 2.5 he constructs $X_t$ out of a $Q$ matrix. When the minimal solution of the Kolmogorov Backward Equations is stochastic, then he claims $X_t$ is a Markov Chain. I have trouble proving that it satisfies the markov property. Here, $X_t$ is constructed as follows (although all this can be found in p. 73 and after in his book)
Let $p(x, y)=\delta_{xy}$ if $c(x)=-q(x, x)=0$ else $p(x, y)=q(x, y)/c(x)$ when $x\neq y$ and $c(x)>0$ else $p(x, y)=0$.
Let $Z_n$ be an MC with this kernel and an initial distribution which has a strictly positive probability for each point in $S$. Conditioned on each $Z_k$ we introduce $\tau_k$ where conditioned on the MC the family of $\tau$ is independent, with all of them being exponentially distributed with parameter $c(Z_k)$. Then we say that $N_t$ is the smallest $n$ such that the sum of the first $n$ $\tau$'s is $>t$. Define $X_t=Z_{N_t}$ which in our case is a global definition up to null sets by a theorem he proves. That is our $X_t$.
Any answer to either the question about the necessity of path regularity, or the Markov property, or both are appreciated.