Infinitesimal Generator of semigroup for markov chain

Question

In Wikipedia the definition of generator of a semigroup is given as follows

For a Feller process $(X_t)_{t\ge0}$ with Feller Semigroup $T=(T_t)_{t\ge0}$ and state space $E$ we define the generator $(A,D(A))$ by:
$$D(A)=\left\{f\in C_\infty :\lim_{t↓0}\frac{T_tf-f}{t}\text{exists as a uniform limit}\right\},$$

I saw another definition of infinitsimal generator in context of markov chains which says that if

$$d\phi(X_t)=A \phi(X_t)dt+dM_t$$

where $M_t$ is a martingale and $\phi \in C_c^{\infty}(\mathcal{X})$ and {$X_t$} is a markov chain on $\mathcal{X}$ then A is the infinitesimal generator.
Are these two definitions equivalent?

Answer 1

I think this theorem is useful, it is from Continuous Time Markov Processes by Liggett:

Theorem: Let $X(t)$ be a Feller process on a state space $S$ and let $A$ be its generator with domain $D(A)$. Let $(P_x)_{x \in S}$ be the collection of path distributions - in case you are not familiar with this general definition, $P_x$ is the measure under which $X$ starts at $x$ almost surely. Then for every $x \in S$ and $f \in D(A)$: $$M(t) := f(X(t)) - \int_{0}^{t} (Af)(X(s)) ds$$ is a martingale under the measure $P_x$.

Note that the Hille-Yosida-theorem states that every Feller process corresponds to a generator and vice versa. However, not every Markov process is a Feller process - which in addition to the Markov property satisfies the following condition:

Feller Property: $\forall f \in \mathcal{C}(S)$ and $t\ge 0$ the function $\phi: S \rightarrow \mathbb{R}$ given by $$\phi(x) = \mathbb{E}_{P^{x}}[f(X(t))]$$ is also in $\mathcal{C}(S)$, i.e. continuous and vanishing at infinity.