# Infinitesimal Generator of semigroup for markov chain

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?

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 $$P_x$$.
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.