I am considering the time homogeneous Ito diffusion:
$$dX_t=b(X_t)dt+\sigma(X_t)dW_t,$$ where $b,\sigma$ currently are only assumed to be global Lipschitz for the existence of solution $X_t$.
The associated transition semigroup $P_tf(x)=E^x[f(X_t)]$ for bounded Borel $f$. It's well-known that the continuous bounded functions $C_b(\mathbb{R}^n)$ is invariant under $P_t$.Hence $P_t$ defines a semigroup on $C_b(\mathbb{R}^n)$ with $C_b^2(\mathbb{R}^n)\subset D(A)$, the domain of the generator of $P_t$.
But it's also well-known that $P_t$ may not be strongly continuous on $C_b(\mathbb{R}^n)$. However, if $C_0(\mathbb{R}^n)$ is invariant under $P_t$ then $P_t$ is a Feller semigroup and the restriction of $P_t$ on $C_0(\mathbb{R}^n)$ is strongly continuous.
On the other hand, there are some criterions that $\{(f,Lf)\mid f\in C_c^\infty(\mathbb{R}^n)\}$, where $L$ be the elliptic operator $L$ associated with $X_t$ generates a Feller semigroup $T_t$ on $C_0(\mathbb{R}^n)$.
May I know is it necessary that $T_t=P_t$ on $C_0(\mathbb{R}^n)$?