Consider the following Ito diffusion $X_t$ satisfying
$$dX_t=b(X_t)dt+\sigma(X_t)dB_t,\quad X_0=x,$$
with Lipschitz coefficients $b,\sigma$.
It can be shown that if $g$ is bounded and continuous, then $u(x)=E^x[g(X_t)]$ is continuous. So any Ito diffusion is Feller continuous.
However, some books define Feller property to be if $g$ is continuous and vanishes at infinity, then $u$ is continuous and vanishes at infinity.
It seems that Brownian Motion satisfies this property. Also some literature shows if the associated generator is uniformly elliptic then it has a transition density and $X_t$ has this property.
May I know whether in general $X_t$ satisfying this property?
Following Danielsen's suggestion, I formulated the following proof, may you help to verify whether it's correct? In particular, I am not sure whether equation (1) is correct, since those two integrals are not fixed constants as $x\to \infty$.
Fix $g$ be continuous and vanishes at infinity, and $t>0$.
Then as $x\to \infty$, $$X^x_t(\omega)=x+\int_0^t b(X_s(\omega))\,ds+\int_0^t \sigma(X_s)\,dB_s(\omega)\to \infty,\tag{1}$$ for almost all $\omega$,
hence $g(X^x_t(\omega))\to 0$ a.s. then by boundedness convergence theorem,
we have $E^x[g(X_t)]=E^0[g(X^x_t)]\to 0$.
So Ito diffusion has the Feller property.