Consider the following Ito diffusion $X_t$ satisfying

$$dX_t=b(X_t)dt+\sigma(X_t)dB_t,\quad X_0=x\in \mathbb{R}^n,$$

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 $C_b$-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. I have also shown that one dimensional Ito diffusion has Feller property using a sufficient condition given in Lorenzi's book.

May I know whether in general the Ito diffusion $X_t$ (SDE with global Lipschitz coefficients) satisfying this property?