Feller property of Ito diffusion 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.
 A: If $b$ and $\sigma$ are Lipschitz continous functions, then $(X_t)_{t \geq 0}$ has the Feller property, i.e. $x \mapsto \mathbb{E}^x(g(X_t))$ is continuous and vanishing at $\infty$ for any function $g$ which is continuous and vanishing at $\infty$. For a proof see for instance René Schiling & Lothar Partzsch: Brownian Motion - An Introduction to Stochastic Processes, Corollary 19.27 and 19.31 (2nd edition).
Your proof, however, is not correct. More precisely, equation $(1)$ is not correct, it should read
$$X_t^x(\omega) = x+\int_0^t b(X_s^{\color{red}{x}}(\omega)) \, ds + \int_0^t \sigma(X_s^{\color{red}{x}}) \, dB_s(\omega);$$
in particular the two integrals on the right-hand side depend on $x$, and therefore we cannot simply let $x \to \infty$.
Remark $(X_t)_{t \geq 0}$ does, in general, not satisfy the Feller property if $\sigma$ and $b$ are not (globally) Lipschitz continuous, but only locally Lipschitz continuous. Consider for instance the solution of the differential equation
$$dX_t = - X_t^3 \, dt \qquad X_0 = x$$
which can be calculated explicitly.
