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](https://en.wikipedia.org/wiki/Feller-continuous_process).


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? 



Feller property for Ito diffusion with Lipschitz coefficients
Under the hypotheses:


1. $b$ and $\sigma$ are globally Lipschitz continuous;
2. $\sigma(x) \sigma(x)^T$ is positive definite for all $x \in \mathbb{R}^n$;


one can prove that the semigroup $P\_t$ associated to $X$ is, in fact, *strong* Feller, meaning that for any $t \ge 0$ the semigroup $P\_t$ maps $B\_b(\mathbb{R}^n)$ (bounded functions) to $C\_b(\mathbb{R}^n)$ ( bounded and continuous functions). This result is typical of SDEs with non-degenerate noise. See, e.g., Chapter 12 of Giuseppe Da Prato (2014). **Introduction to Stochastic Analysis and Malliavin Calculus.** The Feller property is weaker, and as you say, just requires that for any $t \ge 0$ the semigroup $P\_t$ maps $C\_b(\mathbb{R}^n)$ into itself.


Some books discuss the related, but different, concept of Feller-Dynkin processes where $C\_b(\mathbb{R}^n)$ is replaced with $C\_0(\mathbb{R}^n)$, the set of continuous functions vanishing at infinity. This overloading of terminology is unfortunate. Since we have the inclusion $ C\_0(\mathbb{R}^n) \subset C\_b(\mathbb{R}^n)$, the notion of a Feller process is a bit more general than a Feller-Dynkin process.


