# Different definitions of e-property for Markov-Feller chains

Let $X$ be a Polish space. We consider a stochastic kernel $P:X \times \mathcal{B}_X \to [0,1]$ and the Markov semigroup $(P^{\;n})_{n\geq1}$ of iterations of $P$, which satisfy the Chapman–Kolmogorov equation. Routinely, we define a dual operator $U^n$ related to $P^n$ as follows: $$U^n f(x) = \int\limits_{X} f(y)P^n(x,dy) ,$$ for any $f\in C(X)$. We also assume that $P$ has the Feller property; that means that $Uf \in C(X)$ for $f\in C(X)$.

I know three different definitions of so-called e-property and I wonder about the relationships between them.

(i) for all $f\in C(X)$ with bounded support the family $\{U^n f: n\in\mathbb{N}\}$ is uniformly equicontinuous on each compact subset of $X$,

(ii) for all $f\in C(X)$ with bounded support the family $\{U^n f: n\in\mathbb{N}\}$ is equicontinuous in each point of $X$.

(iii) for all Lipschitz functions $f \in C(X)$ with bounded support the family $\{U^n f: n\in\mathbb{N}\}$ is equicontinuous in each point of $X$.

I've proved that (ii) $\Rightarrow$ (i), the implication (ii) $\Rightarrow$ (iii) is obvious.

I think that the implication (i) $\Rightarrow$ (ii) is true only when $X$ is additionally locally compact, am I right or may be it is more general? If one can say something more about (iii)?

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interesting. I also added a tag stochastic processes - seems to be related. –  Ilya Aug 20 '11 at 21:09
It seems to me that the e-property was introduced by Szarek so why don't you drop him an email? –  Tomek Kania Mar 1 at 21:18