$\{X_{i}(\Delta )\}_{i=1}^{\infty }$ is an ergodic process with state space $% \mathbb{R}$, and $X_{i}(\Delta )$ depends on a parameter $\Delta \in (0,1]$. $X_{i}(\Delta )$ has a unique stationary distribution $\pi _{\Delta }$, which also depends on $\Delta $.
Given a smooth and bounded function $f$ on $\mathbb{R}$. For any integer $% n\geq 1$, $\forall x\in \mathbb{R}$, and $\forall\Delta >0$, we define \begin{equation*} F_{n}(\Delta ):=\mathbb{E[}f(X_{n}(\Delta ))|X_{0}(\Delta )=x], \end{equation*} and define \begin{equation*} F(\Delta ):=\int_{\mathbb{R}}f(x)\pi _{\Delta }(dx). \end{equation*} Now suppose that:
(1) for any $n\geq 1$, $F_{n}(\Delta )$ is a smooth function for $\Delta \in (0,1]$.
(2) for any integer $k\geq 0$, \begin{equation*} \text{the limit }\lim_{\Delta \rightarrow 0}\frac{d^{k}F_{n}(\Delta )}{% d\Delta ^{k}}\text{ exists.} \end{equation*}
(3) (geometric ergodicity) there exists $\alpha >0$ such that, for any $n\geq 1$ and $\Delta >0$, \begin{equation*} \left\vert F_{n}(\Delta )-F(\Delta )\right\vert <\exp (-\alpha n\Delta ). \end{equation*}
The Question is to prove: \begin{equation*} \text{the limit }\lim_{\Delta \rightarrow 0}F(\Delta )\text{ exists.} \end{equation*}
My idea: for any $\Delta >0$, by ergodic theorem we have \begin{equation*} \lim_{n\rightarrow \infty }F_{n}(\Delta )=F(\Delta ), \end{equation*} but this convergence is pointwise for $\Delta >0$. However, it is still hard to say something about the limit $\lim_{\Delta \rightarrow 0}F(\Delta )$.
Also, the condition (1) implies that, for any $n\geq 1$, $F_{n}(\Delta )$ can be extended to $% \Delta =0$, i.e., there exists a smooth function $\bar{F}_{n}(\Delta )$ on $% [0,\Delta ]$ such that $\bar{F}_{n}(\Delta )=F_{n}(\Delta )$ for any $\Delta \in (0,1]$.