Let
- $E$ be an at most countable set equipped with the discrete topology $\mathcal E$
- $X=(X_n)_{n\in\mathbb N_0}$ be a discrete Markov chain with values in $(E,\mathcal E)$, distributions $(\operatorname P_x)_{x\in E}$ and transition matrix $$p=\left(p(x,y)\right)_{x,y\in E}:=\left(\operatorname P_x\left[X_1=y\right]\right)\;.$$
- $\tau_x^1:=\inf\left\{n\in\mathbb N:X_n=x\right\}$ and $$\varrho(x,y):=\operatorname P_x\left[\tau_y^1<\infty\right]$$
Suppose $\pi$ is an $p$-invariant measure on $(E,\mathcal E)$, i.e. $$\pi p=\pi\;,$$
where $$\pi p\left(\left\{x\right\}\right):=\sum_{y\in E}\pi\left(\left\{y\right\}\right)p(y,x)\;\;\;\text{for }y\in E\;.$$ Now, let $$\operatorname P_\pi:=\sum_{x\in E}\pi\left(\left\{x\right\}\right)\operatorname P_x\;.$$
Let $X$ be irreducible, i.e. $$\varrho(x,y)>0\;\;\;\text{for all }x,y\in E$$ and positive recurrent, i.e. $$\operatorname E_x\left[\tau_x^1\right]<\infty\;\;\;\text{for all }x\in E\;.$$ Let $$\sigma_x^n:=\sup\left\{m\le n:X_m=x\right\}\in\color{blue}{\mathbb N_0\cup\left\{-\infty\right\}}$$ $\color{blue}{\text{be the last entrance time into }x\in E\text{ until }n\in\mathbb N_0}$. I don't understand why the (weak? elementary?) Markov property implies the $\color{red}{\text{red}}$ equality in \begin{equation} \begin{split} \operatorname P_\pi\left[\sigma_x^n=k\right]&=\operatorname P _\pi\left[X_k=x\text{ and }X_{k+1}\ne x,\ldots,X_n\ne x\right]\\ &=\operatorname P_\pi\left[X_{k+1}\ne x,\ldots,X_n\ne x\mid X_k=x\right]\operatorname P_\pi\left[X_k=x\right]\\ &\color{red}{=\pi\left(\left\{x\right\}\right)\operatorname P_x\left[X_1\ne x,\ldots,X_{n-k}\ne x\right]}\;. \end{split}\tag 1 \end{equation}
The weak Markov property states, that $$\operatorname E_x\left[f\circ (X_{k+n})_{k\in\mathbb N_0}\mid X_0,\ldots,X_k\right]=\operatorname E_{X_k}\left[f\circ X\right]\;\;\;\operatorname P_x\text{-almost surely}\tag 2$$ for all $x\in E$, $k\in\mathbb N_0$ and bounded, $\mathcal E^{\otimes\mathbb N_0}$-measurable $f:E^{\mathbb N_0}\to\mathbb R$.
How can we derive $(1)$ from $(2)$?