I am reading the following paper: http://arxiv.org/pdf/1602.03849v2.pdf
I will explain the general setup below.
Let $x\in X=\mathbb{T}^d$, where $\mathbb{T}^d$ is the d dimensional torus. Let $\rho$ be a Borel prob. measure on SL$_d(\mathbb{Z})$, and set $X_0=x$, and $X_{n+1}=g_{n+1}X_n$ where $(g_n)\subset$ SL$_d(\mathbb{Z})$ is chosen with law $\rho^{\otimes\mathbb{N}}$. Let $\mathbb{P}_x$ be the measure on $X^\mathbb{N}$ associated with the random walk starting at $x$.
For $f$ Borel on $X$, $f\geq 0$, and $x\in X$, define the Markov operator $Pf(x)=\int_{SL_d(\mathbb{Z})}f(gx)d\rho(g).$
The claim with which I can't seem to show is the following:
$\mathbb{E}_x[f(X_n)]=P^nf(x)$, or more generally $\mathbb{E}_x[f(X_n)]=P^{n-k}f(X_k)$ for $k\in\mathbb{N}$, $k\leq n$.
Thank you!