A walk on the integers $\mathbb{Z}=\{\ldots, -2,-1,0,1,2, \ldots\}$ is said to be spatially homogeneous if for $ij$th entry of the transition matrix $P$, we have that $p_{ij}=\mu(j-i)$ for some probability measure $\mu$ on $\mathbb{Z}$. Another way to think of this is that for an initial distribution $\mu$, we have that the transition probability $p(i,j) = \mu(j-i)$.
I want to show that if we assume that $\mu(0) \neq 1$ then no stationary process exists.
One approach that I've tried, is that if we assume the existence of a stationairy distribution $\pi$ where $\pi(0) \neq 1$, then the random variables $X_{n}$ which dictate the position, we have that $E(X_{0})=E(X_{n})$ for any $n$, since $\pi$ is stationary also, we have that $Var(X_{0})=Var(X_{n})$, for any $n$. But I don't quite see a contradiction.