Let $0<p<1$. I want to find all invariant measures $\pi$ for the asymmetric random walk on $Z$ given by the Q-matrix, that is defined as follows:
$q(x,x-1)=1-p, ~~q(x,x)=-1, ~~q(x,x+1)=p~~$ and zero elsewhere
For me a measure $\pi$ on $Z$ is invariant if
$\forall y\in Z, t>0: \pi(y)=\sum_{z\in Z}\pi(z) p_t(z,y)$,
where $p_t(z,y)$ is the probability that the Markov chain starting in $z$ takes the value $y$ at time $t$.
Except of the trivial one, that is $\pi\equiv c$ for $c\geq 0$ I can't find anything. But I'm quite sure that at least there exists no invariant distribution here.
Thanks in advance.