Prove that the random walk X is null recurrent I am having trouble showing the relationship of stationary distribution of this random walk.
Question:
Consider a random walk $X$ on $\mathbb{Z}$ defined by $p(1)=p(0)=p(-1)=1 / 3$ (note that $\left.p_{x, y}=p(y-x)\right)$. Define
$$
\begin{aligned}
T &:=\inf \left\{n \geq 1: X_{n}=0\right\} \\
q_{i} &:=P_{i}(T=\infty)
\end{aligned}
$$
Prove that the random walk $X$ is null recurrent by showing that it cannot have any stationary distributions.
My attempt:
Edit
With the help of Teresa Lisbon, I get that the recursive relationship is $$\pi(k)=\frac{1}{3} \pi(k)+\frac{1}{3} \pi(k-1)+\frac{1}{3} \pi(k+1)$$
This simplifies to the relationship of simple symmetric random walk where:
$$\pi(k)=\frac{1}{2} \pi(k-1)+\frac{1}{2} \pi(k+1)$$
I see that this is a first order difference equation with the answer given as $$\pi_{k}=a+b k$$ for some constants a and b. But now I am a bit stuck as I don't know what is the boundary condition.

Clearly it is a slow down version of simple random walk on $\mathbb{Z}$. So it is recurrent.
Consider the measure $\pi_i=1$ for all $i$. Then
$\pi_i = \frac{1}{3}\pi_{i-1} + \frac{1}{3}\pi_{i+1} + \frac{1}{3}\pi_i$,
which means $\pi = P \pi$, $P$ is the transition matrix.
So $\pi$ is invariant.
By Theorem 1.7.6 and Theorem 1.7.7 from J.R. Norris's Markov Chain, any invariant measure should be a scalar multiple of $\pi$. Since $\sum_{i\in\mathbb{Z}} \pi_i = \infty$, so there can be no invariant distribution, therefore the random walk is null recurrent.
