Suppose $M$ is a $2D$ random walk on a grid of squares where it travels in any direction uniformly, but if the direction traveled is out of bounds, it stays in the same location.
How would one prove that the uniform distribution is the stationary distribution for this case?
I found,
$$\pi_i = \frac{1}{4} (\sum_{j \in N(i)} \pi_j) + \frac{4-n}{4}(\pi_i)$$
where $N(i) =$ the set of neighbors of square i and $n =$ size of $N(i)$, but I'm not sure how to proceed.
edit:
$M$ is clearly irreducible and aperiodic
how does one prove its reversible as well?