Does a simple random walk on $\Bbb Z^2$ admit a stationary distribution?

Question

Can somebody answer me how I can see if a simple symmetric random walk on $\Bbb Z^2$ admits a stationary distribution? I know how to prove that this random walk is recurrent but I don't know how to see if admits a stationary distribution.

Answer 1

A stationary distribution for the simple symmetric random walk on $\mathbb Z^2$ would have to fulfill the discrete Laplace equation. The solutions of the discrete Laplace equation on $\mathbb Z^2$ are the bilinear functions. The only non-negative bilinear function on $\mathbb Z^2$ is the zero function. Thus there can be no stationary distribution for the simple symmetric random walk on $\mathbb Z^2$.