Determining the Stationary Distribution for a Markov Chain with a Countable State Space

Question

I am trying to determine a stationary distribution for the following problem:

Find the stationary distribution of the Markov chain with the countable state space ${0, 1, 2,...,n,...}$, where each point, including $0$, can either return to $0$ with probability $1/2$ or move to the right $n \to n+1$ with probability $1/2$.

I know a stationary distribution must satisfy $\pi P = P$ and from the ergodic theorem $lim_{n \to \infty} p_{ij}^{n}= \pi_{j}.$ How can I apply these principles to determine the stationary distribution in this specific example?

Answer 1

Solve the system $\pi= \pi P$. \begin{align} \pi(0) &= \frac{1}{2} \sum_{i=0}^\infty \pi(i)\\ \pi(1) &= \frac{1}{2} \pi(0)\\ \pi(2) &= \frac{1}{2} \pi(1)\\ &\vdots \end{align}