This question might be super stupid and dumb, but I still have to ask. I thought equilibrium distribution only exists for finite states Markov chain. But my course book did give me a example with state space $S=\{0,1,2,... \}$. 
It talked a lot, but somehow appears to claim there is no equilibrium distribution. Because it violates the condition that $\sum_{i\in S}\pi_i=1$
I am completely confused by it. Does equilibrium distribution for Markov Chain with infinite many state space exist, does it always exist or conditionally exist, or never?
In that particular case there is no equilibrium distribution (rather obviously there is a tendency to drift right)
But suppose instead, for example you had $$X_{n+1}= \left\{\begin{array}{cl} X_{n}-1 & \text{with probability } \frac{X_n}{X_n+2} \\ X_n & \text{with probability } \frac{1}{X_n+2} \\ X_{n}+1 & \text{with probability } \frac{1}{X_n+2} \end{array}\right.$$
then there would be a stable distribution on the non-negative integers
In fact this seems to have a stable distribution of $\pi_j = \dfrac{j +2}{3e j!}$