 Solution

Write $u_n=\mathbb{P}(X_n=0\mid X_0=0)$. The Markov property, conditioning on the state at time $n$, givesPut $r=\alpha +\beta -1$ and ${\pi }_0=(1-\beta )/(2-\alpha -\beta )$. Then ${\pi }_0=r{\pi }_0+1-\beta$, so subtraction yields $u_{n+1}-{\pi }_0=r(u_n-{\pi }_0)$. Iterating this linear recurrence givesFor $r=0$, read the $n=0$ expression using $r^0=1$, or state $u_0=1$ separately; for $n\ge 1$, $u_n={\pi }_0$. Since $|r|<1$, the result also shows convergence in a metric space to the stationary distribution probability ${\pi }_0$.