This was a task in an exam I had in July and I wuld like to know how to solve this task.
Given: Markov chain $(X_n)_{n\ge 0}$ with state space $I=\{0,1,\ldots\}$, initial distribution $\lambda=\delta_0$ and transition probabilities $$p_{00}=1/3,\ p_{01}=2/3,\ p_{i,i-1}=1/3,\ p_{i,i+1}=2/3,\ i\ge1$$ and all other transition probabilities equal $0$.
Let $H=\inf\{n\ge 0: X_n=0\}$ be the hitting time for the state $0$ and $h_i=\mathbb P_i[H<\infty]$ the according hitting probability. Using the weak markov-property we get a (infinite) system of linear equations.
Find this system of equations and show that its general solution is
$$x_i=-1+2^{1-i}+\xi(2-2^{1-i}),i\ge0, \xi \in \mathbb R$$
Which choice of $\xi$ yiels the hitting probabilities $(h_i:i\in I)$?
Theory:
