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I am doing some exercises in E-Li-Vanden-Eijnden's Applied Stochastic Analysis and I meet this problem:

(Exercise 3.19) Consider an irreducible Markov chain $\{X_n\}$ on a finite state space $S$. Let $H\subset S$ and define the first passage time $T_H=\inf\{n:X_n\in H\}$, $h_i=P_i(T_H<\infty)$. Prove that $\boldsymbol{h}=\{h_i\}_{i\in S}$ (column vector) satisfies $(\boldsymbol{I}-\boldsymbol{P})\cdot\boldsymbol{h}=0$ with boundary condition $h_i=1$ for $i\in H$, where $\boldsymbol{P}$ is the transition matrix, and $\boldsymbol{I}$ is the identity matrix.

I have been struggling for this problem for hours and do not have much progress. The only thing that I can do currently is, as suggested by the book, to define the Laplace matrix $\boldsymbol{L}=-(\boldsymbol{I}-\boldsymbol{P})$ by acting on any function $$(\boldsymbol{L}f)(i)=\sum_{j\in S}p_{ij}(f(j)-f(i))$$, where $p_{ij}$ are entries of $\boldsymbol{P}$.

According to this, I write $$(\boldsymbol{I}-\boldsymbol{P})\cdot\boldsymbol{h}=(-\boldsymbol{L}h)(i)=\sum_{i,j\in S}p_{ij}(h(i)-h(j))$$

but starting from here, I get stuck. Can anyone provide any solutions regarding this? Thank you.

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You want to prove that the vector of hitting probabilities satisfies $h = Ph$, with the boundary condition $h_i=1$ for $i\in H$.

If the chain starts in $H$, you're guaranteed to hit it, so the boundary condition follows.

If you start outside $H$, you can condition on the first step (notation $P_i(\cdot) = P(\cdot | X_0 = i)$):

$$h_i=P_i(T_H < \infty) = \sum_{j\in S}P_i(T_H<\infty, X_1 = j)=\sum_{j\in S}P_i(T_H<\infty| X_1 = j)P(X_1=j) = \sum_{j\in S}h_jp_{ij}=\sum_{j\in S}p_{ij}h_j,$$

where we have used that $P_i(T_H<\infty| X_1 = j)=P_j(T_H<\infty)=h_j$ by the Markov property.

If you're interested, $h$ is in fact the minimal solution to this system (in the sense that each vector component is minimal).

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  • $\begingroup$ Thank you for the answer. I have some questions to follow up, since I am completely new to Markov chain. Can you provide more details to the first case? I tried but cannot express it using $p_{ij}$'s. For the second case, is it true that the events $\{T_H<\infty\}$ and $\{X_i=j\}$ are independent? Can you provide more details explaining that $\boldsymbol{h}$ is the minimal solution to the BVP? And did you use Chapman-Kolmogorov in the second case? $\endgroup$ – Mike Feb 23 at 13:27
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    $\begingroup$ 1) I don't think $(I-P)h=0$ is guaranteed to be satisfied for those indexed $h_i$ with $i\in H$ (the chain already starts in $H$). 2) See edit. 3) You should ask another question for this. 4) I just looked up C-K now, and it looks like a simple "marginalisation" / "integration" / "summation" over something, which is basically what I used in the 2nd "=" in the 2nd case. $\endgroup$ – Benjamin Wang Feb 23 at 14:12
  • $\begingroup$ Can you explain more about 1)? I still do not quite understand your point. $\endgroup$ – Mike Feb 23 at 14:34

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