hitting time

Let $(X_{n})_{n\geq 0}$ be a Markov Chain. Then the hitting time for a subset $A$ of $I$ (the indexing set) is the random variable:

H^{A}=\inf\{n\geq 0:X_{n}\in A\}

(set $\inf\varnothing=\infty$ ).

This can be thought of as the time before the chain is first in a state that is a member of $A$ .

Wite $h_{i}^{A}$ for the probability that, starting from $i\in I$ the chain ever hits the set A:

h_{i}^{A}=P(H^{A}<\infty:X_{0}=i)

When A is a closed class, $h_{i}^{A}$ is the absorption probability.