Definition
Let $(X_n)_{n\ge 0}$ to be MarkovChain on statespace $S$.
Hitting time was defined by:
$$ T_j=\inf\{n\ge 0:X_n=j\} $$for random variable $T_j$.
Consequence
$T_j$ represents the # of steps it takes to reach state $j$ from a given current state.
Note that $n=0$ is allowed here, for $X_0=j$ we have $T_j=0$.
Remark
Given $i$, Given $j$, and fix $T_j=n$, if seeking for $P(T_j=n)$ for $(n\in 0,1,2,\cdots)$, we get $T_j$’s distribution.