Finite-chain hitting time and survival law

Let $X$ be the finite state set, $P$ its row-transition matrix, $\mathcal{M}\subseteq X$ the marked set, $U=X\setminus \mathcal{M}$, and $P_{UU}$ the principal submatrix indexed by $U$. For an initial probability row vector $\mu$, the hitting-time survival law is

${\mathrm{Pr}}_{\mu }({\tau }_{\mathcal{M}}>t):={\mu }_U{P}_{UU}^t{\mathbf{1}}_U,t\in {\mathbb{N}}_0,$

where ${\mu }_U$ is the restriction of $\mu$ to $U$ and ${\mathbf{1}}_U$ is the all-ones column vector on $U$. This is the finite sum of probabilities of all length-$t$ paths that remain in $U$; it does not require a path-space construction. For $U=\mathrm{\varnothing }$ the right side is $0$; for $\mathcal{M}=\mathrm{\varnothing }$ it is $1$.

The hitting time ${\tau }_{\mathcal{M}}:=\inf \{t\ge 0:X_t\in \mathcal{M}\}$ is the ${\mathbb{N}}_0\cup \{\mathrm{\infty }\}$-valued variable determined by this survival law. Its extended expectation is ${\mathbb{E}}_{\mu }{\tau }_{\mathcal{M}}={\sum }_{t\ge 0}{\mathrm{Pr}}_{\mu }({\tau }_{\mathcal{M}}>t)$, and a constant-success hitting bound is a specified integer $t$ for which ${\mathrm{Pr}}_{\mu }({\tau }_{\mathcal{M}}\le t)$ is bounded below by a positive constant. These resource conventions are kept separate.