Finite masking diffusion chain

Fix a finite nonempty clean vocabulary $V$, a mask symbol $m\mathrm{\notin }V$, integers $L,N\ge 1$, a grid $0=t_0<t_1<\cdots <t_N=1$, a survival schedule $1={\alpha }_0>{\alpha }_1>\cdots >{\alpha }_N=0$, and a probability vector $\mu$ on $V^L$ (nonnegative entries summing to one). Put $S=(V\cup \{m\}{)}^L$ and $r_j={\alpha }_j\mathrm{/}{\alpha }_{j-1}$ for $1\le j\le N$. On one coordinate, the step-$j$ transition from $v\in V$ stays at $v$ with probability $r_j$ and moves to $m$ with probability $1-r_j$; from $m$ it stays at $m$ with probability one. All other one-coordinate transitions have probability zero. The sequence-state transition from $z$ to $z^{\mathrm{\prime }}$ in $S$ is the product of these one-coordinate probabilities over the $L$ coordinates. Starting with $Z_0=X\sim \mu$, the conditional path mass is $\mathrm{Pr}(Z_0=z_0,\dots ,Z_N=z_N\mid X=x)={\mathbf{1}}_{\{z_0=x\}}{\prod }_{j=1}^N{\prod }_{i=1}^L{a}_j(z_{j-1,i},z_{j,i}),$ where $a_j$ is the stated one-coordinate transition rule. Thus, conditional on the clean sequence, the coordinate trajectories are independent. At each step, the next-coordinate states are independent conditional on the preceding sequence state, with the transition probabilities above; successive states of a coordinate need not be independent. The mask is absorbing.

The following one-coordinate diagram records the three transition formulas at step $j$; each arrow label is its transition probability. The clean-state loop has probability $r_N=0$ at the terminal step $j=N$, so it is absent from the support there.

diagram failed to render:
\begin{tikzpicture}
  \node[draw,circle] (v) at (0,0) {$v\in V$};
  \node[draw,circle] (m) at (3,0) {$m$};
  \draw[->] (v.east) to[bend left=15] node[above] {$1-r_j$} (m.west);
  \draw[->] (v.north) .. controls (-0.8,1.4) and (0.8,1.4) .. node[above] {$r_j$} (v.north);
  \draw[->] (m.south) .. controls (2.2,-1.4) and (3.8,-1.4) .. node[below] {$1$} (m.south);
\end{tikzpicture}

Measure kernel and probability kernel: a probability kernel assigns a probability measure to each state and has measurable evaluation functions; on the finite power-set space above this is exactly a row-stochastic transition matrix.