4 Markov seeds and stochastic transfer
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Use column probability vectors. A finite continuous-time Markov generator \(Q\) has nonnegative off-diagonal entries and column sums zero. The stochastic semigroup is
Theorem 4.1: Finite Markov semigroup
If \(Q\) has nonnegative off-diagonal entries and column sums zero, then \(e^{tQ}\) has nonnegative entries and column sums one for every \(t\ge0\).
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Proof of Theorem 4.1
Choose \(\lambda\ge\max_i(-Q_{ii})\). Then \(B=Q+\lambda I\) has nonnegative entries, so \(e^{tB}\) is entrywise nonnegative by its power series. Hence
is nonnegative. Since \({\boldsymbol{1}}^TQ=0\), differentiation gives
so \({\boldsymbol{1}}^Te^{tQ}={\boldsymbol{1}}^T\).
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Example 4.2: Symmetric random walk
Starting from \(\delta_0\) and applying the transfer
for \(n\) steps gives
when \(k\equiv n\pmod2\), and zero otherwise. Hence
The parity packet is exact: all mass remains in the one residue class permitted by \(n\).