Proposition: Finite generator perturbation

Let $Q,\hat{Q}$ be Markov generators on the same finite state space, and suppose

If initial laws have total-variation distance at most $\eta$, their one-time laws at time $u$ differ by at most $\eta +u\delta$.

Their joint laws at times $w,w+\ell$ differ by at most

A score in $[-v_n,v_n]$ consequently has expectation difference at most

or the trivial bound $2v_n$, whichever is smaller.

Proof. Duhamel's formula expresses the semigroup difference as the integral of

Stochastic multiplication contracts the total variation of signed measures, while multiplication by $Q-\hat{Q}$ costs at most $\delta$ for a probability input. Integration and the initial-law comparison give the first bound.

For a two-time joint law, first compare the law at time $w$, then the conditional transition over the remaining lag. The two allowances add. The bounded-score expectation inequality gives the final assertion.