In the spde literature we have results of the form $$|P_{t}F(x)-\mu(F)|\leq O(g(t)),\text{for all } x\in H, F\in S$$

where $P_t$ is a semigroup, $H$ some Hilbert space, $F\in S$ some function space, $\mu$ is an invariant measure and $g(t)\to 0$ as $t\to \infty$. I would be interested to hear of settings where one has achieved or needed estimates of the form:

$$|P_{t}F(x)-\mu(F)-g_{1}(t)|\leq O(g(t)),\text{for all } x\in H,F\in S,$$

where $g_1(t)\to 0$ but captures some information on the derivative $\partial_t P_t F(x)$, when it exists, as in usual taylor expansion. Or better something akin to a Laurent series where we make the substitution $t\to \frac{1}{t}$.