  We consider the sensitivity, with respect to a parameter \theta, of
parametric families of operators A_{\theta}, vectors \pi_{\theta} corresponding
to the adjoints A_{\theta}^{*} of A_{\theta} via A_{\theta}^{*}\pi_{\theta}=0
and one parameter semigroups t\mapsto e^{tA_{\theta}}. We display formulas
relating weak differentiability of \theta\mapsto \pi_{\theta} (at \theta=0) to
weak differentiability of \theta\mapsto A_{\theta}^{*}\pi_{0} and
[e^{A_{\theta}t}]^{*}\pi_{0}. We give two applications: The first one concerns
the sensitivity of the Ornstein--Uhlenbeck process with respect to its location
parameter. The second one provides new insights regarding the Wright--Fisher
diffusion for small mutation parameter.

Functional Analysis;Analysis of PDEs;Probability