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

Sensitivity analysis of one parameter semigroups exemplified by the Wright--Fisher diffusion