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

We consider the sensitivity, with respect to a parameter θ, of parametric families of operators A_θ, vectors π_θ corresponding to the adjoints A_θ^{*} of A_θ via A_θ^{*}π_θ=0 and one parameter semigroups t\mapsto e^{tA_θ}. We display formulas relating weak differentiability of θ\mapsto π_θ (at θ=0) to weak differentiability of θ\mapsto A_θ^{*}π_{0} and [e^{A_θt}]^{*}π_{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.

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