Computation of sensitivities for the invariant measure of a parameter dependent diffusion

We consider the solution to a stochastic differential equation with a drift function which depends smoothly on some real parameter λ, and admitting a unique invariant measure for any value of λ around λ= 0. Our aim is to compute the derivative with respect to λ of averages with respect to the invariant measure, at λ= 0. We analyze a numerical method which consists in simulating the process at λ= 0 together with its derivative with respect to λ on a long time horizon. We give sufficient conditions implying uniform-in-time square integrability of this derivative. This allows in particular to compute efficiently the derivative with respect to λ of the mean of an observable through Monte Carlo simulations.