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 $\lambda$, and admitting a unique invariant measure for any value of $\lambda$ around $\lambda$ = 0. Our aim is to compute the derivative with respect to $\lambda$ of averages with respect to the invariant measure, at $\lambda$ = 0. We analyze a numerical method which consists in simulating the process at $\lambda$ = 0 together with its derivative with respect to $\lambda$ on 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 $\lambda$ of the mean of an observable through Monte Carlo simulations.