  Given a real function $f$, the rate function for the large deviations of the
diffusion process of drift $\nabla f$ given by the Freidlin-Wentzell theorem
coincides with the time integral of the energy dissipation for the gradient
flow associated with $f$. This paper is concerned with the stability in the
hilbertian framework of this common action functional when $f$ varies. More
precisely, we show that if $(f_h)_h$ is uniformly $\lambda$-convex for some
$\lambda \in \mathbb{R}$ and converges towards $f$ in the sense of Mosco
convergence, then the related functionals $\Gamma$-converge in the strong
topology of curves.
