Γ-convergence for a class of action functionals induced by gradients of convex functions

Given a real function f , the rate function for the large deviations of the di usion process of drift '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 (fh)h is uniformly λ-convex for some λ ∈ R and converges towards f in the sense of Mosco convergence, then the related functionals G-converge in the strong topology of curves.