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 diffusion process of drift del 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 is an element of R and converges towards f in the sense of Mosco convergence, then the related functionals Gamma-converge in the strong topology of curves.