$Γ$-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 $\nabla f$ 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 $λ$-convex for some $λ\in \mathbb{R}$ and converges towards $f$ in the sense of Mosco convergence, then the related functionals $Γ$-converge in the strong topology of curves.

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