# $\Gamma$-convergence for a class of action functionals induced by gradients of convex functions

### Luigi Ambrosio

Scuola Normale Superiore, Pisa, Italy### Aymeric Baradat

Université Lyon 1, Villeurbanne, France### Yann Brenier

École Normale Supérieure, Paris, France

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## Abstract

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.

## Cite this article

Luigi Ambrosio, Aymeric Baradat, Yann Brenier, $\Gamma$-convergence for a class of action functionals induced by gradients of convex functions. Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. 32 (2021), no. 1, pp. 97–108

DOI 10.4171/RLM/928