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Luca Tamanini

Monday 16 December 2019

Luca Tamanini (CEREMADE, Université Paris-Dauphine)
Large Deviations via Gamma-convergence on metric measure spaces

Abstract :
In the Euclidean space (and even on a smooth Riemannian manifold with Ricci curvature bounded from below) it is well known that the Brownian motion satisfies a Large Deviations Principle (LDP) and this can be derived in many different ways, also relying on strong tools such as heat kernel estimates. In this talk we aim at proving that the same is true on RCD(K,\infty) spaces, namely metric measure spaces with Ricci curvature bounded from below in the sense of Lott-Sturm-Villani and essentially Riemannian structure.

The non-smooth structure carries with it several technical difficulties (e.g. lack of local compactness and heat kernel estimates), preventing us to follow the strategies working in the Riemannian framework. Nonetheless, in great generality there exists a strong interplay between LDP, viscosity solutions of the Hamilton-Jacobi equation and Gamma-convergence. For this reason, a particular emphasis will be put on such an interplay, which will allow us to prove a LDP for both the heat kernel and the Brownian motion.

(joint work with N. Gigli and D. Trevisan)