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Viscosity solutions of Hamilton-Jacobi equation in $RCD(K,\infty)$ spaces and applications to large deviations

Authors :
Gigli, Nicola
Tamanini, Luca
Trevisan, Dario
Publication Year :
2022

Abstract

The aim of this paper is twofold. - In the setting of RCD(K,$\infty$) metric measure spaces, we derive uniform gradient and Laplacian contraction estimates along solutions of the viscous approximation of the Hamilton--Jacobi equation. We use these estimates to prove that, as the viscosity tends to zero, solutions of this equation converge to the evolution driven by the Hopf--Lax formula, in accordance with the smooth case. - We then use such convergence to study the small-time Large Deviation Principle for both the heat kernel and the Brownian motion: we obtain the expected behavior under the additional assumption that the space is proper. As an application of the latter point, we also discuss the $\Gamma$-convergence of the Schr\"odinger problem to the quadratic optimal transport problem in proper RCD(K,$\infty$) spaces.

Details

Database :
arXiv
Publication Type :
Report
Accession number :
edsarx.2203.11701
Document Type :
Working Paper