Viscosity Solutions of Hamilton–Jacobi Equation in RCD(K,∞) Spaces and Applications to Large Deviations

The aim of this paper is twofold. In the setting of RCD(K,∞) 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 Γ-convergence of the Schrödinger problem to the quadratic optimal transport problem in proper RCD(K,∞) spaces.