  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 behaviour under the additional assumption that
the space is proper.
