Inserted: 7 jun 2022
Last Updated: 7 jun 2022
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.