Doubly Damped Stochastic Parallel Translations and Hessian Formulas

We study the Hessian of the solutions of time-independent Schr"odinger equations, aiming to obtain as large a class as possible of complete Riemannian manifolds for which the estimate $C(\frac{1}{t}+\frac{d^2}{t^2})$ holds. For this purpose we introduce the doubly damped stochastic parallel transport equation, study them and make exponential estimates on them, deduce a second order Feynman-Kac formula and obtain the desired estimates. Our aim here is to explain the intuition, the basic techniques, and the formulas which might be useful in other studies.