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Doubly Damped Stochastic Parallel Translations and Hessian Formulas
Xue-Mei Li · 2017-10-14 · via math.PR updates on arXiv.org

We study the Hessian of the solutions of time-independent Schrödinger 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.

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