Doubly Damped Stochastic Parallel Translations and Hessian Formulas

TL;DR

This paper develops a stochastic approach to analyze the Hessian of solutions to Schrödinger equations on Riemannian manifolds, introducing new techniques and formulas for better estimates.

Contribution

It introduces the doubly damped stochastic parallel transport equation and derives new second order Feynman-Kac formulas for Hessian estimates.

Findings

Established exponential estimates for the doubly damped stochastic parallel transport.

Derived a second order Feynman-Kac formula for Hessian analysis.

Identified classes of manifolds satisfying specific Hessian bounds.

Abstract

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.