Doubly Damped Stochastic Parallel Translations and Hessian Formulas

  • Xue-Mei LiEmail author
Conference paper
Part of the Springer Proceedings in Mathematics & Statistics book series (PROMS, volume 229)


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.


Heat kernels Weighted laplacian Schrödinger operators Hessian formulas Hessian estimates 

AMS subject classification

60Gxx 60Hxx 58J65 58J70 


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© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of MathematicsImperial College LondonLondonUK

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