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Doubly Damped Stochastic Parallel Translations and Hessian Formulas

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Stochastic Partial Differential Equations and Related Fields (SPDERF 2016)

Part of the book series: Springer Proceedings in Mathematics & Statistics ((PROMS,volume 229))

Abstract

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.

We dedicate this paper to Michael Röckner on the occasion of his 60’s birthday.

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Authors and Affiliations

  1. Department of Mathematics, Imperial College London, London, SW7 2AZ, UK

    Xue-Mei Li

Authors
  1. Xue-Mei Li
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Correspondence to Xue-Mei Li .

Editor information

Editors and Affiliations

  1. Institute for Applied Mathematics, University of Bonn, Bonn, Germany

    Andreas Eberle

  2. Fachbereich Mathematik, Technische Universität Kaiserslautern, Kaiserslautern, Germany

    Martin Grothaus

  3. Faculty of Mathematics, Bielefeld University, Bielefeld, Germany

    Walter Hoh

  4. Faculty of Mathematics, Bielefeld University, Bielefeld, Germany

    Moritz Kassmann

  5. Institut für Mathematik, Technische Universität Berlin, Berlin, Germany

    Wilhelm Stannat

  6. Department of Mathematical Sciences and Research Institute of Mathematics, Seoul National University, Seoul, Korea (Republic of)

    Gerald Trutnau

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Li, XM. (2018). Doubly Damped Stochastic Parallel Translations and Hessian Formulas. In: Eberle, A., Grothaus, M., Hoh, W., Kassmann, M., Stannat, W., Trutnau, G. (eds) Stochastic Partial Differential Equations and Related Fields. SPDERF 2016. Springer Proceedings in Mathematics & Statistics, vol 229. Springer, Cham. https://doi.org/10.1007/978-3-319-74929-7_22

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