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Probability Theory and Related Fields

, Volume 165, Issue 1–2, pp 365–399 | Cite as

A Feynman–Kac–Itô formula for magnetic Schrödinger operators on graphs

  • Batu Güneysu
  • Matthias KellerEmail author
  • Marcel Schmidt
Article

Abstract

In this paper we prove a Feynman–Kac–Itô formula for magnetic Schrödinger operators on arbitrary weighted graphs. To do so, we have to provide a natural and general framework both on the operator theoretic and the probabilistic side of the equation. On the operator side we identify a very general class of potentials that allows the definition of magnetic Schrödinger operators. On the probabilistic side, we introduce an appropriate notion of stochastic line integrals with respect to magnetic potentials. Apart from linking the world of discrete magnetic operators with the probabilistic world through the Feynman–Kac–Itô formula, the insights from this paper gained on both sides should be of an independent interest. As applications of the Feynman–Kac–Itô formula, we prove a Kato inequality, a Golden–Thompson inequality and an explicit representation of the quadratic form domains corresponding to a large class of potentials.

Mathematics Subject Classification

39A70 35J10 47D08 81Q35 

Notes

Acknowledgments

The authors want to express their gratitude to Daniel Lenz for very inspiring and important hints. We are also grateful to Ognjen Milatovic for a very fruitful correspondence and to Hendrik Vogt, Sebastian Haeseler for most helpful discussions. BG has been financially supported by the Sonderforschungsbereich 647: Raum-Zeit-Materie. Moreover, MK acknowledges the financial support of the German Science Foundation (DFG), Golda Meir Fellowship, the Israel Science Foundation (Grant Nos. 1105/10 and 225/10) and BSF Grant No. 2010214. Furthermore, MS acknowledges the financial support of the European Science Foundation (ESF) within the project “Random Geometry of Large Interacting Systems and Statistical Physics”.

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Copyright information

© Springer-Verlag Berlin Heidelberg 2015

Authors and Affiliations

  • Batu Güneysu
    • 1
  • Matthias Keller
    • 2
    Email author
  • Marcel Schmidt
    • 3
  1. 1.Institut für MathematikHumboldt-Universität zu BerlinBerlinGermany
  2. 2.Einstein Institute of MathematicsThe Hebrew University of JerusalemJerusalemIsrael
  3. 3.Mathematisches InstitutFriedrich-Schiller-Universität JenaJenaGermany

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