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A third representation of Feynman–Kac–Itô formula with singular magnetic vector potential

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A Correction to this article was published on 17 April 2021

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Abstract

The Feynman–Kac–Itô (F–K–I) formula is a useful tool to probabilistically analyze the magnetic nonrelativistic Schrödinger semigroup. Hundertmark (Zur Theorie der magnetischen Schödingerhalbgruppe, Dissertation zur Erlangung des Grades eines Doktors der Naturwissenschaften der Fakultät für Mathematik an der Ruhr–Universität Bochum, 1996) gave a second representation for F–K–I formula for more general singular magnetic vector potential A(x). The phase appearing in the integrand exponential consists of the stochastic integrals of the Lyons–Zheng decomposition type using the time reversal operator \(r_{T}\). In this note, we give a simpler third representation by using, instead of the operator \(r_{T}\), the backward Itô stochastic integral \(\int _0^t A(B(s))\cdot {\widehat{\mathrm{d}}}B(s)\).

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Acknowledgements

I am very grateful to Professor Takashi Ichinose and Professor Hidekazu Ito for a number of helpful discussions and warm encouragement during the preparation of this work. My hearty thanks are due to Professor Shigeyoshi Ogawa for useful and enlightening discussions about the subject and the backward Itô stochastic integral at the early stage of this work. I am much indebted to the anonymous referee for very careful reading of the manuscript, and valuable comments as well as suggestions to rectify some unsatisfactory arguments in Sect. 3 to improve the paper.

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  1. Department of General Education, National Institute of Technology (KOSEN), Ishikawa College, Kitachujo, Tsubata, Ishikawa, 929-0392, Japan

    Taro Murayama

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  1. Taro Murayama
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Correspondence to Taro Murayama.

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Murayama, T. A third representation of Feynman–Kac–Itô formula with singular magnetic vector potential. Lett Math Phys 111, 33 (2021). https://doi.org/10.1007/s11005-021-01376-3

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  • DOI: https://doi.org/10.1007/s11005-021-01376-3

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