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A third representation of Feynman-Kac-Itô formula with singular magnetic vector potential. (English) Zbl 1480.60143

Lett. Math. Phys. 111, No. 2, Paper No. 33, 21 p. (2021); correction ibid. 111, No. 2, Paper No. 49, 1 p. (2021).
Summary: The Feynman-Kac-Itô (F-K-I) formula is a useful tool to probabilistically analyze the magnetic nonrelativistic Schrödinger semigroup. D. Hundertmark [Zur Theorie der magnetischen Schrödingerhalbgruppe. Bochum: Univ. Bochum, Math. Fak. (1996; Zbl 0880.47023)] 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 \(\int_0^t A(B(s))\cdot\widehat{\text{d}}B(s)\).

MSC:

60H05 Stochastic integrals
60J65 Brownian motion
81S40 Path integrals in quantum mechanics

Citations:

Zbl 0880.47023
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References:

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