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Diffusions with singular drift related to wave functions
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  • Published: March 1993

Diffusions with singular drift related to wave functions

  • Robert Aebi1 nAff2 

Probability Theory and Related Fields volume 96, pages 107–121 (1993)Cite this article

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Summary

Schrödinger equations are equivalent to pairs of mutually time-reversed non-linear diffusion equations. Here the associated diffusion processes with singular drift are constructed under assumptions adopted from the theory of Schrödinger operators, expressed in terms of a local space-time Sobolev space.

By means of Nagasawa's multiplicative functionalN t s , a Radon-Nikodym derivative on the space of continuous paths, a transformed process is obtained from Wiener measure. Its singular drift is identified by Maruyama's drift transformation. For this a version of Itô's formula for continuous space-time functions with first and second order derivatives in the sense of distributions satisfying local integrability conditions has to be derived.

The equivalence is shown between weak solutions of a diffusion equation with singular creation and killing term and the solutions of a Feynman-Kac integral equation with a locally integrable potential function.

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Author notes
  1. Robert Aebi

    Present address: IMSV, Universität Bern, CH-3012, Bern, Switzerland

Authors and Affiliations

  1. Forschungsinstitut Mathematik, ETH Zürich, CH-8092, Zürich, Switzerland

    Robert Aebi

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  1. Robert Aebi
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Aebi, R. Diffusions with singular drift related to wave functions. Probab. Th. Rel. Fields 96, 107–121 (1993). https://doi.org/10.1007/BF01195885

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  • Received: 11 February 1992

  • Revised: 30 November 1992

  • Issue Date: March 1993

  • DOI: https://doi.org/10.1007/BF01195885

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Mathematics Subject Classification

  • 60J60
  • 58G32
  • 60J70
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