Probability Theory and Related Fields

, Volume 96, Issue 1, pp 107–121 | Cite as

Diffusions with singular drift related to wave functions

  • Robert Aebi
Article
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Revised:

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 s t , 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.

Mathematics Subject Classification

60J60 58G32 60J70 
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Copyright information

© Springer-Verlag 1993

Authors and Affiliations

  • Robert Aebi
    • 1
  1. 1.Forschungsinstitut MathematikETH ZürichZürichSwitzerland

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