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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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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DOI: https://doi.org/10.1007/BF01195885
Mathematics Subject Classification
- 60J60
- 58G32
- 60J70