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Transformations of diffusion and Schrödinger processes. (English) Zbl 0666.60073
A transformation by means of a new type of multiplicative functionals is given, which is a generalization of Doob’s space-time harmonic transformation, in the case of an arbitrary non-harmonic function \(\phi(t,x)\) which may vanish on a subset of \([a,b]\times {\mathbb{R}}^ d\). The transformation induces an additional (singular) drift term \(\nabla \phi /\phi\), like in the case of Doob’s space-time harmonic transformation.
To handle the transformation, an integral equation of singular perturbations and a diffusion equation with singular potentials are discussed and the Feynman-Kac theorem is established for a class of singular potentials. The transformation is applied to Schrödinger processes which are defined following an idea of E. Schrödinger [Sitzungsber. Preuß. Akad. Wiss., Phys.-Math. Kl. H. 8/9, 144-153 (1931; Zbl 0001.37503)].
Reviewer: M.Nagasawa

MSC:
60J60 Diffusion processes
60H25 Random operators and equations (aspects of stochastic analysis)
81P20 Stochastic mechanics (including stochastic electrodynamics)
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