Summary
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 arbitrary non-harmonic function ϕ (t, x) which may vanish on a subset of [a, b]xℤd. The transformation induces an additional (singular) drift term ∇φ/φ, 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 (1931).
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To commemorate the centenary of E. Schrödinger's birth (1887–1961)
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Nagasawa, M. Transformations of diffusion and Schrödinger processes. Probab. Th. Rel. Fields 82, 109–136 (1989). https://doi.org/10.1007/BF00340014
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DOI: https://doi.org/10.1007/BF00340014