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On Itô’s Complex Measure Condition ∗

By Larry Chen, Scott Dobson, Ronald Guenther, Chris Orum, Mina Ossi, Enrique Thomann and Edward Waymire


The complex measure condition was introduced by Itô (1965) as a sufficient condition on the potential term in a one-dimensional Schrödinger equation and/or corresponding linear diffusion equation to obtain a Feynman-Kac path integral formula. In this paper we provide an alternative probabilistic derivation of this condition and extend it to include any other lower order terms, i.e. drift and forcing terms, that may be present. In particular, under a complex measure condition on the lower order terms of the diffusion equation, we derive a representation of mild solutions of the Fourier transform as a functional of a jump Markov process in wavenumber space

Topics: Duality, multiplicative cascade, multi-type branching random
Year: 2003
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