Abstract
The probabilistic counterpart of the theory of strings is the theory of quasi diffusions. The concept of quasi diffusion (generalised diffusion, gap diffusion) generalises the concept of one-dimensional diffusion in that it does not require the speed measure to be strictly positive. This note focuses on some connections between the spectral theory of strings and the excursion theory of quasi diffusions. The main difference in our approach compared with the previous ones is that we are using Krein's theory for “killed” strings as a primary tool instead of dual strings. It is seen that this approach provides a natural setting for various spectral representations for quasi diffusions. In particular, we discuss representations for first hitting time distributions, Lévy measures of inverse local times, and different quantities connected with the Ito excursion law. We consider also the characterisation problem for inverse local times. In fact, it is seen that this is equivalent with the inverse spectral problem for “killed” strings.
Research supported partly by a NSERC grant, while the author was visiting the University of British Columbia, Mathematical Department.
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Küchler, U., Salminen, P. (1989). On spectral measures of strings and excursions of quasi diffusions. In: Azéma, J., Yor, M., Meyer, P.A. (eds) Séminaire de Probabilités XXIII. Lecture Notes in Mathematics, vol 1372. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0083995
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DOI: https://doi.org/10.1007/BFb0083995
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