Exponent of inverse local time for harmonic transformed process
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We are concerned with inverse local time at regular end points for harmonic transform of a one dimensional diffusion process, and consider the corresponding exponents as well as the entrance law and the excursion law associated with inverse local time. In 1964 K. Itô and H. P. McKean showed that the Lévy measure density corresponding to the inverse local time at the regular end point for a recurrent one dimensional diffusion process is represented as the Laplace transform of the spectral measure corresponding to the diffusion process, where the absorbing boundary condition is posed at the end point. We demonstrate that their representation theorem is available for a transient one dimensional diffusion process, and deduce a representation theorem of the Lévy measure density corresponding to the inverse local time for a transient harmonic transformed process. Furthermore, we show a relation between exponents of inverse local time by means of 0-Green functions and those by means of Dirichlet forms, along with correlations between entrance laws of the original diffusion processes and its harmonic transform or between excursion laws and the harmonic transform. Moreover we present a new consideration for harmonic transform of non-minimal processes.
人間文化研究科年報(奈良女子大学大学院人間文化研究科) (31), 127-138, 2016-03-31