Cyclically stationary Brownian local time processes |
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Authors: | Jim Pitman |
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Institution: | (1) Department of Statistics, University of California, 367 Evans Hall # 3860, Berkeley, CA 94720-3860, USA, US |
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Abstract: | Summary. Local time processes parameterized by a circle, defined by the occupation density up to time T of Brownian motion with constant drift on the circle, are studied for various random times T. While such processes are typically non-Markovian, their Laplace functionals are expressed by series formulae related to
similar formulae for the Markovian local time processes subject to the Ray–Knight theorems for BM on the line, and for squares
of Bessel processes and their bridges. For T the time that BM on the circle first returns to its starting point after a complete loop around the circle, the local time
process is cyclically stationary, with same two-dimensional distributions, but not the same three-dimensional distributions,
as the sum of squares of two i.i.d. cyclically stationary Gaussian processes. This local time process is the infinitely divisible
sum of a Poisson point process of local time processes derived from Brownian excursions. The corresponding intensity measure
on path space, and similar Lévy measures derived from squares of Bessel processes, are described in terms of a 4-dimensional
Bessel bridge by Williams’ decomposition of It?’s law of Brownian excursions.
Received: 28 June 1995 |
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Keywords: | Mathematics Subject classification (1991): 60J55 60J65 60G10 |
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