Cyclically Stationary Brownian Local Time Processes

Local time processes parameterized by a circle, de ned 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 rst 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 in nitely divisible sum of a Poisson point process of local time processes derived from Brownian excursions. The corresponding intensity measure on path space, and similar L evy measures derived from squares of Bessel processes, are described in terms of a 4-dimensional Bessel bridge by Williams' decomposition of Itô's law of Brownian excursions.