Large deviations for the occupation time functional of a Poisson system of independent Brownian particles

We study the large deviations and the central limit theorem for the occupation time functional of a Poisson system of independent Brownian particles in , extending the results of Cox and Griffeath (1984) to functional spaces. In the lower (recurrent) dimensions d = 1, 2 we have critical orders T and T/log T, whereas in higher (transient) dimensions we have the usual order T. We give explicit expressions for the corresponding rate functions and covariance functionals and derive some asymptotic microcanonical distributions.