Large deviations for the occupation time functional of a Poisson system of independent Brownian particles
The authors study the large deviations and the central limit theorem for the occupation time functional of a Poisson system of independent Brownian particles in \(\mathbb{R}^ d\). They extend the results of \textit{J. T. Cox} and \textit{D. Griffeath} for random walks [Z. Wahrscheinlichkeitstheorie Verw. Geb. 66, 543-558 (1984; Zbl 0551.60028)] and partially the results of \textit{T.-Y. Lee} [Ann. Probab. 16, No. 4, 1537-1558 (1988; Zbl 0661.60046) and ibid. 17, No. 1, 46-57 (1989; Zbl 0664.60032)] to functional spaces. The order of the large deviations remains the same, namely \(T^{1/2}\) and \(T/ \log T\) in the recurrent dimensions one and two, and \(T\) for the higher transient dimensions. Explicit expressions for the corresponding rate functions and covariance functionals are given and a specific microcanonical principle is considered. In one dimension the function-space rate function is not the standard one; thus an untypical end point is reached via a nonlinear profile.
scientific article; zbMATH DE number 691004