Abstract
Longtime behavior for the occupation time of a super-Brownian motion with immigration governed by the trajectory of another super-Brownian motion is considered. Central limit theorems are obtained for dimensions d⩾3 that lead to some Gaussian random fields: for 3⩽ d⩽5, the field is spatially uniform, which is caused by the randomness of the immigration branching; for d⩾7, the covariance of the limit field is given by the potential operator of the Brownian motion, which is caused by the randomness of the underlying branching; and for d=6, the limit field involves a mixture of the two kinds of fluctuations. Some extensions are made in higher dimensions. An ergodic theorem is proved as well for dimension d=2, which is characterized by an evolution equation.
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