Longtime behavior for the occupation time process of a super-Brownian motion with random immigration

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 greater than or equal to 3 that lead to some Gaussian random fields: for 3 less than or equal to d less than or equal to 5, the field is spatially uniform, which is caused by the randomness of the immigration branching; for d greater than or equal to 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. (C) 2002 Elsevier Science B.V. All rights reserved.