Ergodic theory for a superprocess over a stochastic flow

We study the long time limiting behavior of the occupation time of the superprocess over a stochastic flow introduced by Skoulakis and Adler (2001) [13]. The ergodic theorems for dimensions d = 2 and d ≥ 3 are established. The proofs depend heavily on a characterization of the conditional log-Laplace equation of the occupation time process.