Ergodic theory for a superprocess over a stochastic flow
The paper is connected to the ergodic theory for superprocesses over stochastic flows. A superprocess over a stochastic flow was constructed by \textit{G. Skoulakis} and \textit{R. J. Adler} [Ann. Appl. Probab. 11, No. 2, 488--543 (2001; Zbl 1018.60052)]. Namely, let \(\sigma_1(x)\) and \(\sigma_2(x)\) be \(d\times d\) matrices \((x\in\mathbb R^d)\). Suppose that \(W\) and \(B_1,B_2, \dots\) are independent \(d\)-dimensional Brownian motions. Let us consider a branching particle system on \(\mathbb R^d\) described as follows. Between its branchings, the motion of the \(i\)th particle is defined by the stochastic differential equation \[ {\mathrm d}\xi_i(t) = \sigma_1(\xi_i(t))\,{\mathrm d} W(t) + \sigma_2(\xi_i(t))\,{\mathrm d} B_i(t). \] The particle splits into two or dies with equal probabilities when its standard exponential lifetime runs out, independent of the others. Following the result of Skoulakis and Adler, a suitable scaling limit of the above system gives a continuous superprocess. The authors of the present paper study the long time limiting behaviour of the occupation time of the superprocess over a stochastic flow introduced by Skoulakis and Adler.. The ergodic theorems for dimensions \(d=2\) and \(d\geq 3\) are established. The proofs depend heavily on a characterization of the conditional log-Laplace equation of the occupation time process.
scientific article; zbMATH DE number 5763688
scientific article