A continuous super-Brownian motion in a super-Brownian medium
This paper is devoted to the study of a catalytic super-Brownian motion \(X^\rho\) where the branching catalyst is a given measure-valued path \(\rho_t\), \(t\geq0\). Such a process can be constructed via a generalization of Dynkin' s additive functional approach to superprocesses, provided that the collision local time \(L_{[W,\rho]}\) between a Brownian motion \(W\) and the catalytic medium \(\rho\) exists as a `` nice'' additive functional of \(W\). It is shown that this is the case if \(\rho\) itself is a typical trajectory of a continuous super-Brownian motion in dimension \(d\leq3\), started, for instance, in a Lebesgue measure \(\ell\). Thereby the existence result for \(L_{[W,\rho]}\) of \textit{M. T. Barlow, S. N. Evans} and \textit{E. A. Perkins} [Can. J. Math. 43, No. 5, 897-938 (1991; Zbl 0765.60044)] is generalized to infinite measure paths \(\rho\). Moreover it is shown that \(X^\rho\) then almost surely has (Hölder) continuous sample paths. In the final Section 6, the behavior of \(X^\rho_t\) is studied as \(t\to\infty\) for the case where \(d=1\) and \(\rho\) is a typical super-Brownian path with \(\rho_0=\ell\). If \(X^\rho\) is started in a finite measure \(\mu\), then it is shown that the total mass process \(\langle1,X^\rho_t\rangle\) converges almost surely to a non-trivial limit \(m^\rho\), which has full expectation \(\langle1,\mu\rangle\). In the case where also \(X^\rho_0=\ell\) one has that \(X^\rho_t\to\ell\) in probability. Main ingredients in the proofs are moment estimates and the existence of a space-time Hölder continuous occupation density field for \(\rho\) in dimensions less than 3.
scientific article; zbMATH DE number 1000041