Path Properties of Superprocesses with a General Branching Mechanism

Abstract

We first consider a super Brownian motion $X$ with a general branching mechanism. Using the Brownian snake representation with subordination, we get the Hausdorff dimension of supp $X_t$, the topological support of $X_t$ and, more generally, the Hausdorff dimension of $\Bigcup_{t/in B}\supp X _t$. We also provide estimations on the hitting probability of small balls for those random measures. We then deduce that the support is totally disconnected in high dimension. Eventually, considering a super $\alpha$-stable process with a general branching mechanism, we prove that in low dimension this random measure is absolutely continuous with respect to the Lebesgue measure.