Path properties of superprocesses with a general branching mechanism

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 ∪_t∈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 α-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.