Path properties of superprocesses with a general branching mechanism

We rst consider a super Brownian motion X with a general branching mechanism. Using the Brownian snake representation with subordination, we get the Hausdor di-mension of supp X t, the topological support of X t, and more generally the Hausdor dimension of [ t2B 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 dis-connected in high dimension. Eventually, considering a super -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