The Annals of Probability

Path Properties of Superprocesses with a General Branching Mechanism

Jean-François Delmas

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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.

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Ann. Probab., Volume 27, Number 3 (1999), 1099-1134.

First available in Project Euclid: 29 May 2002

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Zentralblatt MATH identifier

Primary: 60G57: Random measures 60J25: Continuous-time Markov processes on general state spaces 60J55 60J80: Branching processes (Galton-Watson, birth-and-death, etc.)

Superprocesses measure valued processes Brownian snake exit mea-sure hitting probabilities Hausdorff dimension subordinator


Delmas, Jean-François. Path Properties of Superprocesses with a General Branching Mechanism. Ann. Probab. 27 (1999), no. 3, 1099--1134. doi:10.1214/aop/1022677441.

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