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In this work, we characterize cluster-invariant point processes for criticalbranching spatial processes on R d for all large enough d when the motion lawis $\alpha$-stable or has a finite discrete range. More precisely, when themotion is $\alpha$-stable with $\alpha$ $\le$ 2 and the offspring law $\mu$ ofthe branching process has an heavy tail such that $\mu$(k) $\sim$ k--2--$\beta$ , then we need the dimension d to be strictly larger than thecritical dimension $\alpha$/$\beta$. In particular, when the motion is Brownianand the offspring law $\mu$ has a second moment, this critical dimension is 2.Contrary to the previous work of Bramson, Cox and Greven in [BCG97] whose proofused PDE techniques, our proof uses probabilistic tools only.

Invariant measures of critical branching random walks in high dimension