Clustering and invariant measures for spatial branching models with infinite variance

Abstract

We consider two spatial branching models on $\mathbb{R}^d$: branching Brownian motion with a branching law in the domain of normal attraction of a $1 + \beta$ stable law, $0 < \beta \leq 1$, and the corresponding high density limit measure valued diffusion. The longtime behavior of both models depends highly on $\beta$ and $d$. We show that for $d \leq \frac{2}{\beta}$ the only invariant measure is $\delta_0$, the unit mass on the empty configuration. Furthermore, we give a precise condition for convergence toward $\delta_0$. For $d > \frac{2}{\beta}$ it is known that there exists a family $(\nu_{\theta}, \theta \epsilon [0, \infty))$ of nontrivial invariant measures. We show that every invariant measure is a convex combination of the $\nu_{\theta}$. Both results have been known before only under an additional finite mean assumption. For the critical dimension $d = \frac{2}{\beta}$ we show that both models display the phenomenon of diffusive clustering. This means that clusters grow spatially on a random scale. We give a precise description of the clusters via multiple scale analysis. Our methods rely mainly on studying sub- and supersolutions of the reaction diffusion equation $\frac{\partial u}{\partial t} - 1/2 \Delta u + u^{1 + \beta} = 0$.