Convergence of spatial branching processes to $α$-stable CSBPs: Genealogy of semi-pushed fronts

Abstract

We consider inhomogeneous branching diffusions on an infinite domain of $\mathbb{R}^d$. The first aim of this article is to derive a general criterium under which the size process (number of particles) and the genealogy of the particle system become undistinguishable from the ones of an $α$-stable CSBP, with $α\in(1,2)$. The branching diffusion is encoded as a random metric space capturing all the information about the positions and the genealogical structure of the population. Our convergence criterium is based on the convergence of the moments for random metric spaces, which in turn can be efficiently computed through many-to-few formulas. It requires an extension of the method of moments to general CSBPs (with or without finite second moment). In a recent work, Tourniaire introduced a branching Brownian motion which can be thought of as a toy model for fluctuating pushed fronts. The size process was shown to converge to an $α$-stable CSBP and it was conjectured that a genealogical convergence should occur jointly. We prove this result as an application of our general methodology.

Figures (2)

• Figure 1: Illustration of a planar ultrametric matrix $U \in \mathbb{U}_k$. Here, $k = 8$ and the entry $U_{ij}$ corresponds to the coalescence time between the leaves with labels $i$ and $j$, $\tau$ is the distance from the root to the deepest branch point, and $(U_i)_i$ are the distance matrices corresponding to the subtrees that are disconnected by removing the branch point at time $\tau$.
• Figure 2: Left panel. A plane ultrametric matrix in $\mathbb{U}_k$ can be encoded by a plane ultrametric tree. The values $H_i$ are interpreted as the depth of the $i^{th}$ branching point and for $i<j$, $U_{ij} = \max{\{H_i,\cdots,H_{j-1}\}}$ is the distance between the leaves $i$ and $j$. The depth of the tree is given $\tau= \max\{H_1,\cdots, H_{k-1}\}$. In this figure, the subtrees at depth $\tau-\varepsilon$ partition the leaves are partioned into $5$ blocks. Further, $c^{\tau-\varepsilon} = \{1\}\{2,3\}\{4,5,6\}\{7,8\}\{9\}$. Right panel. Recall the trajectorial interpretation of the spine measure outlined in Remark \ref{['rem:trajectorial-construction']} where branching spine processes are diffusing along the branches of the tree. In (\ref{['eq:m_c']}), most of the mass in the integration is concentrated close to the $L$-boundary which indicates that the first branching point occurs close to the boundary. In the gray time window (depth in $[\tau-\varepsilon,\varepsilon]$), 4 branch points accumulate into a single $4$-furcation at the limit.

Theorems & Definitions (99)

• Conjecture 1.1: Informal version
• Definition 1
• Definition 2: Gromov weak toplogy
• Definition 3: Moments
• Proposition 1.1: Convergence of moments
• Definition 4
• Remark 1
• Definition 5
• Proposition 1.2
• Definition 6: Biased $k$-spine measure