Brownian continuum random tree conditioned to be large

TL;DR

The paper develops a comprehensive theory for Brownian continuum random trees conditioned to be large via quadratic CSBPs. It derives local limits under large conditioning using h-transforms and Poisson immigration, yielding a novel backbone structure where an infinite spine is decorated by independent Brownian CRTs, rather than a classical branching backbone. The work systematically builds a rigorous backbone-decomposition framework, culminating in a generalized n-leaves decomposition and precise measurable grafting machinery in a robust Gromov–Hausdorff-type topology. These results generalize Kesten-tree-type limits, connect to Doob h-transforms, and provide foundational topology for decorated random trees with potential applications in branching-process genealogies and continuum random-tree theory.

Abstract

We consider a Feller diffusion (Zs, s $\ge$ 0) (with diffusion coefficient $\sqrt$ 2$β$ and drift $θ$ $\in$ R) that we condition on {Zt = at}, where at is a deterministic function, and we study the limit in distribution of the conditioned process and of its genealogical tree as t $\rightarrow$ +$\infty$. When at does not increase too rapidly, we recover the standard size-biased process (and the associated genealogical tree given by the Kesten's tree). When at behaves as $α$$β$ 2 t 2 when $θ$ = 0 or as $α$ e 2$β$|$θ$|t when $θ$ = 0, we obtain a new process whose distribution is described by a Girsanov transformation and equivalently by a SDE with a Poissonian immigration. Its associated genealogical tree is described by an infinite discrete skeleton (which does not satisfy the branching property) decorated with Brownian continuum random trees given by a Poisson point measure. As a by-product of this study, we introduce several sets of trees endowed with a Gromovtype distance which are of independent interest and which allow here to define in a formal and measurable way the decoration of a backbone with a family of continuum random trees.

Figures (6)

• Figure 1: A tree $({\mathbf t},\mathbf v)\in {\mathbb T}^{(3)}_{\mathrm{plan}}$ with $\mathbf v=({\varrho},1,3,2)$
• Figure 2: The trees ${\mathbf T}_1^{(4)}$, ${\mathbf T}_2^{(4)}$, ${\mathbf T}_3^{(4)}$ and ${\mathbf T}_4^{(4)}$ obtained from the sequences $(K_1=1,K_2=1,K_3=2)$ and $(\varepsilon_1={\rm g},\varepsilon_2={\rm d},\varepsilon_3={\rm d})$. The dashed lines represent the levels $\xi_1^{(4)},\xi_2^{(4)},\xi_3^{(4)}$.
• Figure 3: Example of restrictions of a tree $T$ with a marked spine $S$ (in bold).
• Figure 4: A discrete trees spanned by the leaves $\{1, 2, 3\}$.
• Figure 5: The splitting of the left hand tree with respect to $\mathbf v=\{{\varrho},1,2,3\}$. In this instance, $T_{\{1,2\}}$ and $T_{\{1,3\}}$ are reduced to their own root.

Theorems & Definitions (119)

• Proposition 1.1: Representation using a Poisson immigration, case $\theta=0$
• Theorem 1.2: Generalized $n$-leaves decomposition, case $\theta=0$
• Remark 2.1: Scaling property of $Z$
• Lemma 2.2
• Lemma 2.3: Entrance law and transition densities of $Z$
• proof
• Lemma 2.4
• proof
• Lemma 2.5: Martin boundary
• Remark 2.6: Equivalent condition for the moderate regime