Brownian continuum random tree conditioned to be large

TL;DR

This paper studies the asymptotic behavior of a conditioned Feller diffusion and its genealogical tree, revealing new processes and tree structures involving Brownian continuum random trees decorated by Poisson point measures.

Contribution

It introduces a new class of conditioned diffusion processes and their genealogical trees, characterized by Girsanov transformations and Poissonian immigration, with novel tree structures.

Findings

Recovered standard size-biased process and Kesten's tree.

Derived new processes with Girsanov transformations.

Described genealogical trees with infinite skeletons decorated by CR trees.

Abstract

We consider a Feller diffusion (Zs, s $\ge$ 0) (with diffusion coefficient $\sqrt$ 2$\beta$ and drift $\theta$ $\in$ 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 $\alpha$$\beta$ 2 t 2 when $\theta$ = 0 or as $\alpha$ e 2$\beta$|$\theta$|t when $\theta$ = 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.