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Preprints, Working Papers, ... Year : 2024

Brownian continuum random tree conditioned to be large


We consider a Feller diffusion (Zs, s ≥ 0) (with diffusion coefficient √ 2β and drift θ ∈ 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 → +∞. 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 αβ 2 t 2 when θ = 0 or as α e 2β|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.
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Dates and versions

hal-03581326 , version 1 (19-02-2022)
hal-03581326 , version 2 (02-07-2024)



Romain Abraham, Jean-Franç Ois Delmas, Hui He. Brownian continuum random tree conditioned to be large. 2024. ⟨hal-03581326v2⟩
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