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Some properties of stable snakes

Eleanor Archer, Ariane Carrance, Laurent Ménard

Abstract

We prove some technical results relating to the Brownian snake on a stable Lévy tree. This includes some estimates on the range of the snake, estimates on its occupation measure around its minimum and also a proof of the fact that the snake and the height function of the associated tree have no common increase points.

Some properties of stable snakes

Abstract

We prove some technical results relating to the Brownian snake on a stable Lévy tree. This includes some estimates on the range of the snake, estimates on its occupation measure around its minimum and also a proof of the fact that the snake and the height function of the associated tree have no common increase points.
Paper Structure (16 sections, 25 theorems, 162 equations, 3 figures)

This paper contains 16 sections, 25 theorems, 162 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Prerequisites on stable trees and snakes
  3. Stable trees
  4. Stable Lévy processes and the Itô excursion measure
  5. Stable height processes and exploration processes
  6. Stable trees
  7. Stable snake with Brownian spatial displacements
  8. Definition
  9. Excursion measures for snakes
  10. Spinal decompositions
  11. Exit measures and the special Markov property
  12. Re-rooting at the minimum
  13. Snake range and lifetime
  14. Computation of moments
  15. Occupation measure
  16. ...and 1 more sections

Key Result

Theorem 1.1

Fix $(a,b) \varsubsetneq \mathbb R$. For $\lambda \geq 0$ and $x \in (a,b)$ we set The mapping $x \mapsto v_{\lambda,a,b} (x)$ satisfies the ODE

Figures (3)

  • Figure 1: The times $T_t$, $T'_t$ and $L_t$.
  • Figure 2: The sizes of the hubs on the left of this branch are encoded by ${\hat{\rho}}^{[L,T]}$, and the subtrees emanating from them are coded by the Itô measure, with an extra restriction on their height.
  • Figure 3: Relation between times appearing in the event \ref{['eqn:event for increase points']}. Not to scale.

Theorems & Definitions (40)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Proposition 2.1
  • proof
  • Lemma 2.2: Lemma 4.2.4 of LeGDuqMono
  • Proposition 2.3: Lemma 1 in riera2022structure; cf also Lemma 3.4 in DLG05
  • Proposition 2.4: cf Theorem 2.2 in legall-weill
  • proof
  • ...and 30 more