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Local times and excursions for self-similar Markov trees

Jean Bertoin, Armand Riera, Alejandro Rosales-Ortiz

TL;DR

The paper develops a comprehensive framework for local times and excursions on self-similar Markov trees (ssMt). It constructs level-time measures $L(x,\mathrm{d}t)$ supported on level sets, links the decoration along a root-to-typical-point path to a positive self-similar Markov process conditioned to end at $x$, and proves that $L(x,\mathrm{d}t)$ converges to the harmonic measure $\mu$ as $x\downarrow0$. It further shows that the level-1 set $\mathcal{L}(1)$, equipped with the $L(1,\mathrm{d}t)$-induced metric, forms a continuous branching tree and provides a detailed excursion theory: excursions away from level sets are described by a Poisson point process built from a spine/decorated-subtree framework. The results generalize Itô-type excursion theory to the setting of Markov processes indexed by ssMt and establish deep links between spine decompositions, local times, harmonic measures, and the branching structure of level sets.

Abstract

This work builds upon the recent monograph [5] on self-similar Markov trees. A self-similar Markov tree is a random real tree equipped with a function from the tree to $[0,\infty)$ that we call the decoration. Here, we construct local time measures $L(x,dt)$ at every level $x>0$ of the decoration for a large class of self-similar Markov trees. This enables us to mark at random a typical point in the tree at which the decoration is $x$. We identify the law of the decoration along the branch from the root to this tagged point in terms of a remarkable (positive) self-similar Markov process. We also show that after a proper normalization, $L(x,dt)$ converges as $x\to 0+$ to the harmonic measure $μ$ on the tree. Finally, we point out that using a local time measure instead of the usual length measure $λ$ to compute distances on the tree turn the latter into a continuous branching tree. This is relevant to analyze the excusions of the decoration away from a given level. Many results of the present work shall be compared with the recent ones in [22,23] about local times and excursions of a Markov process indexed by Lévy tree.

Local times and excursions for self-similar Markov trees

TL;DR

The paper develops a comprehensive framework for local times and excursions on self-similar Markov trees (ssMt). It constructs level-time measures supported on level sets, links the decoration along a root-to-typical-point path to a positive self-similar Markov process conditioned to end at , and proves that converges to the harmonic measure as . It further shows that the level-1 set , equipped with the -induced metric, forms a continuous branching tree and provides a detailed excursion theory: excursions away from level sets are described by a Poisson point process built from a spine/decorated-subtree framework. The results generalize Itô-type excursion theory to the setting of Markov processes indexed by ssMt and establish deep links between spine decompositions, local times, harmonic measures, and the branching structure of level sets.

Abstract

This work builds upon the recent monograph [5] on self-similar Markov trees. A self-similar Markov tree is a random real tree equipped with a function from the tree to that we call the decoration. Here, we construct local time measures at every level of the decoration for a large class of self-similar Markov trees. This enables us to mark at random a typical point in the tree at which the decoration is . We identify the law of the decoration along the branch from the root to this tagged point in terms of a remarkable (positive) self-similar Markov process. We also show that after a proper normalization, converges as to the harmonic measure on the tree. Finally, we point out that using a local time measure instead of the usual length measure to compute distances on the tree turn the latter into a continuous branching tree. This is relevant to analyze the excusions of the decoration away from a given level. Many results of the present work shall be compared with the recent ones in [22,23] about local times and excursions of a Markov process indexed by Lévy tree.
Paper Structure (16 sections, 32 theorems, 163 equations, 3 figures)

This paper contains 16 sections, 32 theorems, 163 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Self-similar Markov trees in a nutshell
  3. Some background on local times of Lévy processes
  4. Local time measures and level sets
  5. Conditioning on the terminal value
  6. Finite terminal value
  7. Infinite terminal value
  8. Local time measures on decoration level sets
  9. Construction of the local time measure
  10. Total local times and their mean values
  11. Decoration from the root to a typical level point
  12. Local time approximation of the harmonic measure
  13. Excursions away from a level set
  14. Excursions debuting on the ancestral segment
  15. Branching structure of level sets of individuals
  16. ...and 1 more sections

Key Result

Lemma 3.1

There exists a sequence $(\varepsilon_n)_{n\geq 1}$ of positive real numbers with $\varepsilon_n\to0$, such that for any $x,y\in \mathbb{R}$ and $t'>0$, As a consequence, $P_x$-a.s., the sequence of random measures converges towards the Stieltjes measure $\ell(y, \mathrm{d} t)$ in the sense of weak convergence of finite measures on $\mathbb{R}_+$.

Figures (3)

  • Figure 1: 3D simulation of a self-similar Markov tree $(T,g)$: the tree $T$ is embedded in a horizontal plane and vertical coordinates indicate the values taken by the decoration $g$ on $T$ .
  • Figure 2: 3D simulation of a self-similar Markov tree $(T,g)$ started from $1$. In red are represented the points where the decoration takes a fixed value $x$ (here $x=1/2$).
  • Figure 3: Illustration of Theorem \ref{['T1']}. The spine to a typical $L(x,\mathrm{d}t)$ point and the associated decoration is represented in color, and the rest of the decorated tree in gray. Here the decoration starts at the root at $1$ and we take $x=1/2$. The distribution of the decoration along this spine is distributed $Q_{1,x}$.

Theorems & Definitions (65)

  • Lemma 3.1
  • proof
  • Lemma 3.2
  • Remark 3.3
  • proof
  • Proposition 3.4
  • proof
  • Lemma 3.5
  • proof
  • Corollary 3.6
  • ...and 55 more