Introduction to self-similar growth-fragmentations Quan Shi CIMAT, - PowerPoint PPT Presentation

Introduction to self-similar growth-fragmentations Quan Shi CIMAT, 11-15 December, 2017 Quan Shi Growth-Fragmentations CIMAT, 11-15 December, 2017 1 / 34

Literature Jean Bertoin, Compensated fragmentation processes and limits of dilated fragmentations , Ann. Probab. 44 (2016), no. 2, 1254–1284. MR 3474471 , Markovian growth-fragmentation processes , Bernoulli 23 (2017), no. 2, 1082–1101. MR 3606760 Jean Bertoin, Timothy Budd, Nicolas Curien, and Igor Kortchemski, Martingales in self-similar growth-fragmentations and their connections with random planar maps , Preprint, arXiv:1605.00581v1 [math.PR], 2016. Jean Bertoin, Nicolas Curien, and Igor Kortchemski, Random planar maps & growth-fragmentations , Preprint, arXiv:1507.02265v2 [math.PR], July 2015. ◮ Lecture notes available at my personal webpage: https://sites.google.com/site/qshimath Quan Shi Growth-Fragmentations CIMAT, 11-15 December, 2017 2 / 34

Overview 1. Background: fragmentation processes Quan Shi Growth-Fragmentations CIMAT, 11-15 December, 2017 3 / 34

Overview 1. Background: fragmentation processes 2. Construction of growth-fragmentation processes Quan Shi Growth-Fragmentations CIMAT, 11-15 December, 2017 3 / 34

Overview 1. Background: fragmentation processes 2. Construction of growth-fragmentation processes 3. Properties of self-similar growth-fragmentations Quan Shi Growth-Fragmentations CIMAT, 11-15 December, 2017 3 / 34

Overview 1. Background: fragmentation processes 2. Construction of growth-fragmentation processes 3. Properties of self-similar growth-fragmentations 4. Martingales in self-similar growth-fragmentations Quan Shi Growth-Fragmentations CIMAT, 11-15 December, 2017 3 / 34

1. Background: fragmentation processes Quan Shi Growth-Fragmentations CIMAT, 11-15 December, 2017 4 / 34

Background ◮ Fragmentation: “the process or state of breaking or being broken into fragments”. Quan Shi Growth-Fragmentations CIMAT, 11-15 December, 2017 5 / 34

Background ◮ Fragmentation: “the process or state of breaking or being broken into fragments”. ◮ Fragmentation phenomena can be observed widely in nature: biology and population genetics, aerosols, droplets, mining industry, etc. Quan Shi Growth-Fragmentations CIMAT, 11-15 December, 2017 5 / 34

Background ◮ Fragmentation: “the process or state of breaking or being broken into fragments”. ◮ Fragmentation phenomena can be observed widely in nature: biology and population genetics, aerosols, droplets, mining industry, etc. ◮ The first studies of fragmentation from a probabilistic point of view are due to Kolmogorov [1941] and Filippov [1961]. Quan Shi Growth-Fragmentations CIMAT, 11-15 December, 2017 5 / 34

Background ◮ Fragmentation: “the process or state of breaking or being broken into fragments”. ◮ Fragmentation phenomena can be observed widely in nature: biology and population genetics, aerosols, droplets, mining industry, etc. ◮ The first studies of fragmentation from a probabilistic point of view are due to Kolmogorov [1941] and Filippov [1961]. ◮ The general framework of the theory of stochastic fragmentation processes was built mainly by Bertoin [2001, 2002]. See Bertoin [2006] for a comprehensive monograph. Quan Shi Growth-Fragmentations CIMAT, 11-15 December, 2017 5 / 34

Background ◮ Fragmentation: “the process or state of breaking or being broken into fragments”. ◮ Fragmentation phenomena can be observed widely in nature: biology and population genetics, aerosols, droplets, mining industry, etc. ◮ The first studies of fragmentation from a probabilistic point of view are due to Kolmogorov [1941] and Filippov [1961]. ◮ The general framework of the theory of stochastic fragmentation processes was built mainly by Bertoin [2001, 2002]. See Bertoin [2006] for a comprehensive monograph. ◮ Fragmentations are relevant to other areas of probability theory, such as branching processes, coalescent processes, multiplicative cascades and random trees. Quan Shi Growth-Fragmentations CIMAT, 11-15 December, 2017 5 / 34

Model: Self-similar fragmentation processes [Bertoin 2002] ◮ Index of self-similarity: α ∈ R . ◮ Dislocation measure: ν sigma-finite measure on [1 / 2 , 1), such that � [1 / 2 , 1) (1 − y ) ν ( dy ) < ∞ . ◮ For every y ∈ [1 / 2 , 1), a fragment of size x > 0 splits into two fragments of size xy , x (1 − y ) at rate x α ν ( dy ). Quan Shi Growth-Fragmentations CIMAT, 11-15 December, 2017 6 / 34

Model: Self-similar fragmentation processes [Bertoin 2002] ◮ Index of self-similarity: α ∈ R . ◮ Dislocation measure: ν sigma-finite measure on [1 / 2 , 1), such that � [1 / 2 , 1) (1 − y ) ν ( dy ) < ∞ . ◮ For every y ∈ [1 / 2 , 1), a fragment of size x > 0 splits into two fragments of size xy , x (1 − y ) at rate x α ν ( dy ). ◮ Record the sizes of the fragments at time t ≥ 0 by a measure on R + � X ( t ) = δ X i ( t ) . i ≥ 1 The process X is a self-similar fragmentation (without erosion) with characteristics ( α, ν ). Quan Shi Growth-Fragmentations CIMAT, 11-15 December, 2017 6 / 34

Examples: ◮ Splitting intervals [BrennanDurrett1986]: ◮ U i : i.i.d. uniform random variables on (0 , 1), arrive one after another at rate 1. ◮ At time t , (0 , 1) is separated into intervals I 1 ( t ) , I 2 ( t ) , . . . ◮ F ( t ) := � i ≥ 1 δ I i ( t ) is a self-similar fragmentation with characteristics (1 , ν ), x ∈ [ 1 where ν ( d x ) = 2 d x , 2 , 1). Quan Shi Growth-Fragmentations CIMAT, 11-15 December, 2017 7 / 34

Examples: ◮ Splitting intervals [BrennanDurrett1986]: ◮ U i : i.i.d. uniform random variables on (0 , 1), arrive one after another at rate 1. ◮ At time t , (0 , 1) is separated into intervals I 1 ( t ) , I 2 ( t ) , . . . ◮ F ( t ) := � i ≥ 1 δ I i ( t ) is a self-similar fragmentation with characteristics (1 , ν ), x ∈ [ 1 where ν ( d x ) = 2 d x , 2 , 1). 1.8 ◮ The Brownian fragmentation 1.6 ◮ Normalized Brownian excursion: 1.4 B : [0 , 1] → R + . 1.2 ◮ O ( t ) := { x ∈ (0 , 1) : B ( x ) > t } . 1 ◮ F ( t ) := � I : component of O ( t ) δ | I | xs 1 xs 2 0.8 ◮ F is a self-similar fragmentation 0.6 with characteristics ( − 1 2 , ν B ), where ν B ( d x ) = 0.4 2 x ∈ [ 1 √ 2 , 1). 2 π x 3 (1 − x ) 3 d x , 0.2 t 2 t 3 t 1 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Quan Shi Growth-Fragmentations CIMAT, 11-15 December, 2017 7 / 34

2. Construction of growth-fragmentation processes Quan Shi Growth-Fragmentations CIMAT, 11-15 December, 2017 8 / 34

Growth-fragmentation processes ◮ (Markovian) growth-fragmentation processes [Bertoin 2017] describe the evolution of the sizes of a family of particles, which can grow larger or smaller with time, and occasionally split in a conservative manner. Quan Shi Growth-Fragmentations CIMAT, 11-15 December, 2017 9 / 34

Growth-fragmentation processes ◮ (Markovian) growth-fragmentation processes [Bertoin 2017] describe the evolution of the sizes of a family of particles, which can grow larger or smaller with time, and occasionally split in a conservative manner. ◮ Applications of the model: Random planar maps [Bertoin&Curien&Kortchemski, 2015+; Bertoin&Budd&Curien&Kortchemski, 2016+] Quan Shi Growth-Fragmentations CIMAT, 11-15 December, 2017 9 / 34

Growth-fragmentation processes ◮ (Markovian) growth-fragmentation processes [Bertoin 2017] describe the evolution of the sizes of a family of particles, which can grow larger or smaller with time, and occasionally split in a conservative manner. ◮ Applications of the model: Random planar maps [Bertoin&Curien&Kortchemski, 2015+; Bertoin&Budd&Curien&Kortchemski, 2016+] ◮ Simulation by I.Kortchemski & N.Curien : https://www.normalesup.org/ kortchem/images/tribord.gif Quan Shi Growth-Fragmentations CIMAT, 11-15 December, 2017 9 / 34

Construction of growth-fragmentations [Bertoin 2017] ◮ Starfishes: Quan Shi Growth-Fragmentations CIMAT, 11-15 December, 2017 10 / 34

Construction of growth-fragmentations [Bertoin 2017] ◮ Asexual reproduction of starfishes: Quan Shi Growth-Fragmentations CIMAT, 11-15 December, 2017 10 / 34

Construction of growth-fragmentations [Bertoin 2017] ◮ Asexual reproduction of starfishes: ◮ Growth: The size of the ancestor evolves according to a positive self-similar Markov process (pssMp) X , with only negative jumps. Quan Shi Growth-Fragmentations CIMAT, 11-15 December, 2017 10 / 34

Construction of growth-fragmentations [Bertoin 2017] ◮ Asexual reproduction of starfishes: ◮ Growth: The size of the ancestor evolves according to a positive self-similar Markov process (pssMp) X , with only negative jumps. ◮ Regeneration: At each jump time t ≥ 0 with − y := X ( t ) − X ( t − ) < 0, a daughter is born with initial size y . The size evolution of the daughter has the same distribution as X (but started at y ), and is independent of other daughters. ◮ Granddaughters are born at the jumps of each daughter, and so on.