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

▶
introduction to self similar growth fragmentations
SMART_READER_LITE
LIVE PREVIEW

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

Nov 22, 2023 453 likes •1.39k views

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

slide-1
SLIDE 1

Introduction to self-similar growth-fragmentations

Quan Shi CIMAT, 11-15 December, 2017

Quan Shi Growth-Fragmentations CIMAT, 11-15 December, 2017 1 / 34

slide-2
SLIDE 2

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

slide-3
SLIDE 3

Overview

  • 1. Background: fragmentation processes

Quan Shi Growth-Fragmentations CIMAT, 11-15 December, 2017 3 / 34

slide-4
SLIDE 4

Overview

  • 1. Background: fragmentation processes
  • 2. Construction of growth-fragmentation processes

Quan Shi Growth-Fragmentations CIMAT, 11-15 December, 2017 3 / 34

slide-5
SLIDE 5

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

slide-6
SLIDE 6

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

slide-7
SLIDE 7
  • 1. Background: fragmentation processes

Quan Shi Growth-Fragmentations CIMAT, 11-15 December, 2017 4 / 34

slide-8
SLIDE 8

Background

◮ Fragmentation: “the process or state of breaking or being broken into

fragments”.

Quan Shi Growth-Fragmentations CIMAT, 11-15 December, 2017 5 / 34

slide-9
SLIDE 9

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

slide-10
SLIDE 10

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

slide-11
SLIDE 11

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

slide-12
SLIDE 12

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

slide-13
SLIDE 13

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 xαν(dy).

Quan Shi Growth-Fragmentations CIMAT, 11-15 December, 2017 6 / 34

slide-14
SLIDE 14

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 xαν(dy).

◮ Record the sizes of the fragments at time t ≥ 0 by a measure on R+

X(t) =

  • i≥1

δXi(t). 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

slide-15
SLIDE 15

Examples:

◮ Splitting intervals [BrennanDurrett1986]:

◮ Ui: 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 I1(t), I2(t), . . . ◮ F(t) :=

i≥1 δIi (t) is a self-similar fragmentation with characteristics (1, ν),

where ν(dx) = 2dx, x ∈ [ 1

2, 1).

Quan Shi Growth-Fragmentations CIMAT, 11-15 December, 2017 7 / 34

slide-16
SLIDE 16

Examples:

◮ Splitting intervals [BrennanDurrett1986]:

◮ Ui: 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 I1(t), I2(t), . . . ◮ F(t) :=

i≥1 δIi (t) is a self-similar fragmentation with characteristics (1, ν),

where ν(dx) = 2dx, x ∈ [ 1

2, 1).

◮ The Brownian fragmentation

◮ Normalized Brownian excursion:

B : [0, 1] → R+.

◮ O(t) := {x ∈ (0, 1) : B(x) > t}. ◮ F(t) :=

I: component of O(t) δ|I|

◮ F is a self-similar fragmentation

with characteristics (− 1

2, νB),

where νB(dx) =

2

√

2πx3(1−x)3 dx,

x ∈ [ 1

2, 1).

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8

xs2 t1 t3 t2 xs1

Quan Shi Growth-Fragmentations CIMAT, 11-15 December, 2017 7 / 34

slide-17
SLIDE 17
  • 2. Construction of growth-fragmentation processes

Quan Shi Growth-Fragmentations CIMAT, 11-15 December, 2017 8 / 34

slide-18
SLIDE 18

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

slide-19
SLIDE 19

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

slide-20
SLIDE 20

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

slide-21
SLIDE 21

Construction of growth-fragmentations [Bertoin 2017]

◮ Starfishes:

Quan Shi Growth-Fragmentations CIMAT, 11-15 December, 2017 10 / 34

slide-22
SLIDE 22

Construction of growth-fragmentations [Bertoin 2017]

◮ Asexual reproduction of starfishes:

Quan Shi Growth-Fragmentations CIMAT, 11-15 December, 2017 10 / 34

slide-23
SLIDE 23

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

slide-24
SLIDE 24

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.

slide-25
SLIDE 25

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.

Quan Shi Growth-Fragmentations CIMAT, 11-15 December, 2017 10 / 34

slide-26
SLIDE 26

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.

Quan Shi Growth-Fragmentations CIMAT, 11-15 December, 2017 10 / 34

slide-27
SLIDE 27

X∅

X: a pssMp with negative jumps

Simulation by B.Dadoun

Quan Shi Growth-Fragmentations CIMAT, 11-15 December, 2017 11 / 34

slide-28
SLIDE 28

X∅ x b1

X: a pssMp with negative jumps

Simulation by B.Dadoun

Quan Shi Growth-Fragmentations CIMAT, 11-15 December, 2017 11 / 34

slide-29
SLIDE 29

X∅ x b1

X: a pssMp with negative jumps

Simulation by B.Dadoun

Quan Shi Growth-Fragmentations CIMAT, 11-15 December, 2017 11 / 34

slide-30
SLIDE 30

X∅ b1 X1

X: a pssMp with negative jumps

Simulation by B.Dadoun

Quan Shi Growth-Fragmentations CIMAT, 11-15 December, 2017 11 / 34

slide-31
SLIDE 31

X∅ b1 X1 b2

X: a pssMp with negative jumps

Simulation by B.Dadoun

Quan Shi Growth-Fragmentations CIMAT, 11-15 December, 2017 11 / 34

slide-32
SLIDE 32

X∅ b1 X1 b2 X2

X: a pssMp with negative jumps

Simulation by B.Dadoun

Quan Shi Growth-Fragmentations CIMAT, 11-15 December, 2017 11 / 34

slide-33
SLIDE 33

X∅ b1 X1 b2 X2 b21

X: a pssMp with negative jumps

Simulation by B.Dadoun

Quan Shi Growth-Fragmentations CIMAT, 11-15 December, 2017 11 / 34

slide-34
SLIDE 34

X∅ b1 X1 b2 X2 b21 X21

X: a pssMp with negative jumps

Simulation by B.Dadoun

Quan Shi Growth-Fragmentations CIMAT, 11-15 December, 2017 11 / 34

slide-35
SLIDE 35

X∅ b1 X1 b2 X2 b21 X21

Record the sizes of all individuals at time t ≥ 0 by a point measure on R+: X(t) :=

  • u∈U

δXu(t−bu)1{bu≤t,Xu(t−bu)>0}, t ≥ 0.

Quan Shi Growth-Fragmentations CIMAT, 11-15 December, 2017 11 / 34

slide-36
SLIDE 36

Self-similar growth-fragmentations

◮ The population is indexed by the Ulam-Harris tree: U = ∞ i=0 Ni. By

convention N0 = {∅}. An element u = (n1, . . . , ni) ∈ Ni, then uk := (n1, . . . , ni, k) for k ∈ N.

Quan Shi Growth-Fragmentations CIMAT, 11-15 December, 2017 12 / 34

slide-37
SLIDE 37

Self-similar growth-fragmentations

◮ The population is indexed by the Ulam-Harris tree: U = ∞ i=0 Ni. By

convention N0 = {∅}. An element u = (n1, . . . , ni) ∈ Ni, then uk := (n1, . . . , ni, k) for k ∈ N.

◮ Construct a cell system, that is a family

(Xu, bu), u ∈ U. Xu: evolution of the size of u as time grows. bu: birth time of u.

Quan Shi Growth-Fragmentations CIMAT, 11-15 December, 2017 12 / 34

slide-38
SLIDE 38

Self-similar growth-fragmentations

◮ The population is indexed by the Ulam-Harris tree: U = ∞ i=0 Ni. By

convention N0 = {∅}. An element u = (n1, . . . , ni) ∈ Ni, then uk := (n1, . . . , ni, k) for k ∈ N.

◮ Construct a cell system, that is a family

(Xu, bu), u ∈ U. Xu: evolution of the size of u as time grows. bu: birth time of u.

  • 1. Initialization: b∅ := 0,

X∅

d

= Px (the law of the pssMp X started from x).

  • 2. Induction: enumerate the jump times of Xu by (tk, k ∈ N) and let

yk := −∆Xu(tk) > 0. Then buk = bu + tk, Xuk

d

= Pyk , k ∈ N. The daughters (Xuk, k ∈ N) are independent.

Quan Shi Growth-Fragmentations CIMAT, 11-15 December, 2017 12 / 34

slide-39
SLIDE 39

Self-similar growth-fragmentations

◮ The population is indexed by the Ulam-Harris tree: U = ∞ i=0 Ni. By

convention N0 = {∅}. An element u = (n1, . . . , ni) ∈ Ni, then uk := (n1, . . . , ni, k) for k ∈ N.

◮ Construct a cell system, that is a family

(Xu, bu), u ∈ U. Xu: evolution of the size of u as time grows. bu: birth time of u.

  • 1. Initialization: b∅ := 0,

X∅

d

= Px (the law of the pssMp X started from x).

  • 2. Induction: enumerate the jump times of Xu by (tk, k ∈ N) and let

yk := −∆Xu(tk) > 0. Then buk = bu + tk, Xuk

d

= Pyk , k ∈ N. The daughters (Xuk, k ∈ N) are independent.

◮ (Xu, u ∈ U) is a Crump-Mode-Jagers branching process [Jagers, 1983]. ◮ The law of (Xu, u ∈ U) is determined by the pssMp X.

Quan Shi Growth-Fragmentations CIMAT, 11-15 December, 2017 12 / 34

slide-40
SLIDE 40

Self-similar growth-fragmentation processes

◮ Record the sizes of all individuals alive at time t ≥ 0 by a point measure on

R+: X(t) :=

  • u∈U

δXu(t−bu)1{bu≤t,Xu(t−bu)>0}, t ≥ 0, where δ denotes the Dirac measure. The process X is called a self-similar growth-fragmentation associated with X.

Quan Shi Growth-Fragmentations CIMAT, 11-15 December, 2017 13 / 34

slide-41
SLIDE 41

Self-similar growth-fragmentation processes

◮ Record the sizes of all individuals alive at time t ≥ 0 by a point measure on

R+: X(t) :=

  • u∈U

δXu(t−bu)1{bu≤t,Xu(t−bu)>0}, t ≥ 0, where δ denotes the Dirac measure. The process X is called a self-similar growth-fragmentation associated with X.

◮ X does not contain any information of the genealogy. ◮ The law of the pssMp X determines the law of X.

Quan Shi Growth-Fragmentations CIMAT, 11-15 December, 2017 13 / 34

slide-42
SLIDE 42

Self-similar growth-fragmentation processes

◮ Record the sizes of all individuals alive at time t ≥ 0 by a point measure on

R+: X(t) :=

  • u∈U

δXu(t−bu)1{bu≤t,Xu(t−bu)>0}, t ≥ 0, where δ denotes the Dirac measure. The process X is called a self-similar growth-fragmentation associated with X.

◮ X does not contain any information of the genealogy. ◮ The law of the pssMp X determines the law of X.

Question

◮ Is X(t) a Radon measure on R+?

Quan Shi Growth-Fragmentations CIMAT, 11-15 December, 2017 13 / 34

slide-43
SLIDE 43

Positive self-similar Markov processes

Let X be a positive self-similar Markov process (pssMp) with no positive jumps, and Px be the law of X starting from X(0) = x.

◮ α ∈ R is the index of self-similarity: if X(0) = x, then for every r > 0

(rX(r αt), t ≥ 0) have the law of Prx and Px.

Quan Shi Growth-Fragmentations CIMAT, 11-15 December, 2017 14 / 34

slide-44
SLIDE 44

Positive self-similar Markov processes

Let X be a positive self-similar Markov process (pssMp) with no positive jumps, and Px be the law of X starting from X(0) = x.

◮ α ∈ R is the index of self-similarity: if X(0) = x, then for every r > 0

(rX(r αt), t ≥ 0) have the law of Prx and Px.

◮ For the homogeneous case α = 0: there exists a L´

evy process ξ such that X (0)(t) = exp(ξt), t ≥ 0.

Quan Shi Growth-Fragmentations CIMAT, 11-15 December, 2017 14 / 34

slide-45
SLIDE 45

Positive self-similar Markov processes

Let X be a positive self-similar Markov process (pssMp) with no positive jumps, and Px be the law of X starting from X(0) = x.

◮ α ∈ R is the index of self-similarity: if X(0) = x, then for every r > 0

(rX(r αt), t ≥ 0) have the law of Prx and Px.

◮ For the homogeneous case α = 0: there exists a L´

evy process ξ such that X (0)(t) = exp(ξt), t ≥ 0.

◮ ξ (without positive jumps, not killed) is characterized by its Laplace exponent

Φ : [0, ∞) → R, E

  • eqξ(t)

= eΦ(q)t, for all t, q ≥ 0.

◮ The function Φ is convex, given by the L´

evy-Khintchine formula: Φ(q) = 1 2σ2q2 + cq +

  • (−∞,0)

(eqx − 1 + q(1 − ex)) Λ(dx), q ≥ 0, where σ2 ≥ 0, c ∈ R, and the L´ evy measure Λ is a measure on (−∞, 0) with

  • (−∞,0)

(|x|2 ∧ 1)Λ(dx) < ∞.

Quan Shi Growth-Fragmentations CIMAT, 11-15 December, 2017 14 / 34

slide-46
SLIDE 46

Positive self-similar Markov processes

◮ Lamperti’s transform [Lamperti 1972]: when α = 0:

X (α)

t

= exp(ξτ (α)(t)), 0 ≤ t < ζ(α), where (τ (α)(t), t ≥ 0) is an explicit time-change that depends on α: τ (α)(t) := inf

  • s ≥ 0 :

s exp(−αξr)dr > t

  • ,

t ≥ 0.

Quan Shi Growth-Fragmentations CIMAT, 11-15 December, 2017 15 / 34

slide-47
SLIDE 47

Positive self-similar Markov processes

◮ Lamperti’s transform [Lamperti 1972]: when α = 0:

X (α)

t

= exp(ξτ (α)(t)), 0 ≤ t < ζ(α), where (τ (α)(t), t ≥ 0) is an explicit time-change that depends on α: τ (α)(t) := inf

  • s ≥ 0 :

s exp(−αξr)dr > t

  • ,

t ≥ 0.

◮ The lifetime of X (α):

ζ(α) := ∞ exp(−αξr)dr ∈ (0, ∞]

Quan Shi Growth-Fragmentations CIMAT, 11-15 December, 2017 15 / 34

slide-48
SLIDE 48

Positive self-similar Markov processes

◮ Lamperti’s transform [Lamperti 1972]: when α = 0:

X (α)

t

= exp(ξτ (α)(t)), 0 ≤ t < ζ(α), where (τ (α)(t), t ≥ 0) is an explicit time-change that depends on α: τ (α)(t) := inf

  • s ≥ 0 :

s exp(−αξr)dr > t

  • ,

t ≥ 0.

◮ The lifetime of X (α):

ζ(α) := ∞ exp(−αξr)dr ∈ (0, ∞]

◮ The law of the pssMp X is characterized by (Φ, α).

Quan Shi Growth-Fragmentations CIMAT, 11-15 December, 2017 15 / 34

slide-49
SLIDE 49

Lamperti’s time-change

X (0)(t) = eξ(t)

[ξ: L´ evy process]

Simulation by B.Dadoun

Quan Shi Growth-Fragmentations CIMAT, 11-15 December, 2017 16 / 34

slide-50
SLIDE 50

Lamperti’s time-change

X (0)(t) = eξ(t)

[ξ: L´ evy process]

X (α)(t) = X (0) t

  • X (α)(s)

α ds

  • Simulation by B.Dadoun

Quan Shi Growth-Fragmentations CIMAT, 11-15 December, 2017 16 / 34

slide-51
SLIDE 51

A coupling construction

Repeat Lamperti’s time-change for each trajectory from the homogeneous case: X (α)

u

(t) := X (0)

u (τ (α) u

(t)) with τ (α)

u

(t) := inf

  • r ≥ 0 :

r

0 X (0) u (s)−αds ≥ t

  • α = 0

Simulation by B.Dadoun

Quan Shi Growth-Fragmentations CIMAT, 11-15 December, 2017 17 / 34

slide-52
SLIDE 52

A coupling construction

Repeat Lamperti’s time-change for each trajectory from the homogeneous case: X (α)

u

(t) := X (0)

u (τ (α) u

(t)) with τ (α)

u

(t) := inf

  • r ≥ 0 :

r

0 X (0) u (s)−αds ≥ t

  • α = 0.5

Simulation by B.Dadoun

Quan Shi Growth-Fragmentations CIMAT, 11-15 December, 2017 17 / 34

slide-53
SLIDE 53
  • 3. Properties of self-similar growth-fragmentations

Quan Shi Growth-Fragmentations CIMAT, 11-15 December, 2017 18 / 34

slide-54
SLIDE 54

Cumulant function for the homogeneous case

◮ X: a homogeneous growth-fragmentation associated with a pssMp X with

characteristics (Φ, α = 0).

Quan Shi Growth-Fragmentations CIMAT, 11-15 December, 2017 19 / 34

slide-55
SLIDE 55

Cumulant function for the homogeneous case

◮ X: a homogeneous growth-fragmentation associated with a pssMp X with

characteristics (Φ, α = 0).

◮ Define the cumulant κ: [0, ∞) → (−∞, ∞] by

κ(q) := Φ(q) +

  • (−∞,0)

(1 − ex)q Λ(dx), q ≥ 0.

Quan Shi Growth-Fragmentations CIMAT, 11-15 December, 2017 19 / 34

slide-56
SLIDE 56

Cumulant function for the homogeneous case

◮ X: a homogeneous growth-fragmentation associated with a pssMp X with

characteristics (Φ, α = 0).

◮ Define the cumulant κ: [0, ∞) → (−∞, ∞] by

κ(q) := Φ(q) +

  • (−∞,0)

(1 − ex)q Λ(dx), q ≥ 0.

◮ κ is convex and possibly takes value at ∞. But κ(q) < ∞ for all q ≥ 2.

Quan Shi Growth-Fragmentations CIMAT, 11-15 December, 2017 19 / 34

slide-57
SLIDE 57

Cumulant function for the homogeneous case

◮ X: a homogeneous growth-fragmentation associated with a pssMp X with

characteristics (Φ, α = 0).

◮ Define the cumulant κ: [0, ∞) → (−∞, ∞] by

κ(q) := Φ(q) +

  • (−∞,0)

(1 − ex)q Λ(dx), q ≥ 0.

Theorem (Bertoin, 2016)

Suppose that κ(q) < ∞. If α = 0, then for every t ≥ 0, Ex

R+

y qXt(dy)

  • = xq exp(κ(q)t).

Quan Shi Growth-Fragmentations CIMAT, 11-15 December, 2017 19 / 34

slide-58
SLIDE 58

Non-explosion condition

◮ X: a self-similar growth-fragmentation associated with a pssMp X with

characteristics (Φ, α).

Quan Shi Growth-Fragmentations CIMAT, 11-15 December, 2017 20 / 34

slide-59
SLIDE 59

Non-explosion condition

◮ X: a self-similar growth-fragmentation associated with a pssMp X with

characteristics (Φ, α).

◮ X is called non-explosive, if

for every t ≥ 0 and a > 0, X(t) has finite mass on [a, ∞).

Quan Shi Growth-Fragmentations CIMAT, 11-15 December, 2017 20 / 34

slide-60
SLIDE 60

Non-explosion condition

◮ X: a self-similar growth-fragmentation associated with a pssMp X with

characteristics (Φ, α).

◮ X is called non-explosive, if

for every t ≥ 0 and a > 0, X(t) has finite mass on [a, ∞).

◮ [Bertoin&Stephenson, 2016]: if α = 0 and κ(q) > 0 for all q ≥ 0, then X

explodes in finite time.

Quan Shi Growth-Fragmentations CIMAT, 11-15 December, 2017 20 / 34

slide-61
SLIDE 61

Non-explosion condition

◮ X: a self-similar growth-fragmentation associated with a pssMp X with

characteristics (Φ, α).

◮ X is called non-explosive, if

for every t ≥ 0 and a > 0, X(t) has finite mass on [a, ∞).

◮ [Bertoin&Stephenson, 2016]: if α = 0 and κ(q) > 0 for all q ≥ 0, then X

explodes in finite time.

Theorem (Bertoin, 2017)

Suppose that there exists q > 0 such that κ(q) ≤ 0. Then for every t ≥ 0, there is the inequality Ex

R+

y qXt(dy)

  • ≤ xq.

Quan Shi Growth-Fragmentations CIMAT, 11-15 December, 2017 20 / 34

slide-62
SLIDE 62

Proof of the theorem

◮ Using Lamperti’s transform, we have

Ex

  • X(t)q +
  • 0≤s≤t

|∆X(s)|q ≤ xq, for all t ≥ 0.

Quan Shi Growth-Fragmentations CIMAT, 11-15 December, 2017 21 / 34

slide-63
SLIDE 63

Proof of the theorem

◮ Using Lamperti’s transform, we have

Ex

  • X(t)q +
  • 0≤s≤t

|∆X(s)|q ≤ xq, for all t ≥ 0.

◮ Consequence: fix t ≥ 0, we have a supermartingale:

Σn :=

  • |u|≤n−1,bu≤t

Xu(t − bu)q +

  • |v|=n,bv≤s≤t

|∆Xv(s − bv)|q, n ≥ 1.

Quan Shi Growth-Fragmentations CIMAT, 11-15 December, 2017 21 / 34

slide-64
SLIDE 64

Proof of the theorem

◮ Using Lamperti’s transform, we have

Ex

  • X(t)q +
  • 0≤s≤t

|∆X(s)|q ≤ xq, for all t ≥ 0.

◮ Consequence: fix t ≥ 0, we have a supermartingale:

Σn :=

  • |u|≤n−1,bu≤t

Xu(t − bu)q +

  • |v|=n,bv≤s≤t

|∆Xv(s − bv)|q, n ≥ 1.

◮

Ex

  • u∈U:bu≤t

Xu(t − bu)q ≤ Ex[Σ∞] ≤ Ex[Σ0] = Ex

  • X∅(t)q +
  • 0≤s≤t

|∆X∅(s)|q ≤ xq.

Quan Shi Growth-Fragmentations CIMAT, 11-15 December, 2017 21 / 34

slide-65
SLIDE 65

Properties of self-similar growth-fragmentations

Let X be a self-similar growth-fragmentation driven by X started from X(0) = δx. Suppose that the non-explosion condition holds.

◮ [The branching property]

Write X(s) =

i≥1 δXi(s). Conditionally on σ(X(r), r ≤ s), we have

(X(t + s), t ≥ 0)

d

=

  • i≥1

X(i)(t), where (X(i), i ≥ 1) are independent self-similar growth-fragmentations driven by X, with X(i) started at X(i)(0) = δXi(s).

◮ X is self-similar with index α: X(0) = δx. For every θ > 0,

(

  • i≥1

δθXi(θαt), t ≥ 0)

d

= X started from δθx.

Quan Shi Growth-Fragmentations CIMAT, 11-15 December, 2017 22 / 34

slide-66
SLIDE 66

Extinction

Theorem (Bertoin 2017 )

Suppose that α < 0 and that there exists q > 0 such that κ(q) < 0. Then the extinction time inf{t ≥ 0 : X(α)(t) = 0} is Px-a.s. finite for every x > 0.

Quan Shi Growth-Fragmentations CIMAT, 11-15 December, 2017 23 / 34

slide-67
SLIDE 67

Extinction

Theorem (Bertoin 2017 )

Suppose that α < 0 and that there exists q > 0 such that κ(q) < 0. Then the extinction time inf{t ≥ 0 : X(α)(t) = 0} is Px-a.s. finite for every x > 0.

Proof.

◮ inf{t ≥ 0 : X(α)(t) = 0} = sup

  • b(α)

u

+ ζ(α)

u

: u ∈ U

  • Quan Shi

Growth-Fragmentations CIMAT, 11-15 December, 2017 23 / 34

slide-68
SLIDE 68

Extinction

Theorem (Bertoin 2017 )

Suppose that α < 0 and that there exists q > 0 such that κ(q) < 0. Then the extinction time inf{t ≥ 0 : X(α)(t) = 0} is Px-a.s. finite for every x > 0.

Proof.

◮ inf{t ≥ 0 : X(α)(t) = 0} = sup

  • b(α)

u

+ ζ(α)

u

: u ∈ U

  • ◮ Y(α)

u

: the ancestral lineage of u.

◮ identity: b(α) u

+ ζ(α)

u

= ∞ Y(0)

u (s)−αds

Quan Shi Growth-Fragmentations CIMAT, 11-15 December, 2017 23 / 34

slide-69
SLIDE 69

Extinction

Theorem (Bertoin 2017 )

Suppose that α < 0 and that there exists q > 0 such that κ(q) < 0. Then the extinction time inf{t ≥ 0 : X(α)(t) = 0} is Px-a.s. finite for every x > 0.

Proof.

◮ inf{t ≥ 0 : X(α)(t) = 0} = sup

  • b(α)

u

+ ζ(α)

u

: u ∈ U

  • ◮ Y(α)

u

: the ancestral lineage of u.

◮ identity: b(α) u

+ ζ(α)

u

= ∞ Y(0)

u (s)−αds ◮ Y(0) u (t)q ≤ X (0) 1 (t)q ≤ eκ(q)t

e−κ(q)t

i≥1 X (0) i

(t)q ≤ Ceκ(q)t

Quan Shi Growth-Fragmentations CIMAT, 11-15 December, 2017 23 / 34

slide-70
SLIDE 70
  • 4. Martingales in self-similar growth-fragmentations

Quan Shi Growth-Fragmentations CIMAT, 11-15 December, 2017 24 / 34

slide-71
SLIDE 71

Hypothesis

◮ Recall that κ is convex, so it has at most two roots.

Quan Shi Growth-Fragmentations CIMAT, 11-15 December, 2017 25 / 34

slide-72
SLIDE 72

Hypothesis

◮ Recall that κ is convex, so it has at most two roots. ◮ Suppose that the Cram´

er’s hypothesis holds: ω+ > ω− > 0, s.t. κ(ω−) = κ(ω+) = 0 and κ′(ω−) > −∞. [H]

Quan Shi Growth-Fragmentations CIMAT, 11-15 December, 2017 25 / 34

slide-73
SLIDE 73

Hypothesis

◮ Recall that κ is convex, so it has at most two roots. ◮ Suppose that the Cram´

er’s hypothesis holds: ω+ > ω− > 0, s.t. κ(ω−) = κ(ω+) = 0 and κ′(ω−) > −∞. [H]

◮ [H] implies the non-explosion condition (there exists q > 0 with κ(q) < 0).

Quan Shi Growth-Fragmentations CIMAT, 11-15 December, 2017 25 / 34

slide-74
SLIDE 74

The genealogical martingales

◮ (− log Xu(0), u ∈ U) is a branching random walk. ◮ If Φ(q) < 0 and κ(q) < ∞, then

m(q) := E1

|u|=1

e−q(− log Xu(0)) = E1

0<s<ζ

|∆X(s)|q = 1 − κ(q) Φ(q).

Lemma

Suppose that [H] holds. 1. M+(n) := x−ω+

|u|=n

Xu(0)ω+ is a martingale that converges Px-a.s. to 0.

  • 2. for any p ∈ [1, ω+

ω− ), the martingale

M−(n) := x−ω−

|u|=n

Xu(0)ω− converges Px-a.s. and in Lp(Px). Its terminal value M−(∞) > 0.

Quan Shi Growth-Fragmentations CIMAT, 11-15 December, 2017 26 / 34

slide-75
SLIDE 75

Mellin transform of the potential measure

◮ Fact: For a pssMp X (α) with characteristics (Φ, α):

Ex ∞ X (α)(t)q+αdt

  • = −

1 Φ(q)xq. whenever Φ(q) < 0.

Quan Shi Growth-Fragmentations CIMAT, 11-15 December, 2017 27 / 34

slide-76
SLIDE 76

Mellin transform of the potential measure

◮ Fact: For a pssMp X (α) with characteristics (Φ, α):

Ex ∞ X (α)(t)q+αdt

  • = −

1 Φ(q)xq. whenever Φ(q) < 0.

Proposition (BBCK, 2016+)

For every q with κ(q) < 0: Ex ∞ ∞

  • i=1

X (α)

i

(t)q+α dt

  • = −

1 κ(q)xq.

Quan Shi Growth-Fragmentations CIMAT, 11-15 December, 2017 27 / 34

slide-77
SLIDE 77

Mellin transform of the potential measure

◮ Fact: For a pssMp X (α) with characteristics (Φ, α):

Ex ∞ X (α)(t)q+αdt

  • = −

1 Φ(q)xq. whenever Φ(q) < 0.

Proposition (BBCK, 2016+)

For every q with κ(q) < 0: Ex ∞ ∞

  • i=1

X (α)

i

(t)q+α dt

  • = −

1 κ(q)xq. Remarks: Consider a growth-fragmentation ˜ X associated with another pssMp ˜ X with characteristics (˜ Φ, ˜ α).

◮ X d

= ˜ X (Φ = ˜ Φ and α = ˜ α) ⇒ X

d

= ˜ X; but X

d

= ˜ X ⇒ X

d

= ˜ X.

Quan Shi Growth-Fragmentations CIMAT, 11-15 December, 2017 27 / 34

slide-78
SLIDE 78

Mellin transform of the potential measure

◮ Fact: For a pssMp X (α) with characteristics (Φ, α):

Ex ∞ X (α)(t)q+αdt

  • = −

1 Φ(q)xq. whenever Φ(q) < 0.

Proposition (BBCK, 2016+)

For every q with κ(q) < 0: Ex ∞ ∞

  • i=1

X (α)

i

(t)q+α dt

  • = −

1 κ(q)xq. Remarks: Consider a growth-fragmentation ˜ X associated with another pssMp ˜ X with characteristics (˜ Φ, ˜ α).

◮ X d

= ˜ X (Φ = ˜ Φ and α = ˜ α) ⇒ X

d

= ˜ X; but X

d

= ˜ X ⇒ X

d

= ˜ X.

◮ If X d

= ˜ X, then κ = ˜ κ and α = ˜ α.

Quan Shi Growth-Fragmentations CIMAT, 11-15 December, 2017 27 / 34

slide-79
SLIDE 79

Mellin transform of the potential measure

◮ Fact: For a pssMp X (α) with characteristics (Φ, α):

Ex ∞ X (α)(t)q+αdt

  • = −

1 Φ(q)xq. whenever Φ(q) < 0.

Proposition (BBCK, 2016+)

For every q with κ(q) < 0: Ex ∞ ∞

  • i=1

X (α)

i

(t)q+α dt

  • = −

1 κ(q)xq. Remarks: Consider a growth-fragmentation ˜ X associated with another pssMp ˜ X with characteristics (˜ Φ, ˜ α).

◮ X d

= ˜ X (Φ = ˜ Φ and α = ˜ α) ⇒ X

d

= ˜ X; but X

d

= ˜ X ⇒ X

d

= ˜ X.

◮ If X d

= ˜ X, then κ = ˜ κ and α = ˜ α.

◮ Conversely, if κ = ˜

κ and α = ˜ α, then X

d

= ˜

  • X. [Pitman&Winkel, 2015; S.

2017]

Quan Shi Growth-Fragmentations CIMAT, 11-15 December, 2017 27 / 34

slide-80
SLIDE 80

Many-to-one formula

◮ Define the intensity measure µx t of X(t) on R+ such that for all f ∈ C ∞ c (R+):

µx

t , f :=

  • R+

f (y)µx

t (dy) = Ex

∞

  • i=1

f (Xi(t))

  • .

Quan Shi Growth-Fragmentations CIMAT, 11-15 December, 2017 28 / 34

slide-81
SLIDE 81

Many-to-one formula

◮ Define the intensity measure µx t of X(t) on R+ such that for all f ∈ C ∞ c (R+):

µx

t , f :=

  • R+

f (y)µx

t (dy) = Ex

∞

  • i=1

f (Xi(t))

  • .

◮

Φ−(q) := κ(q + ω−), q ≥ 0

  • is the Laplace exponent of a L´

evy process η−, and η− drifts to −∞.

Theorem (BBCK, 2016+)

µx

t (dy) =

x y ω− ρ−

t (x, dy),

y > 0, where ρ−

t (x, ·) be the transition kernel of a pssMp Y −(t) with characteristics

(Φ−, α) starting from x > 0.

Quan Shi Growth-Fragmentations CIMAT, 11-15 December, 2017 28 / 34

slide-82
SLIDE 82

Many-to-one formula

◮ Define the intensity measure µx t of X(t) on R+ such that for all f ∈ C ∞ c (R+):

µx

t , f :=

  • R+

f (y)µx

t (dy) = Ex

∞

  • i=1

f (Xi(t))

  • .

◮

Φ−(q) := κ(q + ω−), q ≥ 0

  • is the Laplace exponent of a L´

evy process η−, and η− drifts to −∞.

Theorem (BBCK, 2016+)

µx

t (dy) =

x y ω− ρ−

t (x, dy),

y > 0, where ρ−

t (x, ·) be the transition kernel of a pssMp Y −(t) with characteristics

(Φ−, α) starting from x > 0. That is: Ex ∞

  • i=1

f (Xi(t))

  • = Ex
  • f (Y −(t))
  • x

Y −(t) ω− 1{Y −(t)∈(0,∞)}

  • Quan Shi

Growth-Fragmentations CIMAT, 11-15 December, 2017 28 / 34

slide-83
SLIDE 83

Proof of many-to-one formula

◮ For θ such that κ(θ) < 0: κ(· + θ) is the Laplace exponent of a L´

evy process (with killing rate −κ(θ)).

Quan Shi Growth-Fragmentations CIMAT, 11-15 December, 2017 29 / 34

slide-84
SLIDE 84

Proof of many-to-one formula

◮ For θ such that κ(θ) < 0: κ(· + θ) is the Laplace exponent of a L´

evy process (with killing rate −κ(θ)).

◮ Define

˜ ρt(x, ·), f := x−θEx ∞

  • i=1

f (Xi(t))Xi(t)θ .

Quan Shi Growth-Fragmentations CIMAT, 11-15 December, 2017 29 / 34

slide-85
SLIDE 85

Proof of many-to-one formula

◮ For θ such that κ(θ) < 0: κ(· + θ) is the Laplace exponent of a L´

evy process (with killing rate −κ(θ)).

◮ Define

˜ ρt(x, ·), f := x−θEx ∞

  • i=1

f (Xi(t))Xi(t)θ .

◮

µx

t (dy) =

x y θ ˜ ρt(x, dy), y > 0.

◮ ˜

ρt is the transition kernel of a pssMp ˜ Y with characteristics (κ(· + θ), α):

Quan Shi Growth-Fragmentations CIMAT, 11-15 December, 2017 29 / 34

slide-86
SLIDE 86

Proof of many-to-one formula

◮ For θ such that κ(θ) < 0: κ(· + θ) is the Laplace exponent of a L´

evy process (with killing rate −κ(θ)).

◮ Define

˜ ρt(x, ·), f := x−θEx ∞

  • i=1

f (Xi(t))Xi(t)θ .

◮

µx

t (dy) =

x y θ ˜ ρt(x, dy), y > 0.

◮ ˜

ρt is the transition kernel of a pssMp ˜ Y with characteristics (κ(· + θ), α):

◮ ˜

ρt+s(x, ·), f =

  • (0,∞) ˜

ρt(y, ·), f ˜ ρs(x, dy) Chapman-Kolmogorov equation

◮ ˜

ρt(x, ·), f = ˜ ρxαt(1, ·), f (x·) the self-similarity.

◮ For every q with κ(q + θ) < 0:

∞ dt

  • (0,∞)

y q+α˜ ρt(1, dy) = x−θEx ∞ ∞

  • i=1

Xi(t)θ+q+α dt

  • = −

1 κ(θ + q) =

Quan Shi Growth-Fragmentations CIMAT, 11-15 December, 2017 29 / 34

slide-87
SLIDE 87

Growth-fragmentation equations

The intensity measure µx

t of X(t): for all f ∈ C ∞ c (R+),

µx

t , f :=

  • R+

f (y)µx

t (dy) = Ex

∞

  • i=1

f (Xi(t))

  • .

Quan Shi Growth-Fragmentations CIMAT, 11-15 December, 2017 30 / 34

slide-88
SLIDE 88

Growth-fragmentation equations

The intensity measure µx

t of X(t): for all f ∈ C ∞ c (R+),

µx

t , f :=

  • R+

f (y)µx

t (dy) = Ex

∞

  • i=1

f (Xi(t))

  • .

Proposition (Bertoin&Watson, 2016; BBCK, 2016+)

Suppose that [H] holds. Then µx

t , f = f (x) +

t µx

s , Gf ds,

where Gf (y) := y α 1 2σ2f ′′(y)y 2 + (c + 1 2σ2)f ′(y)y +

  • (−∞,0)
  • f (yez) + f (y(1 − ez)) − f (y) + y(1 − ez)f ′(y)
  • Λ(dz)
  • .

Quan Shi Growth-Fragmentations CIMAT, 11-15 December, 2017 30 / 34

slide-89
SLIDE 89

Growth-fragmentation equations

The intensity measure µx

t of X(t): for all f ∈ C ∞ c (R+),

µx

t , f :=

  • R+

f (y)µx

t (dy) = Ex

∞

  • i=1

f (Xi(t))

  • .

Proposition (Bertoin&Watson, 2016; BBCK, 2016+)

Suppose that [H] holds. Then µx

t , f = f (x) +

t µx

s , Gf ds,

where Gf (y) := y α 1 2σ2f ′′(y)y 2 + (c + 1 2σ2)f ′(y)y +

  • (−∞,0)
  • f (yez) + f (y(1 − ez)) − f (y) + y(1 − ez)f ′(y)
  • Λ(dz)
  • .

Proof: Use the many-to-one formula and the infinitesimal generator of the pssMp Y −.

Quan Shi Growth-Fragmentations CIMAT, 11-15 December, 2017 30 / 34

slide-90
SLIDE 90

Temporal martingales

◮ Suppose that [H] holds. For the smaller root ω− of κ, define

M−(t) :=

∞

  • i=1

Xi(t)ω−, t ≥ 0.

Theorem (BBCK, 2016+)

◮ When α ≥ 0, M− is a uniformly integrable Px-martingale; ◮ When α < 0, M− is a Px-supermartingale which converges to 0 in L1(Px).

Proof:

◮ By many-to-one formula: Ex[M−(t)] = xω−P

  • Y −(t) ∈ (0, ∞)
  • ;

◮ If α ≥ 0, then P

  • Y −(t) ∈ (0, ∞)
  • ≡ 1;

◮ If α < 0, then limt→∞ P

  • Y −(t) ∈ (0, ∞)
  • = 0;

◮ We conclude by the branching property of X.

Quan Shi Growth-Fragmentations CIMAT, 11-15 December, 2017 31 / 34

slide-91
SLIDE 91

Temporal martingales

◮ Suppose that [H] holds. For the larger root ω+ of κ, define

M+(t) :=

∞

  • i=1

Xi(t)ω+, t ≥ 0.

Theorem (BBCK, 2016+)

◮ When α > 0, M+ is a Px-supermartingale which converges to 0 in L1(Px); ◮ When α ≤ 0, M+ is a Px-martingale.

Quan Shi Growth-Fragmentations CIMAT, 11-15 December, 2017 32 / 34

slide-92
SLIDE 92

Summary

◮ pssMp X ⇒ cell system (Xu, u ∈ U) ⇒ growth-fragmentation X ◮ the law of X is characterized by the cumulant κ and the index of

self-similarity α

◮ non-explosion condition: there exists κ(q) ≤ 0. ◮ X satisfies the branching property and the self-similarity. ◮ the many-to-one formula ◮ the intensity measure of X solves a (deterministic) growth-fragmentation

equation

◮ If κ(ω−) = κ(ω+) = 0: two martingales M− and M+

Quan Shi Growth-Fragmentations CIMAT, 11-15 December, 2017 33 / 34

slide-93
SLIDE 93

Thank you!

Quan Shi Growth-Fragmentations CIMAT, 11-15 December, 2017 34 / 34