Introduction to self-similar growth-fragmentations

Transcription

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

2 Literature Jean Bertoin, Compensated fragmentation processes and limits of dilated fragmentations, Ann. Probab. 44 (2016), no. 2, MR , Markovian growth-fragmentation processes, Bernoulli 23 (2017), no. 2, MR Jean Bertoin, Timothy Budd, Nicolas Curien, and Igor Kortchemski, Martingales in self-similar growth-fragmentations and their connections with random planar maps, Preprint, arxiv: v1 [math.pr], Jean Bertoin, Nicolas Curien, and Igor Kortchemski, Random planar maps & growth-fragmentations, Preprint, arxiv: v2 [math.pr], July Lecture notes available at my personal webpage: Quan Shi Growth-Fragmentations CIMAT, December, / 34

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

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

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

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, December, / 34

7 1. Background: fragmentation processes Quan Shi Growth-Fragmentations CIMAT, December, / 34

8 Background Fragmentation: the process or state of breaking or being broken into fragments. Quan Shi Growth-Fragmentations CIMAT, December, / 34

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, December, / 34

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, December, / 34

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, December, / 34

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, December, / 34

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 y)ν(dy) <. [1/2,1) 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, December, / 34

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 y)ν(dy) <. [1/2,1) 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) = i 1 δ Xi (t). The process X is a self-similar fragmentation (without erosion) with characteristics (α, ν). Quan Shi Growth-Fragmentations CIMAT, December, / 34

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), I 2(t),... F (t) := i 1 δ I i (t) is a self-similar fragmentation with characteristics (1, ν), where ν(dx) = 2dx, x [ 1 2, 1). Quan Shi Growth-Fragmentations CIMAT, December, / 34

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), I 2(t),... F (t) := i 1 δ I i (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, ν B), 0.4 where ν B (dx) = 2 dx, x [ 1, 1) πx 3 (1 x) 3 2 t t 1 2 t xs 1 xs 2 Quan Shi Growth-Fragmentations CIMAT, December, / 34

17 2. Construction of growth-fragmentation processes Quan Shi Growth-Fragmentations CIMAT, December, / 34

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, December, / 34

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, December, / 34

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: kortchem/images/tribord.gif Quan Shi Growth-Fragmentations CIMAT, December, / 34

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

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

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, December, / 34

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.

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, December, / 34

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, December, / 34

27 X X : a pssmp with negative jumps Simulation by B.Dadoun Quan Shi Growth-Fragmentations CIMAT, December, / 34

28 x X b 1 X : a pssmp with negative jumps Simulation by B.Dadoun Quan Shi Growth-Fragmentations CIMAT, December, / 34

29 x X b 1 X : a pssmp with negative jumps Simulation by B.Dadoun Quan Shi Growth-Fragmentations CIMAT, December, / 34

30 X X 1 b 1 X : a pssmp with negative jumps Simulation by B.Dadoun Quan Shi Growth-Fragmentations CIMAT, December, / 34

31 X X 1 b 1 b 2 X : a pssmp with negative jumps Simulation by B.Dadoun Quan Shi Growth-Fragmentations CIMAT, December, / 34

32 X X 2 X 1 b 1 b 2 X : a pssmp with negative jumps Simulation by B.Dadoun Quan Shi Growth-Fragmentations CIMAT, December, / 34

33 X X 2 X 1 b 1 b 2 b 21 X : a pssmp with negative jumps Simulation by B.Dadoun Quan Shi Growth-Fragmentations CIMAT, December, / 34

34 X X 2 X 21 X 1 b 1 b 2 b 21 X : a pssmp with negative jumps Simulation by B.Dadoun Quan Shi Growth-Fragmentations CIMAT, December, / 34

35 X X 2 X 21 X 1 b 1 b 2 b 21 Record the sizes of all individuals at time t 0 by a point measure on R + : X(t) := δ Xu(t b u)1 {bu t,x u(t b u)>0}, t 0. u U Quan Shi Growth-Fragmentations CIMAT, December, / 34

36 Self-similar growth-fragmentations The population is indexed by the Ulam-Harris tree: U = i=0 Ni. By convention N 0 = { }. An element u = (n 1,..., n i ) N i, then uk := (n 1,..., n i, k) for k N. Quan Shi Growth-Fragmentations CIMAT, December, / 34

37 Self-similar growth-fragmentations The population is indexed by the Ulam-Harris tree: U = i=0 Ni. By convention N 0 = { }. An element u = (n 1,..., n i ) N i, then uk := (n 1,..., n i, k) for k N. Construct a cell system, that is a family (X u, b u ), u U. X u : evolution of the size of u as time grows. b u : birth time of u. Quan Shi Growth-Fragmentations CIMAT, December, / 34

38 Self-similar growth-fragmentations The population is indexed by the Ulam-Harris tree: U = i=0 Ni. By convention N 0 = { }. An element u = (n 1,..., n i ) N i, then uk := (n 1,..., n i, k) for k N. Construct a cell system, that is a family (X u, b u ), u U. X u : evolution of the size of u as time grows. b u : birth time of u. 1. Initialization: b := 0, X d = P x (the law of the pssmp X started from x). 2. Induction: enumerate the jump times of X u by (t k, k N) and let y k := X u(t k ) > 0. Then b uk = b u + t k, X uk d = P yk, k N. The daughters (X uk, k N) are independent. Quan Shi Growth-Fragmentations CIMAT, December, / 34

39 Self-similar growth-fragmentations The population is indexed by the Ulam-Harris tree: U = i=0 Ni. By convention N 0 = { }. An element u = (n 1,..., n i ) N i, then uk := (n 1,..., n i, k) for k N. Construct a cell system, that is a family (X u, b u ), u U. X u : evolution of the size of u as time grows. b u : birth time of u. 1. Initialization: b := 0, X d = P x (the law of the pssmp X started from x). 2. Induction: enumerate the jump times of X u by (t k, k N) and let y k := X u(t k ) > 0. Then b uk = b u + t k, X uk d = P yk, k N. The daughters (X uk, k N) are independent. (X u, u U) is a Crump-Mode-Jagers branching process [Jagers, 1983]. The law of (X u, u U) is determined by the pssmp X. Quan Shi Growth-Fragmentations CIMAT, December, / 34

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 δ Xu(t b u)1 {bu t,x u(t b u)>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, December, / 34

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 δ Xu(t b u)1 {bu t,x u(t b u)>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, December, / 34

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 δ Xu(t b u)1 {bu t,x u(t b u)>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, December, / 34

43 Positive self-similar Markov processes Let X be a positive self-similar Markov process (pssmp) with no positive jumps, and P x 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 α t), t 0) have the law of P rx and P x. Quan Shi Growth-Fragmentations CIMAT, December, / 34

44 Positive self-similar Markov processes Let X be a positive self-similar Markov process (pssmp) with no positive jumps, and P x 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 α t), t 0) have the law of P rx and P x. For the homogeneous case α = 0: there exists a Lévy process ξ such that X (0) (t) = exp(ξ t ), t 0. Quan Shi Growth-Fragmentations CIMAT, December, / 34

45 Positive self-similar Markov processes Let X be a positive self-similar Markov process (pssmp) with no positive jumps, and P x 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 α t), t 0) have the law of P rx and P x. For the homogeneous case α = 0: there exists a Lévy process ξ such that X (0) (t) = exp(ξ t ), t 0. ξ (without positive jumps, not killed) is characterized by its Laplace exponent Φ : [0, ) R, [ E e qξ(t)] = e Φ(q)t, for all t, q 0. The function Φ is convex, given by the Lévy-Khintchine formula: Φ(q) = 1 2 σ2 q 2 + cq + (e qx 1 + q(1 e x )) Λ(dx), q 0, (,0) where σ 2 0, c R, and the Lévy measure Λ is a measure on (, 0) with ( x 2 1)Λ(dx) <. (,0) Quan Shi Growth-Fragmentations CIMAT, December, / 34

46 Positive self-similar Markov processes Lamperti s transform [Lamperti 1972]: when α 0: X (α) t = exp(ξ τ (α) (t)), 0 t < ζ (α), where (τ (α) (t), t 0) is an explicit time-change that depends on α: { s } τ (α) (t) := inf s 0 : exp( αξ r )dr > t, t 0. 0 Quan Shi Growth-Fragmentations CIMAT, December, / 34

47 Positive self-similar Markov processes Lamperti s transform [Lamperti 1972]: when α 0: X (α) t = exp(ξ τ (α) (t)), 0 t < ζ (α), where (τ (α) (t), t 0) is an explicit time-change that depends on α: { s } τ (α) (t) := inf s 0 : exp( αξ r )dr > t, t 0. 0 The lifetime of X (α) : ζ (α) := 0 exp( αξ r )dr (0, ] Quan Shi Growth-Fragmentations CIMAT, December, / 34

48 Positive self-similar Markov processes Lamperti s transform [Lamperti 1972]: when α 0: X (α) t = exp(ξ τ (α) (t)), 0 t < ζ (α), where (τ (α) (t), t 0) is an explicit time-change that depends on α: { s } τ (α) (t) := inf s 0 : exp( αξ r )dr > t, t 0. 0 The lifetime of X (α) : ζ (α) := 0 exp( αξ r )dr (0, ] The law of the pssmp X is characterized by (Φ, α). Quan Shi Growth-Fragmentations CIMAT, December, / 34

49 Lamperti s time-change X (0) (t) = e ξ(t) [ξ: Lévy process] Simulation by B.Dadoun Quan Shi Growth-Fragmentations CIMAT, December, / 34

50 Lamperti s time-change X (0) (t) = e ξ(t) ( t X (α) (t) = X (0) ( X (α) (s) ) ) α ds 0 [ξ: Lévy process] Simulation by B.Dadoun Quan Shi Growth-Fragmentations CIMAT, December, / 34

51 A coupling construction Repeat Lamperti s time-change for each trajectory { from the homogeneous case: X u (α) (t) := X u (0) (τ u (α) (t)) with τ u (α) } r (t) := inf r 0 : u (s) α ds t α = 0 0 X (0) Simulation by B.Dadoun Quan Shi Growth-Fragmentations CIMAT, December, / 34

52 A coupling construction Repeat Lamperti s time-change for each trajectory from the homogeneous o case: n R r (0) (α) (0) (α) (α) Xu (t) := Xu (τu (t)) with τu (t) := inf r 0 : 0 Xu (s) α ds t α = 0.5 Simulation by B.Dadoun Quan Shi Growth-Fragmentations CIMAT, December, / 34

53 3. Properties of self-similar growth-fragmentations Quan Shi Growth-Fragmentations CIMAT, December, / 34

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, December, / 34

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) + (1 e x ) q Λ(dx), q 0. (,0) Quan Shi Growth-Fragmentations CIMAT, December, / 34

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) + (1 e x ) q Λ(dx), q 0. (,0) κ is convex and possibly takes value at. But κ(q) < for all q 2. Quan Shi Growth-Fragmentations CIMAT, December, / 34

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) + (1 e x ) q Λ(dx), q 0. (,0) Theorem (Bertoin, 2016) Suppose that κ(q) <. If α = 0, then for every t 0, [ ] E x y R q X t (dy) = x q exp(κ(q)t). + Quan Shi Growth-Fragmentations CIMAT, December, / 34

58 Non-explosion condition X: a self-similar growth-fragmentation associated with a pssmp X with characteristics (Φ, α). Quan Shi Growth-Fragmentations CIMAT, December, / 34

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, December, / 34

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, December, / 34

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 [ ] E x y R q X t (dy) x q. + Quan Shi Growth-Fragmentations CIMAT, December, / 34

62 Proof of the theorem Using Lamperti s transform, we have [ E x X (t) q + X (s) q] x q, for all t 0. 0 s t Quan Shi Growth-Fragmentations CIMAT, December, / 34

63 Proof of the theorem Using Lamperti s transform, we have [ E x X (t) q + X (s) q] x q, for all t 0. 0 s t Consequence: fix t 0, we have a supermartingale: Σ n := X u (t b u ) q + X v (s b v ) q, n 1. u n 1,b u t v =n,b v s t Quan Shi Growth-Fragmentations CIMAT, December, / 34

64 Proof of the theorem Using Lamperti s transform, we have [ E x X (t) q + X (s) q] x q, for all t 0. 0 s t Consequence: fix t 0, we have a supermartingale: Σ n := X u (t b u ) q + X v (s b v ) q, n 1. u n 1,b u t v =n,b v s t E x [ u U:b u t X u (t b u ) q] E x [Σ ] E x [Σ 0 ] = E x [ X (t) q + 0 s t X (s) q] x q. Quan Shi Growth-Fragmentations CIMAT, December, / 34

65 Properties of self-similar growth-fragmentations Let X be a self-similar growth-fragmentation driven by X started from X(0) = δ x. Suppose that the non-explosion condition holds. [The branching property] Write X(s) = i 1 δ X i (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) = δ Xi (s). X is self-similar with index α: X(0) = δ x. For every θ > 0, ( i 1 δ θxi (θ α t), t 0) d = X started from δ θx. Quan Shi Growth-Fragmentations CIMAT, December, / 34

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 P x -a.s. finite for every x > 0. Quan Shi Growth-Fragmentations CIMAT, December, / 34

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 P x -a.s. finite for every x > 0. Proof. inf{t 0 : X (α) (t) = 0} = sup { } b u (α) + ζ u (α) : u U Quan Shi Growth-Fragmentations CIMAT, December, / 34

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 P x -a.s. finite for every x > 0. Proof. inf{t 0 : X (α) (t) = 0} = sup Y u (α) : the ancestral lineage of u. identity: b u (α) + ζ u (α) = u (s) α ds 0 Y (0) { } b u (α) + ζ u (α) : u U Quan Shi Growth-Fragmentations CIMAT, December, / 34

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 P x -a.s. finite for every x > 0. Proof. inf{t 0 : X (α) (t) = 0} = sup { } b u (α) + ζ u (α) : u U Y u (α) : the ancestral lineage of u. identity: b u (α) + ζ u (α) = Y (0) 0 u (s) α ds ( Y u (0) (t) q X (0) 1 (t) q e κ(q)t e κ(q)t ) i 1 X (0) i (t) q Ce κ(q)t Quan Shi Growth-Fragmentations CIMAT, December, / 34

70 4. Martingales in self-similar growth-fragmentations Quan Shi Growth-Fragmentations CIMAT, December, / 34

71 Hypothesis Recall that κ is convex, so it has at most two roots. Quan Shi Growth-Fragmentations CIMAT, December, / 34

72 Hypothesis Recall that κ is convex, so it has at most two roots. Suppose that the Cramér s hypothesis holds: ω + > ω > 0, s.t. κ(ω ) = κ(ω + ) = 0 and κ (ω ) >. [H] Quan Shi Growth-Fragmentations CIMAT, December, / 34

73 Hypothesis Recall that κ is convex, so it has at most two roots. Suppose that the Cramér 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, December, / 34

74 The genealogical martingales ( log X u (0), u U) is a branching random walk. If Φ(q) < 0 and κ(q) <, then [ m(q) := E 1 e q( log Xu(0))] [ = E 1 X (s) q] = 1 κ(q) Φ(q). Lemma u =1 Suppose that [H] holds. 1. M + (n) := x ω+ 0<s<ζ X u (0) ω+ u =n is a martingale that converges P x -a.s. to for any p [1, ω+ ω ), the martingale M (n) := x ω X u (0) ω u =n converges P x -a.s. and in L p (P x ). Its terminal value M ( ) > 0. Quan Shi Growth-Fragmentations CIMAT, December, / 34

75 Mellin transform of the potential measure Fact: For a pssmp X (α) with characteristics (Φ, α): E x [ 0 ] X (α) (t) q+α dt = 1 Φ(q) x q. whenever Φ(q) < 0. Quan Shi Growth-Fragmentations CIMAT, December, / 34

76 Mellin transform of the potential measure Fact: For a pssmp X (α) with characteristics (Φ, α): E x [ Proposition (BBCK, 2016+) For every q with κ(q) < 0: 0 ] X (α) (t) q+α dt = 1 Φ(q) x q. whenever Φ(q) < 0. [ ( E x X (α) i (t) q+α) ] dt = 1 κ(q) x q. 0 i=1 Quan Shi Growth-Fragmentations CIMAT, December, / 34

77 Mellin transform of the potential measure Fact: For a pssmp X (α) with characteristics (Φ, α): E x [ Proposition (BBCK, 2016+) For every q with κ(q) < 0: 0 ] X (α) (t) q+α dt = 1 Φ(q) x q. whenever Φ(q) < 0. [ ( E x X (α) i (t) q+α) ] dt = 1 κ(q) x q. 0 i=1 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, December, / 34

78 Mellin transform of the potential measure Fact: For a pssmp X (α) with characteristics (Φ, α): E x [ Proposition (BBCK, 2016+) For every q with κ(q) < 0: 0 ] X (α) (t) q+α dt = 1 Φ(q) x q. whenever Φ(q) < 0. [ ( E x X (α) i (t) q+α) ] dt = 1 κ(q) x q. 0 i=1 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, December, / 34

79 Mellin transform of the potential measure Fact: For a pssmp X (α) with characteristics (Φ, α): E x [ Proposition (BBCK, 2016+) For every q with κ(q) < 0: 0 ] X (α) (t) q+α dt = 1 Φ(q) x q. whenever Φ(q) < 0. [ ( E x X (α) i (t) q+α) ] dt = 1 κ(q) x q. 0 i=1 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, December, / 34

80 Many-to-one formula Define the intensity measure µ x t of X(t) on R + such that for all f C c (R + ): [ ] µ x t, f := f (y)µ x t (dy) = E x f (X i (t)). R + i=1 Quan Shi Growth-Fragmentations CIMAT, December, / 34

81 Many-to-one formula Define the intensity measure µ x t of X(t) on R + such that for all f C c (R + ): [ ] µ x t, f := f (y)µ x t (dy) = E x f (X i (t)). R + ( Φ (q) := κ(q + ω ), q 0 ) is the Laplace exponent of a Lévy process η, and η drifts to. Theorem (BBCK, 2016+) µ x t (dy) = i=1 ( ) ω x ρ t (x, dy), y > 0, y 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, December, / 34

82 Many-to-one formula Define the intensity measure µ x t of X(t) on R + such that for all f C c (R + ): [ ] µ x t, f := f (y)µ x t (dy) = E x f (X i (t)). R + ( Φ (q) := κ(q + ω ), q 0 ) is the Laplace exponent of a Lévy process η, and η drifts to. Theorem (BBCK, 2016+) µ x t (dy) = i=1 ( ) ω x ρ t (x, dy), y > 0, y where ρ t (x, ) be the transition kernel of a pssmp Y (t) with characteristics (Φ, α) starting from x > 0. [ ] [ ( ) ω x That is: E x f (X i (t)) = E x f (Y (t)) Y 1 (t) {Y (t) (0, )}] i=1 Quan Shi Growth-Fragmentations CIMAT, December, / 34

83 Proof of many-to-one formula For θ such that κ(θ) < 0: κ( + θ) is the Laplace exponent of a Lévy process (with killing rate κ(θ)). Quan Shi Growth-Fragmentations CIMAT, December, / 34

84 Proof of many-to-one formula For θ such that κ(θ) < 0: κ( + θ) is the Laplace exponent of a Lévy process (with killing rate κ(θ)). Define [ ρ t (x, ), f := x θ E x f (X i (t))x i (t) θ]. i=1 Quan Shi Growth-Fragmentations CIMAT, December, / 34

85 Proof of many-to-one formula For θ such that κ(θ) < 0: κ( + θ) is the Laplace exponent of a Lévy process (with killing rate κ(θ)). Define [ ρ t (x, ), f := x θ E x f (X i (t))x i (t) θ]. i=1 µ x t (dy) = ( ) θ x ρ t (x, dy), y > 0. y ρ t is the transition kernel of a pssmp Ỹ with characteristics (κ( + θ), α): Quan Shi Growth-Fragmentations CIMAT, December, / 34

86 Proof of many-to-one formula For θ such that κ(θ) < 0: κ( + θ) is the Laplace exponent of a Lévy process (with killing rate κ(θ)). Define [ ρ t (x, ), f := x θ E x f (X i (t))x i (t) θ]. i=1 µ x t (dy) = ( ) θ x ρ t (x, dy), y > 0. y ρ t is the transition kernel of a pssmp Ỹ with characteristics (κ( + θ), α): ρt+s(x, ), f = ρt(y, ), f ρs(x, dy) Chapman-Kolmogorov equation (0, ) ρt(x, ), f = ρ x α t(1, ), f (x ) the self-similarity. For every q with κ(q + θ) < 0: 0 dt (0, ) [ y q+α ρ t(1, dy) = x θ ( E x 0 i=1 X i (t) θ+q+α) ] dt 1 = κ(θ + q) = Quan Shi Growth-Fragmentations CIMAT, December, / 34

87 Growth-fragmentation equations The intensity measure µ x t of X(t): for all f Cc (R + ), [ ] µ x t, f := f (y)µ x t (dy) = E x f (X i (t)). R + i=1 Quan Shi Growth-Fragmentations CIMAT, December, / 34

88 Growth-fragmentation equations The intensity measure µ x t of X(t): for all f Cc (R + ), [ ] µ x t, f := f (y)µ x t (dy) = E x f (X i (t)). R + Proposition (Bertoin&Watson, 2016; BBCK, 2016+) Suppose that [H] holds. Then i=1 where µ x t, f = f (x) + t 0 µ x s, Gf ds, ( 1 Gf (y) := y α 2 σ2 f (y)y 2 + (c σ2 )f (y)y ( ) ) + f (ye z ) + f (y(1 e z )) f (y) + y(1 e z )f (y) Λ(dz). (,0) Quan Shi Growth-Fragmentations CIMAT, December, / 34

89 Proof: Use the many-to-one formula and the infinitesimal generator of the pssmp Y. Quan Shi Growth-Fragmentations CIMAT, December, / 34 Growth-fragmentation equations The intensity measure µ x t of X(t): for all f Cc (R + ), [ ] µ x t, f := f (y)µ x t (dy) = E x f (X i (t)). R + Proposition (Bertoin&Watson, 2016; BBCK, 2016+) Suppose that [H] holds. Then i=1 where µ x t, f = f (x) + t 0 µ x s, Gf ds, ( 1 Gf (y) := y α 2 σ2 f (y)y 2 + (c σ2 )f (y)y ( ) ) + f (ye z ) + f (y(1 e z )) f (y) + y(1 e z )f (y) Λ(dz). (,0)

90 Temporal martingales Suppose that [H] holds. For the smaller root ω of κ, define M (t) := X i (t) ω, t 0. i=1 Theorem (BBCK, 2016+) When α 0, M is a uniformly integrable P x -martingale; When α < 0, M is a P x -supermartingale which converges to 0 in L 1 (P x ). Proof: By many-to-one formula: E x [M (t)] = x ω P ( Y (t) (0, ) ) ; If α 0, then P ( Y (t) (0, ) ) 1; If α < 0, then lim t P ( Y (t) (0, ) ) = 0; We conclude by the branching property of X. Quan Shi Growth-Fragmentations CIMAT, December, / 34

91 Temporal martingales Suppose that [H] holds. For the larger root ω + of κ, define M + (t) := X i (t) ω+, t 0. i=1 Theorem (BBCK, 2016+) When α > 0, M + is a P x -supermartingale which converges to 0 in L 1 (P x ); When α 0, M + is a P x -martingale. Quan Shi Growth-Fragmentations CIMAT, December, / 34

92 Summary pssmp X cell system (X u, 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, December, / 34

93 Thank you! Quan Shi Growth-Fragmentations CIMAT, December, / 34