The genealogy of self-similar fragmentations with negative index as a continuum random tree

Transcription

1 The genealogy of self-similar fragmentations with negative index as a continuum random tree Electron. J. Probab. 9, (2004) ejpecp Bénédicte Haas & Grégory Miermont LPMA, Université Paris VI & DMA, École Normale Supérieure Paris, France 6th BS/IMSC, Barcelona, July 26-31, 2004 The genealogy of self-similar fragmentations with negative index as a continuum random tree p.1/20

2 Introduction Fragmentation processes describe a massive object that collapses into pieces as time goes. The genealogy of self-similar fragmentations with negative index as a continuum random tree p.2/20

3 Introduction Fragmentation processes describe a massive object (with initial mass 1) that collapses into pieces as time goes. The genealogy of self-similar fragmentations with negative index as a continuum random tree p.2/20

4 Introduction Fragmentation processes describe a massive object (with initial mass 1) that collapses into pieces as time goes. State space { } S = s = (s 1,s 2,...) : s 1 s , s i 1 i=1 The genealogy of self-similar fragmentations with negative index as a continuum random tree p.2/20

5 Introduction Fragmentation processes describe a massive object (with initial mass 1) that collapses into pieces as time goes. State space { } S = s = (s 1,s 2,...) : s 1 s , s i 1 i=1 This allows some loss of mass. The genealogy of self-similar fragmentations with negative index as a continuum random tree p.2/20

6 Self-similar fragmentations Definition (Bertoin, 2002) A Markovian S-valued process (F(t),t 0) starting at (1, 0,...) is a ranked self-similar fragmentation with index α R if it is continuous in probability The genealogy of self-similar fragmentations with negative index as a continuum random tree p.3/20

7 Self-similar fragmentations Definition (Bertoin, 2002) A Markovian S-valued process (F(t),t 0) starting at (1, 0,...) is a ranked self-similar fragmentation with index α R if it is continuous in probability and satisfies the following fragmentation property. The genealogy of self-similar fragmentations with negative index as a continuum random tree p.3/20

8 Self-similar fragmentations Definition (Bertoin, 2002) A Markovian S-valued process (F(t),t 0) starting at (1, 0,...) is a ranked self-similar fragmentation with index α R if it is continuous in probability and satisfies the following fragmentation property. the fragments present at time t subsequently evolve independently of the others, in a way similar to that of the original mass 1 fragment, up to a space-time renormalization depending on the mass of each considered fragment. The genealogy of self-similar fragmentations with negative index as a continuum random tree p.3/20

9 Self-similar fragmentations Definition (Bertoin, 2002) A Markovian S-valued process (F(t),t 0) starting at (1, 0,...) is a ranked self-similar fragmentation with index α R if it is continuous in probability and satisfies the following fragmentation property. For every t,t 0, given F(t) = (x 1,x 2,...), F(t + t ) has the same law as the decreasing rearrangement of the sequences x 1 F (1) (x 1 α t ),x 2 F (2) (x 2 α t ),... where the F (i) s are independent copies of F. The genealogy of self-similar fragmentations with negative index as a continuum random tree p.3/20

10 A crucial example Let e be (twice) a standard brownian excursion. The genealogy of self-similar fragmentations with negative index as a continuum random tree p.4/20

11 A crucial example Let e be (twice) a standard brownian excursion. For t 0 let I t := {s [0, 1] : e(s) > t}. The genealogy of self-similar fragmentations with negative index as a continuum random tree p.4/20

12 A crucial example Let e be (twice) a standard brownian excursion. For t 0 let I t := {s [0, 1] : e(s) > t}. Rank the lengths of interval components of I t to get the Brownian fragmentation F B (t) S,t 0. The genealogy of self-similar fragmentations with negative index as a continuum random tree p.4/20

13 A crucial example Let e be (twice) a standard brownian excursion. For t 0 let I t := {s [0, 1] : e(s) > t}. Rank the lengths of interval components of I t to get the Brownian fragmentation F B (t) S,t 0. Then easy scaling properties of Brownian motion show: Proposition (F B (t),t 0) is a self-similar fragmentation with index α = 1/2. The genealogy of self-similar fragmentations with negative index as a continuum random tree p.4/20

14 A crucial example Let e be (twice) a standard brownian excursion. For t 0 let I t := {s [0, 1] : e(s) > t}. Rank the lengths of interval components of I t to get the Brownian fragmentation F B (t) S,t 0. Then easy scaling properties of Brownian motion show: Proposition (F B (t),t 0) is a self-similar fragmentation with index α = 1/2. Notice α < 0, so fragments keep on collapsing faster and faster, implying some loss of mass (even extinction in finite time). The genealogy of self-similar fragmentations with negative index as a continuum random tree p.4/20

15 A crucial example t t=1 t=1/2 0 1 The genealogy of self-similar fragmentations with negative index as a continuum random tree p.5/20

16 A crucial example t t=1 t=1/2 0 1 This one (mass m) is breaking into two fragments with sum of masses m. The genealogy of self-similar fragmentations with negative index as a continuum random tree p.5/20

17 Continuum Random Trees Definition An R-tree is a metric space (T,d) such that for v,w T, The genealogy of self-similar fragmentations with negative index as a continuum random tree p.6/20

18 Continuum Random Trees Definition An R-tree is a metric space (T,d) such that for v,w T, 1. there exists a unique geodesic [[v,w]] going from v to w, i.e. there exists a unique isometry ϕ v,w : [0,d(v,w)] T with ϕ v,w (0) = v and ϕ v,w (d(v,w)) = w, and its image is called [[v,w]], and... The genealogy of self-similar fragmentations with negative index as a continuum random tree p.6/20

19 Continuum Random Trees Definition An R-tree is a metric space (T,d) such that for v,w T, 1. there exists a unique geodesic [[v,w]] going from v to w, i.e. there exists a unique isometry ϕ v,w : [0,d(v,w)] T with ϕ v,w (0) = v and ϕ v,w (d(v,w)) = w, and its image is called [[v,w]], and Any simple path going from v to w is [[v,w]] (tree property). The genealogy of self-similar fragmentations with negative index as a continuum random tree p.6/20

20 Example: the Brownian tree This is a tree The genealogy of self-similar fragmentations with negative index as a continuum random tree p.7/20

21 Example: the Brownian tree t 1 root 0 1 U1 The genealogy of self-similar fragmentations with negative index as a continuum random tree p.7/20

22 Example: the Brownian tree t 1 2 root 0 U1 U2 1 The genealogy of self-similar fragmentations with negative index as a continuum random tree p.7/20

23 Example: the Brownian tree t root 0 U4 U1 U3 U2 U5 1 The genealogy of self-similar fragmentations with negative index as a continuum random tree p.7/20

24 Example: the Brownian tree t root 0 U4 U1 U3 U2 U5 1 And so on: the limiting Brownian CRT = [0, 1], with D(s,s ) = e(s) + e(s ) 2 inf s s u s s e(u). The genealogy of self-similar fragmentations with negative index as a continuum random tree p.7/20

25 Example: the Brownian tree Therefore, the Brownian fragmentation F B can be alternatively obtained from the Brownian CRT T B, by recording in decreasing order the sizes of tree components of the forest {v T : d(root,v) > t}, where the sizes are the µ-masses of these components, µ= Lebesgue measure on [0, 1]. The genealogy of self-similar fragmentations with negative index as a continuum random tree p.8/20

26 Example: the Brownian tree Therefore, the Brownian fragmentation F B can be alternatively obtained from the Brownian CRT T B, by recording in decreasing order the sizes of tree components of the forest {v T : d(root,v) > t}, where the sizes are the µ-masses of these components, µ= Lebesgue measure on [0, 1]. Is ANY self-similar fragmentation F, which vanishes in finite time (i.e. α < 0), of this form, for some kind of measured R-tree T F? Answer: indeed. The genealogy of self-similar fragmentations with negative index as a continuum random tree p.8/20

27 Fragmentation trees Theorem 1 (Haas & M., 2004) Let F be a s.s.f. with no erosion The genealogy of self-similar fragmentations with negative index as a continuum random tree p.9/20

28 Fragmentation trees Theorem 1 (Haas & M., 2004) Let F be a s.s.f. with no erosion This means that every process F i (t),t 0, is pure-jump: the fragments cannot melt continuously. The genealogy of self-similar fragmentations with negative index as a continuum random tree p.9/20

29 Fragmentation trees Theorem 1 (Haas & M., 2004) Let F be a s.s.f. with no erosion and no sudden loss of mass, The genealogy of self-similar fragmentations with negative index as a continuum random tree p.9/20

30 Fragmentation trees Theorem 1 (Haas & M., 2004) Let F be a s.s.f. with no erosion and no sudden loss of mass,meaning the function M(t) = i=1 F i(t) is continuous, The genealogy of self-similar fragmentations with negative index as a continuum random tree p.9/20

31 Fragmentation trees Theorem 1 (Haas & M., 2004) Let F be a s.s.f. with no erosion and no sudden loss of mass,meaning the function M(t) = i=1 F i(t) is continuous, then there exists a random measured R-tree (a CRT) (T F,µ) such that F has the same law as the process obtained as in the above construction, from the tree T F. The genealogy of self-similar fragmentations with negative index as a continuum random tree p.9/20

32 Fragmentation trees Theorem 1 (Haas & M., 2004) Let F be a s.s.f. with no erosion and no sudden loss of mass,meaning the function M(t) = i=1 F i(t) is continuous, then there exists a random measured R-tree (a CRT) (T F,µ) such that F has the same law as the process obtained as in the above construction, from the tree T F. To see this: discretize space. The genealogy of self-similar fragmentations with negative index as a continuum random tree p.9/20

33 Fragmentations and partitions of N Suppose the massive object is given with a probability mass measure µ. The genealogy of self-similar fragmentations with negative index as a continuum random tree p.10/20

34 Fragmentations and partitions of N Suppose the massive object is given with a probability mass measure µ. One can sample n (resp. an infinite number of) points independently according to µ and create an exchangeable partition of {1, 2,...,n} (resp. N). The genealogy of self-similar fragmentations with negative index as a continuum random tree p.10/20

35 Fragmentations and partitions of N Suppose the massive object is given with a probability mass measure µ. One can sample n (resp. an infinite number of) points independently according to µ and create an exchangeable partition of {1, 2,...,n} (resp. N) {1,2,3,4,5,6,7} {1,3},{2,5,7},{4},{6} The genealogy of self-similar fragmentations with negative index as a continuum random tree p.10/20

36 Fragmentations and partitions of N Suppose the massive object is given with a probability mass measure µ. One can sample n (resp. an infinite number of) points independently according to µ and create an exchangeable partition of {1, 2,...,n} (resp. N) {1,2,3,4,5,6,7} {1,3},{2,5,7},{4},{6} This gives a partition-valued fragmentation Π(t), t 0. The genealogy of self-similar fragmentations with negative index as a continuum random tree p.10/20

37 A tree from a partition For i N let D i = inf{t 0 : {i} Π(t)} (death time, height of leaf i) for finite B N let D B = inf{t 0 : #(B Π(t)) 1} first splitting time of B. The genealogy of self-similar fragmentations with negative index as a continuum random tree p.11/20

38 A tree from a partition For i N let D i = inf{t 0 : {i} Π(t)} (death time, height of leaf i) for finite B N let D B = inf{t 0 : #(B Π(t)) 1} first splitting time of B. then one may construct (Aldous 1993 general CRT theory) a random measured R-tree T F,µ so that the subtree of T F spanned by n uniform µ-picked vertices has same law as the R-tree determined by the D B,B {1,...,n}: the branchpoint of the leaves of B has height D B (height of i if B = {i}). The genealogy of self-similar fragmentations with negative index as a continuum random tree p.11/20

39 A tree from a partition For i N let D i = inf{t 0 : {i} Π(t)} (death time, height of leaf i) for finite B N let D B = inf{t 0 : #(B Π(t)) 1} first splitting time of B. then one may construct (Aldous 1993 general CRT theory) a random measured R-tree T F,µ so that the subtree of T F spanned by n uniform µ-picked vertices has same law as the R-tree determined by the D B,B {1,...,n}: the branchpoint of the leaves of B has height D B (height of i if B = {i}). µ is the weak limit of the empirical measure on the first n leaves. The genealogy of self-similar fragmentations with negative index as a continuum random tree p.11/20

40 Question Therefore: a self-similar fragmentation with α < 0 is a random measured R-tree. One wants to look at possible properties of this random metric space. The genealogy of self-similar fragmentations with negative index as a continuum random tree p.12/20

41 Question Therefore: a self-similar fragmentation with α < 0 is a random measured R-tree. One wants to look at possible properties of this random metric space. The Hausdorff dimension is a natural quantity to consider. For the Brownian tree the dimension is 2. For the Duquesne-Le Gall β-stable tree (the associated fragmentation process is studied in M., 2003), the dimension is β/(β 1). We obtain it as a Corollary of Theorem 2 (Haas & M., 2004) Under mild further hypotheses on F (F B matches these), the Hausdorff dimension of T F is α 1 1, a.s. The genealogy of self-similar fragmentations with negative index as a continuum random tree p.12/20

42 Question Therefore: a self-similar fragmentation with α < 0 is a random measured R-tree. One wants to look at possible properties of this random metric space. The Hausdorff dimension is a natural quantity to consider. For the Brownian tree the dimension is 2. For the Duquesne-Le Gall β-stable tree (the associated fragmentation process is studied in M., 2003), the dimension is β/(β 1). We obtain it as a Corollary of Theorem 2 (Haas & M., 2004) Under mild further hypotheses on F (F B matches these), the Hausdorff dimension of T F is α 1 1, a.s. Notice that α 1 is thus qualitatively different from 1 < α < 0. The genealogy of self-similar fragmentations with negative index as a continuum random tree p.12/20

43 Idea of proof Upper bound: based on known exponential control of the tail of the death time of a marked fragment (Haas, 2002). The genealogy of self-similar fragmentations with negative index as a continuum random tree p.13/20

44 Idea of proof Upper bound: based on known exponential control of the tail of the death time of a marked fragment (Haas, 2002). Lower Bound: one natural method is to apply Frostman s energy method with the most natural measure on T F : the mass measure µ. The genealogy of self-similar fragmentations with negative index as a continuum random tree p.13/20

45 Idea of proof Upper bound: based on known exponential control of the tail of the death time of a marked fragment (Haas, 2002). Lower Bound: one natural method is to apply Frostman s energy method with the most natural measure on T F : the mass measure µ. One would like to show The genealogy of self-similar fragmentations with negative index as a continuum random tree p.13/20

46 Idea of proof Upper bound: based on known exponential control of the tail of the death time of a marked fragment (Haas, 2002). Lower Bound: one natural method is to apply Frostman s energy method with the most natural measure on T F : the mass measure µ. One would like to show T F µ(dx)µ(dy) T F d(x,y) γ < for all γ < α 1 1, where L 1,L 2 are independent µ-distributed. The genealogy of self-similar fragmentations with negative index as a continuum random tree p.13/20

47 Idea of proof Upper bound: based on known exponential control of the tail of the death time of a marked fragment (Haas, 2002). Lower Bound: one natural method is to apply Frostman s energy method with the most natural measure on T F : the mass measure µ. One would like to show [ ] µ(dx)µ(dy) E T F d(x,y) γ T F = E [ 1 d(l 1,L 2 ) γ ] < for all γ < α 1 1, where L 1,L 2 are independent µ-distributed. The genealogy of self-similar fragmentations with negative index as a continuum random tree p.13/20

48 Idea of proof d(l 1,L 2 ) is manageable: find the first time D {1,2} when 1 and 2 separate, evaluate the sizes λ 1,λ 2 of fragments containing 1 and 2 at time D {1,2} The genealogy of self-similar fragmentations with negative index as a continuum random tree p.14/20

49 Idea of proof d(l 1,L 2 ) is manageable: find the first time D {1,2} when 1 and 2 separate, evaluate the sizes λ 1,λ 2 of fragments containing 1 and 2 at time D {1,2}, then (by the fragmentation property) d(l 1,L 2 ) has same distribution as λ α 1 D + λ α 2 D, where D, D are independent, independent of λ i,i = 1, 2 with law that of D {1}. The genealogy of self-similar fragmentations with negative index as a continuum random tree p.14/20

50 Idea of proof d(l 1,L 2 ) is manageable: find the first time D {1,2} when 1 and 2 separate, evaluate the sizes λ 1,λ 2 of fragments containing 1 and 2 at time D {1,2}, then (by the fragmentation property) d(l 1,L 2 ) has same distribution as λ α 1 D + λ α 2 D, where D, D are independent, independent of λ i,i = 1, 2 with law that of D {1}. Therefore E[d(L 1,L 2 ) γ ] 2E[D γ ]E[λ αγ 1 ;λ 1 < λ 2 ]. The genealogy of self-similar fragmentations with negative index as a continuum random tree p.14/20

51 Idea of proof d(l 1,L 2 ) is manageable: find the first time D {1,2} when 1 and 2 separate, evaluate the sizes λ 1,λ 2 of fragments containing 1 and 2 at time D {1,2}, then (by the fragmentation property) d(l 1,L 2 ) has same distribution as λ α 1 D + λ α 2 D, where D, D are independent, independent of λ i,i = 1, 2 with law that of D {1}. Therefore E[d(L 1,L 2 ) γ ] 2E[D γ ]E[λ αγ 1 ;λ 1 < λ 2 ]. Rewrite (λ 1,λ 2 ) = λ(d {1,2} )(l 1,l 2 ), where λ(t) is the size of the fragment containing L 1,L 2 before separation (t < D {1,2} ). The genealogy of self-similar fragmentations with negative index as a continuum random tree p.14/20

52 Idea of proof Crucial tool: the dislocation measure of F : a σ-finite measure ν on S s.t. S (1 s 1)ν(ds) <. Informally, a fragment with mass x breaks in fragments xs (with s S) at rate x α ν(ds). The knowledge of α and ν determines the law of a(n erosionless) F (Bertoin, 2002). It is the jump measure of F. If F has no sudden loss of mass then i s i = 1, ν-a.e. The genealogy of self-similar fragmentations with negative index as a continuum random tree p.15/20

53 Idea of proof Were it not for the x α term (i.e. when α = 0), the size F (t) of the fragment containing L 1 would be exp( ξ(t)),t 0 where ξ is a subordinator with the size-biased Lévy measure m(dx) = s i ν( log(s i ) dx). i=1 The genealogy of self-similar fragmentations with negative index as a continuum random tree p.16/20

54 Idea of proof Were it not for the x α term (i.e. when α = 0), the size F (t) of the fragment containing L 1 would be exp( ξ(t)),t 0 where ξ is a subordinator with the size-biased Lévy measure m(dx) = s i ν( log(s i ) dx). i=1 One gets F by a Lamperti time-change (Bertoin, 2002): F (t) d = exp(ξ(ρ(t))) t 0 where ρ(t) = inf{s 0 : s 0 du exp( αξ(u))}, notice ρ explodes in finite time (α < 0). The genealogy of self-similar fragmentations with negative index as a continuum random tree p.16/20

55 Idea of proof Similarly, λ(t) is a time-changed version of an exponential of subordinator with Lévy measure s 2 iν( log(s i ) dx) i killed at an exponential (rate k) time (corresponding to the separation of L 1 and L 2 ), The genealogy of self-similar fragmentations with negative index as a continuum random tree p.17/20

56 Idea of proof Similarly, λ(t) is a time-changed version of an exponential of subordinator with Lévy measure s 2 iν( log(s i ) dx) i killed at an exponential (rate k) time (corresponding to the separation of L 1 and L 2 ), and then the relative sizes after separation are independent of λ(d {1,2} ) and distributed as E[f(l 1,l 2 )] = 1 k i j f(s i,s j )s i s j ν(ds). S The genealogy of self-similar fragmentations with negative index as a continuum random tree p.17/20

57 Idea of proof Similarly, λ(t) is a time-changed version of an exponential of subordinator with Lévy measure s 2 iν( log(s i ) dx) i killed at an exponential (rate k) time (corresponding to the separation of L 1 and L 2 ), and then the relative sizes after separation are independent of λ(d {1,2} ) and distributed as E[f(l 1,l 2 )] = 1 k i j f(s i,s j )s i s j ν(ds). Then one can S compute everything needed in E[d(L 1,L 2 ) γ ] for γ < α 1 1, but... The genealogy of self-similar fragmentations with negative index as a continuum random tree p.17/20

58 We are not done yet! Indeed, the result is not always <, it is in the case ν finite and there exists N 2 with ν( N i=1 s i < 1) = 0 (at most N-ary tree). The idea is then to truncate the tree to make it look like a fragmentation tree with the two properties above, and then make the truncation resemble the initial tree more and more. The genealogy of self-similar fragmentations with negative index as a continuum random tree p.18/20

59 Possible developments Finer analysis of fragmentation trees : level sets, local times? Case α 0 (ultrametrics rather than Brownian CRT s-looking trees)? Extension x α f(x) with regularity assumptions of f near 0 is easy. Encoding functions and their Hölder properties (partially done in the paper). The genealogy of self-similar fragmentations with negative index as a continuum random tree p.19/20

60 Zi ainde Thank you! The genealogy of self-similar fragmentations with negative index as a continuum random tree p.20/20