Path-properties of the tree-valued Fleming-Viot process

Transcription

1 Path-properties of the tree-valued Fleming-Viot process Peter Pfaffelhuber Joint work with Andrej Depperschmidt and Andreas Greven Luminy,

2 The Moran model time t As every population model, the Moran model (of size N) gives rise to a tree-valued process (X N t ) t 0 Without proof, we assume that (X N t ) t 0 N === (X t ) t 0 in an appropriate sense We call (X t ) t 0 the tree-valued Fleming-Viot dynamics

3 The Moran model time t s As every population model, the Moran model (of size N) gives rise to a tree-valued process (X N t ) t 0 Without proof, we assume that (X N t ) t 0 N === (X t ) t 0 in an appropriate sense We call (X t ) t 0 the tree-valued Fleming-Viot dynamics

4 The Moran model time t s As every population model, the Moran model (of size N) gives rise to a tree-valued process (X N t ) t 0 Without proof, we assume that (X N t ) t 0 N === (X t ) t 0 in an appropriate sense We call (X t ) t 0 the tree-valued Fleming-Viot dynamics

5 The tree-valued Fleming-Viot dynamics Theorem The process (X t ) t 0 exists as limit of tree-valued Moran models and is unique Its state space is the set of (equivalence classes of) metric measure spaces {(X, r, µ) : µ M 1 (X )}, equipped with the Gromov-Prohorov topology It can be described by a martingale problem Almost surely, (Xt ) t 0 has continuous sample paths (Xt ) t 0 is compact for all t > 0 For many functions Φ, the quadratic variation of (Φ(X t )) t 0 can be computed

6 The Kingman coalescent as the equilibrium of (X t ) t 0 Theorem Let X be the Kingman coalescent, a random tree with: Start with many lines If there are k lines left, wait S k exp ( k 2) and merge two randomly chosen lines Stop upon reaching one line Then, X t t === X Important property: subtree with n leaves tree started with n lines

7 Goal Lift properties of Kingman coalescent X to the paths of (X t ) t 0 (when started in equilibrium) Example: Let N t ε be the number of ancestors of the time-t population at time t ε It is well-known that almost surely εn ε 2 ε 0 0 Is it also true that almost surely sup εnε t 2 ε 0 0? t 0

8 A law of large numbers for the number of ancestors Theorem Let N ε be the the number of ancestors of the Kingman coalescent X at time ε Then, almost surely, εn ε 2 ε 0 0

9 A law of large numbers for the number of ancestors εn ε 2 ε 0 0 Proof: Recall S k exp ( k 2) is the time the coalescent has k lines The assertion is the same as n (S n+1 + S n+2 + )n 2 0 }{{} =:T n With E[T n ] = 2/n and E[(T n E[T n ]) 4 ] 1 n 6, we find P( T n n 2 > ε) n4 E[(T n E[T n ]) 4 ] ε 4 1 ε 4 n 2 The result follows from the Borel-Cantelli-Lemma

10 A law of large numbers for the number of ancestors Theorem Let N t ε be the the number of ancestors of the time-t population X t, at time t ε Then, almost surely, sup εnε t 2 ε 0 0 t 0

11 A law of large numbers for the number of ancestors sup εnε t 2 ε 0 0 t 0 Proof: Let T t n = S t n + S t n+1 + and S t n is the time the time-t tree spends with n lines It suffices to show sup 0 t 1 T t nn 2 n 0 Using moment calculations, ( P sup Tn k/n2 k=0,,n 2 ) n 2 > ε 1 ε 8 n 2

12 A law of large numbers for the number of ancestors It suffices to show sup 0 t 1 Tnn t 2 n 0 We claim that {T t n > s} T t δ n time t time t δ > s δ} ( P sup 0 t 1 ( P ) Tnn t > 2 + ε sup 0 t 1 T tn2 /n 2 n >nancestors of time t population > 2 + ε n ( t tn2 )) }{{ n 2 } 1/n 2 1 ε 8 n 2

13 Small family sizes Theorem Let N t (x, ε) be the number of ancestors of the time-t population with families of size at least x Then, almost surely sup εn (xε, ε) 2e 2x ε x< Open problem: Is it also true, that sup sup t 0 0 x< εn t (xε, ε) 2e 2x ε 0 0?

14 A law of large numbers for the tree metric Theorem Let F t 1 (ε),, F t N ε (ε) be the family sizes of the ancestors 1,, N t ε in X t Then, almost surely, 1 ε N t ε (Fi t (ε)) 2 = lim i=1 N 1 εn 2 N u,v=1 leaves in X t 1 {r t(u,v)<ε} ε 0 1

15 A law of large numbers for the tree metric Lemma lim N λ N 2 N u,v=1 leaves in X 1 {rt(u,v)<1/λ} λ 1 Ψ λ (X ) λ 1 with 1 Ψ λ (X ) := (λ + 1) lim N N 2 N u,v=1 leaves in X e λrt(u,v), where r t (u, v) is the time to the most recent common ancestor of u and v in X

16 A law of large numbers for the tree metric Theorem For Ψ λ as above, almost surely, Ψ λ (X ) 1 λ 0 ( ) Proof: For random leaves U, V, U, V from X, E[Ψ λ (X )] = E[(λ + 1)e λr(u,v ) ] = 1, E[(Ψ λ (X ) 1) 2 ] = (λ + 1) 2( E[e λ(r(u,v )+r(u,v )) ] 1 ) = some calculations on tree with 4 leaves 2λ 2 λ 1 = (λ + 3)(2λ + 1)(2λ + 3) 2λ, E[ ( Ψ λ (X ) 1) 4 ] = λ 3 4λ 2

17 A law of large numbers for the tree metric Theorem For Ψ λ as above, in probability, Ψ λ (X t ) 1 λ 0 sup 0 t T ( ) ( ) fdd-convergence in ( ) Lemma tightness in C R [0, ) sup E[(Ψ λ (X t ) Ψ λ (X 0 )) 4 ] t 2 λ>0

18 A law of large numbers for the tree metric Proof of Lemma: Recall E[Ψ λ (X t )] = (λ + 1)E[e λrt(u,v ) ] By the dynamics of the Moran model, d dt E[Ψ λ(x t )] = λ + 1 dt ( dt E[e λ 0 e λrt(u,v ) ] }{{} resampling between U and V + (1 dt) E[e λ(rt(u,v )+dt) e λrt(u,v ) ] }{{} tree growth = (λ + 1)(1 E[(λ + 1)e λrt(u,v ) ]) = (λ + 1)(E[Ψ λ (X t )] 1) )

19 A law of large numbers for the tree metric Proof of Lemma: Recall Similarly, d dt (E[Ψ λ(x t )] 1) = (λ + 1)(E[Ψ λ (X t )] 1) E[Ψ λ (X t ) 1 F s ] = e (λ+1)(t s) (Ψ λ (X s ) 1) and ( e (λ+1)t( )) Ψ λ (X t ) 1 is a martingale t 0

20 A law of large numbers for the tree metric Proof of Lemma: E[(Ψ λ (X t ) Ψ λ (X 0 )) 2 ] = E[2Ψ λ (X 0 )(Ψ λ (X t ) Ψ λ (X 0 ))] [ = 2E Ψ λ (X 0 ) ( [ e (λ+1)t E e (λ+1)t( ) ] Ψ λ (X t ) 1 F 0 }{{} martingale ( )] Ψ λ (X 0 ) 1 [ ] = 2E Ψ λ (X 0 )(Ψ λ (X 0 ) 1)(1 e (λ+1)t ) λ 1 e (λ+1)t λ t

21 A law of large numbers for the tree metric Proof of Lemma: E[(Ψ λ (X t ) Ψ λ (X 0 )) 4 ] = E[2Ψ λ (X 0 )(Ψ λ (X t ) Ψ λ (X 0 ))] [ = 2E Ψ λ (X 0 ) ( [ e (λ+1)t E e (λ+1)t( ) ] Ψ λ (X t ) 1 F 0 }{{} martingale ( )] Ψ λ (X 0 ) 1 [ = 2E Ψ λ (X 0 )(Ψ λ (X 0 ) 1)(1 e (λ+1)t )] λ 1 e (λ+1)t 2λ t 2

22 A Brownian motion in the Fleming-Viot dynamics Theorem Let W λ = (W λ (t)) t 0 be given by W λ (t) := λ t 0 (Ψ λ (X s ) 1)ds Then, W λ λ === W, where W = (W t ) t 0 is a Brownian motion

23 A Brownian motion in the Fleming-Viot dynamics W λ = λ Proof (part): E[W λ (t) 2 ] = 2λ 2 t 0 0 = 2λ 2 t λ 0 λ t 0 ( Ψ λ (X s ) 1 }{{} mean = 0 variance 1/2λ s 0 s 0 s 0 )ds λ === W E [ E[Ψ λ (X s ) 1 F r ](Ψ λ (X r ) 1) ] dr ds e (λ+1)(s r) E[(Ψ λ (X r ) 1) 2 ]dr ds e (λ+1)r dr ds λ t

24 Summary and outlook More about formalising trees (and Gromov-Prohorov convergence) and construction of tree-valued processes (via well-posed martingale problem) can be said All result also hold in models with mutation and selection (individuals also carry types which are (dis)favored to get offspring) All Theorems affect properties near the tree top do similar properties hold for branching trees?