Quasi stationary distributions and Fleming Viot Processes

Size: px

Start display at page:

Download "Quasi stationary distributions and Fleming Viot Processes"

Milton Taylor
5 years ago
Views:

1 Quasi stationary distributions and Fleming Viot Processes Pablo A. Ferrari Nevena Marić Universidade de São Paulo 1

2 Example Continuous time Markov chain in Λ={0, 1, 2} with transition rates q(1, 0) = q(1, 2) = q(2, 1) = 1. q(0, 1) = q(0, 2) = 0 (0 is absorbing state). If one starts with (say) chains in state 1, which proportion of the survival chains will be in state 1 by time 1? And by time? 2

3 Example 2.1 of Burdzy, Holyst and March Λ={0, 1, 2} and q(1, 0) = q(1, 2) = q(2, 1) = 1. and ν(1) = ν(2) = =1+φ = φ (golden number) φ = =

4 Quasi stationary distributions (qsd) Irreducible jump Markov process with rates Q =(q(x, y)) on Λ {0}. P t (x, y) transition matrix. Λ countable and 0 absorbing state. Z t is ergodic with a unique invariant measure δ 0 Law starting with µ conditioned to non absorption until time t: ϕ µ y Λ t (x) = µ(y)p t(y, x) 1 y Λ µ(y)p t(y, 0), x Λ. A quasi stationary distribution (qsd) is a probability measure ν on Λ satisfying ϕ ν t = ν 4

5 ν is a left eigenvector for the restriction of the matrix Q to Λ with eigenvalue λ ν = y Λ ν(y)q(y, 0): ν must satisfy the system ( ν(y) q(y, x) = ) ν(y)q(y, 0) ν(x), x Λ. y Λ y Λ\{x} y Λ νq = λ ν ν ν(y)[q(y, x)+q(y, 0)ν(x)] = 0, x Λ. y Λ recall q(x, x) = ν(y)[q(y, x)+q(y, 0)ν(x)] = ν(x) (balance equations) y Λ {0}\{x} y Λ\{x} q(x, y) ( q(x, y)+q(x, 0)ν(y) ) 5

6 Yaglom limit for µ: lim t ϕµ t (y), y Λ if it exists and it is a probability on Λ. Λ finite, Darroch and Seneta (1967): there exists a unique qsd ν for Q and that the Yaglom limit converges to ν independently of the initial distribution. Λ infinite: neither existence nor uniqueness of qsd are guaranteed. Example: asymmetric random walk Seneta: p = q(i, i +1)=1 q(i, i 1), for i 0. In this case there are infinitely many qsd when p<1/2 and none when p 1/2. Minimal qsd (for p<1/2): ( p ν(x) x 1 p ) x/2 6

7 Existence For Λ = N under the condition lim x P(R >t Z 0 = x) =1 where R absorption time, foreacht>0 existence of qsd Ee θr < for some θ>0. (Ferrari, Kesten, Martínez and Picco [6]) 7

8 Existence Ergodicity coefficient of Q: α = α(q) := z Λ Maximal absorbing rate of Q: inf q(x, z) x Λ\{z} C = C(Q) :=supq(x, 0) x Λ Theorem 1. If α>c then there exists a unique qsd ν for Q and the Yaglom limit converges to ν for any initial measure µ. Jacka and Roberts [10]: under α>cuniqueness and Yaglom limit. 8

9 The Fleming-Viot process (fv). System of N>0 particles evolving on Λ. Particles move independently with rates Q between absorptions. When a particle is absorbed, it chooses one of the other particles uniformly and jumps instantaneously to its position. Generator (Master equation): Lf(ξ) = N i=1 y Λ\{ξ(i)} [ q(ξ(i),y)+q(ξ(i), 0) η(ξ,y) ] (f(ξ i,y ) f(ξ)) N 1 where ξ i,y (j) =y for j = i and ξ i,y (j) =ξ(j) otherwiseand η(ξ,y) := N 1{ξ(i) =y}. i=1 9

10 Empirical profile and conditioned process ξ t process in Λ (1,...,N) ; η t {η N Λ : x η(x) =N} unlabeled process, η t (x) =numberofξ particles in state x at time t. Theorem 2. Let µ probability on Λ. Assume (ξ N,µ 0 (i), i=1,...,n) iid with law µ. Then, for t>0 and x Λ, lim N lim N Eη N,µ t (x) N η N,µ t (x) N = ϕ µ t (x) = ϕ µ t (x), in Probability Fleming and Viot [8], Burdzy, Holyst and March [1], Grigorescu and Kang [9] and Löbus [12] in a Brownian motion setting. 10

11 Ergodicity of fv Λ finite, fv Markov in finite state space Hence ergodic (there exists unique stationary measure and the process converges to the stationary measure). For Λ infinite: Theorem 3. If α>0, then for each N the fv process with N particlesisergodic. 11

12 Stationary empirical profile and qsd Assume ergodicity. Let η N be distributed with the unique invariant measure. Theorem 4. α>c.foreachx Λ, the following limits exist lim N η N (x) N and ν is the unique qsd for Q. = ν(x), in Probability 12

13 Sketch of proofs Existence part of Theorem 1 is a corollary of Theorem 4. Uniqueness: Jacka and Robert. Theorem 3: stationary version of the process from the past as in perfect simulation. 13

14 Theorems 2 and 4 based on asymptotic independence. ϕ t unique solution of d dt ϕµ t (x) = ϕ µ t (y)[q(y, x)+q(y, 0)ϕ µ t (x)], y Λ x Λ η t satisfies d ( η N,µ dt E t (x) ) N = ( η N,µ t (y) E N y Λ (q(y, x)+q(y, 0) ηn,µ t (x) )) N 1 We prove: E[η N,µ t (y) η N,µ t (x)] Eη N,µ t (y) Eη N,µ t (x) = O(N) 14

15 qsd satisfies ν(y)[q(y, x)+q(y, 0)ν(x)] = 0, x Λ. y Λ η N invariant for fv satisfies: y Λ ( η N (y) ( E q(y, x)+q(y, 0) ηn (x) )) N N 1 = 0, x Λ. Under α>c: E[η N (y) η N (x)] Eη N (y) Eη N (x) =O(N) Variance order 1/N, setting x = y. Finally we show (ϕ N,µ t,n N) and(ρ N,N N) are tight. 15

16 Comments Fleming-Viot permits to show existence of a qsd in the α>c case (new). Compared with Brownian motion in a bounded region with absorbing boundary (Burdzy, Holyst and March [1], Grigorescu and Kang [9] and Löbus [12] and other related works): Existence of the fv process immediate here. they prove the convergence without factorization. We prove: vanishing correlations sufficient for convergence of expectations and in probability. To prove tightness classify ξ particles in types. Tightness proof needs α>c as the vanishing correlations proof. 16

17 Graphical construction of fv process 17

18 Graphical construction of fv process To each particle i =1,...,N, associate 3 marked Poisson processes: Regeneration times. PP (α): (a i n) n Z,marks(A i n) n Z Internal times. PP ( q α): (b i n) n Z, marks ((B i n(x), x Λ), n Z) Voter times. PP (C): (c i n) n Z, marks ((C i n, (F i n(x), x Λ)), n Z) 18

19 Law of marks: P(A i n = y) =α(y)/α, y Λ; P(Bn(x) i q(x, y) α(y) =y) =, x Λ, y Λ \{x}; q α P(Bn(x) i =x) =1 y Λ\{x} P(Bi n(x) =y). P (Fn(x) i q(x, 0) =1)= =1 P (F i C n(x) =0),x Λ. P (Cn i = j) = 1 N 1, j i. Call ω a realization of the marked PP. 19

20 Construction of ξ N,ξ [s,t] = ξn,ξ [s,t],ω Order Poisson times. Initial configuration ξ at time s. Configuration does not change between Poisson events. At each regeneration time a i n particle i adopts state A i n regardless the current configuration. If at the internal time b i n the state of particle i is x, thenat time b i n particle i adopts state B i n(x) regardless the state of the other particles. If at the voter time c i n the state of particle i is x and F i n(x) =1, then at time c i n particle i adopts the state of particle C i n;if F i n(x) = 0, then particle i does not change state. The final configuration is ξ N,ξ [s,t]. 20

21 Lemma 1. The process (ξ N,ξ [s,t],t s) is fv with initial condition ξ N,ξ [s,s] = ξ. 21

22 Generalized duality Define ω i [s, t] ={m ω : m involved in the definition of ξ N,ξ [s,t],ω (i)}, Generalized duality equation: No explicit formula for H. ξ N,ξ [s,t],ω (i) =H(ωi [s, t],ξ). (1) For any time s, ξ N,ξ [s,t](i) depends only on the finite number of Poisson events contained in ω i [s, t]. 22

23 Theorem 3. If α>0 the fv process is ergodic. Proof If number of marks in ω i [,t] is finite, then ξ N t,ω(i) =: lim s H(ωi [s, t],ξ), i {1,...,N}, t R is well defined and does not depend on ξ. By construction (ξ N t,t R) isastationaryfv process. The law of ξ N t is unique invariant measure. Number of points in ω i [,t] is finite if there is [s(ω),s(ω)+1] in the past of t with one regeneration mark for each k and no voter marks. 23

24 Particle correlations in the fv process Proposition 1. Let x, y Λ. For all t>0 ( η N E t (x)ηt N (y) ) N 2 ( η N E t (x) ) ( η N E t (y) ) < 1 N N N e2ct (2) Assume α>c.letη N be distributed according to the unique invariant measure for the fv process with N particles. Then ( η N (x)η N (y) ) E N 2 ( η N (x) ) ( η N (y) ) 1 E E < N N N α α C (3) 24

25 Coupling 4-fold coupling (ω i [s, t],ω j [s, t], ˆω i [s, t], ˆω j [s, t]) ω i [s, t] =ˆω i [s, t] ˆω j [s, t] ω i [s, t] = implies ω j [s, t] =ˆω j [s, t] marginal process (ˆω i [s, t], ˆω j [s, t])havethesamelawastwo independent processes with the same marginals as (ω i [s, t],ω j [s, t]). 25

26 P(ξ N,ξ t (j) =x, ξ N,ξ t (i) =y) P(ξ N,ξ t (j) =x)p(ξ N,ξ t (i) =y) ( ) = E 1{H(ω j,ξ)=x, H(ω i,ξ)=y)} 1{H(ˆω j,ξ)=x), H(ˆω i,ξ)=y)} If then ω i ω j = ω j (s, t) =ˆω j (s, t) andω i (s, t) =ˆω i (s, t) Hence, P(ξ N,ξ t (j) =x, ξ N,ξ t (i) =y) P(ξ N,ξ t P(ω i ω j ). (j) =x)p(ξ N,ξ t (i) =y) 26

27 Lemma 2. (s =0) P(ω i ω j ) 1 N 1 C α C (1 e2(c α)t ) (4) Proof: P(ω i ω j ) 2C N 1 t 0 E ˆΨ i [s, t] E ˆΨ j [s, t]ds ˆΨ i [s, t] Random walk that grows with rate Cx and decreases with rate αx. Expectation is bounded by e (t s)(c α). which gives the result. P(ω i ω j ) 2C N 1 t 0 e 2(C α)s ds 27

28 Proof of Proposition 1 Take η and ξ such that η(x) = j 1{ξ(j) =x}. Then ( η N,η t (x)η N,η t (y) ) E N 2 Eη N,η t = 1 N 2 N i=1 (x) Eη N,η t (y) N 2 = 1 ( N N 2 i=1 N j=1 N j=1 P(ξ N,ξ t P(ξ N,ξ t Using this, and (4) with α = 0 we get (2). Assume α>c.takingt = in (4) we get (3). (i) =x, ξ N,ξ t (j) =y) ) (i) =x)p(ξ N,ξ t (j) =y) 28

29 Tightness Proposition 2. For all t>0, x Λ, i =1,...,N and µ, Eη N,µ t (x) N e Ct z Λ µ(z)p t (z,x). As a consequence (Eη N,µ t /N, N N) is tight. 29

30 Assume α>0 and define µ α on Λ by µ α (x) = α x α, x Λ, where α x =inf z q(z,x). For z,x Λ define R λ (z,x) = 0 λe λt P t (z,x)dt. Proposition 3. Assume α>c and let η N distributed with invariant measure for fv. Then for x Λ, ρ N (x) C α C µ αr (α C) (x) As a consequence, the family of measures (η N /N, N N) is tight. 30

31 Types Particle i is type 0attimet if it has not been absorbed in the time interval [0,t]. If at absorption time s particle i jumps over particle j which has type k, thenattimes particle i changes its type to k +1. P(ξ N,µ t (i) =x, type(i, t) =0)= z Λ µ(z)p t (z,x). A t (x, k) =:P(ξ N,µ t (i) =x, type(i, t) =k) 31

32 Proof of Proposition 2 A t (x, k) (Ct)k k! Recursive hypothesis: µ(z)p t (z,x) (5) z Λ By (5) the statement is true for k =0. A t (x, k +1) t 0 C y Λ A s (y, k) P t s (y, x) ds. Using recursive hypothesis, = t 0 C (Cs)k k! = (Ct)k+1 (k +1)! µ(z) z Λ y Λ µ(z)p t (z,x). z Λ by Chapman-Kolmogorov. This proves (5). P s (z,y)p t s (y, x)ds 32

33 Proof of Proposition 3 Under the hypothesis α>cthe process ((ξt N (i), type(i, t)),i=1,...,n), t R) is Markovian constructed in a stationary way A(x, k) :=P(ξs N (i) =x, type(i, s) =k) does not depend on s. Last regeneration mark of site i before time s happened at time s Tα,whereT i α i is exponential of rate α. Then, A(x, 0) = 0 αe αt z Λ µ α (z)p t (z,x)dt = µ α R α (x). 33

34 Similar reasoning implies A(x, k) 0 e αt C z Λ A(z,k 1)P s (z,x) dt. = C α A k 1R α (x) ( C α ) kµα Rα k+1 (x). R k λ (z,x) expectation of P τ k (z,x), τ k sum of k independent exponential λ. Multiplying and dividing by (α C), P(ξ N s (i) =x) C α C ( C ) k ( 1 C ) µ α Rα k+1 (x) α α k=0 Expectation of µ α R K α, K geometric with p =1 (C/α). P(ξ N s (i) =x) C α C µ αr α C (x). 34

35 References [1] Burdzy, K., Holyst, R., March, P. (2000) A Fleming-Viot particle representation of the Dirichlet Laplacian, Comm. Math. Phys. 214, [2] Cavender, J. A. (1978) Quasi-stationary distributions of birth and death processes. Adv. Appl. Prob. 10, [3] Darroch, J.N., Seneta, E. (1967) On quasi-stationary distributions in absorbing continuous-time finite Markov chains, J. Appl. Prob. 4, [4] Ferrari, P.A. (1990) Ergodicity for spin systems with stirrings, Ann. Probab. 18, 4: [5] Fernández, R., Ferrari, P.A., Garcia, N. L. (2001) Loss network representation of Peierls contours. Ann. Probab. 29, no. 2,

36 [6] Ferrari, P.A, Kesten, H., Martínez,S., Picco, P. (1995) Existence of quasi stationary distributions. A renewal dynamical approach, Ann. Probab. 23, 2: [7] Ferrari, P.A., Martínez, S., Picco, P. (1992) Existence of non-trivial quasi-stationary distributions in the birth-death chain, Adv. Appl. Prob 24, [8] Fleming,.W.H., Viot, M. (1979) Some measure-valued Markov processes in population genetics theory, Indiana Univ. Math. J. 28, [9] Grigorescu, I., Kang,M. (2004) Hydrodynamic limit for a Fleming-Viot type system, Stoch. Proc. Appl. 110, no.1: [10] Jacka, S.D., Roberts, G.O. (1995) Weak convergence of conditioned processes on a countable state space, J. Appl. Prob. 32,

37 [11] Kipnis, C., Landim, C. (1999) Scaling Limits of Interacting Particle Systems, Springer-Verlag, Berlin. [12] Löbus, J.-U. (2006) A stationary Fleming-Viot type Brownian particle system. Preprint. [13] Nair, M.G., Pollett, P. K. (1993) On the relationship between µ-invariant measures and quasi-stationary distributions for continuous-time Markov chains, Adv. Appl. Prob. 25, [14] Seneta, E. (1981) Non-Negative Matrices and Markov Chains. Springer-Verlag, Berlin. [15] Seneta, E., Vere-Jones, D. (1966) On quasi-stationary distributions in discrete-time Markov chains with a denumerable infinity of states, J. Appl. Prob. 3, [16] Vere-Jones, D. (1969) Some limit theorems for evanescent processes, Austral. J. Statist 11,

38 [17] Yaglom, A. M. (1947) Certain limit theorems of the theory of branching stochastic processes (in Russian). Dokl. Akad. Nauk SSSR (n.s.) 56,

SIMILAR MARKOV CHAINS

SIMILAR MARKOV CHAINS by Phil Pollett The University of Queensland MAIN REFERENCES Convergence of Markov transition probabilities and their spectral properties 1. Vere-Jones, D. Geometric ergodicity in