On the behavior of the population density for branching random. branching random walks in random environment. July 15th, 2011

Transcription

1 On the behavior of the population density for branching random walks in random environment Makoto Nakashima Kyoto University July 15th, 2011

2 Branching Processes offspring distribution P(N) = {{q(k)} k=0 [0, 1]N : q(k) = 1} k=0

3 Branching Random Walks in Random Environment on Z d In this talk, we consider branching random walk in time-space random environment defined by the following rule: 1 There exists one particle at the origin at time 0 2 Each particle at site x at integer time t independently chooses a nearest neighbor site, moves there, and is independently replaced by k particles with probability q t,x (k), where q t,x = (q t,x (k)) k N are offspring distributions given as P(N)-valued iid random variables in time and space

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12 Remark 1 When q t,x q, this process is branching random walks on Z d and the total number of particles is a Galton-Watson process with offspring distribution q 2 If the space is a single point, then this process is called Smith-Wilkinson model which is a branching processes in random environment

13 In this talk, we will consider the population density of particles, ie Population density ρ t (x) = N t,x 1{N t > 0}, N t where we define N t, N t,x as follows: N t = {particles at time t}, N t,x = {particles at site x at time t} Remark {ρ t ( )} t N are random probability measures on Z d R d on {surv}, where {surv} def = {N t > 0, t N}

14 Known Results When environment is constant (q t,x q), we have the following theorem about the behavior of ρ t (x) (central limit theorem for BRW) Theorem 1(Biggins 90) Suppose that m > 1 and k 1 (k log k)q(k) <, where m = k 1 k q(k) Then, ρ t ( t ) ν( ), as on {surv}, that is, for all f C b (R d ), lim f ( x )ρ t (dx) = f (x)ν(dx), t R d t R d as on {surv}, where ν is a Gaussian measure with mean 0 and covariance 1 d I

15 We have the following question: Question How about the behavior of population density for BRWRE? More precisely, does CLT hold for BRWRE?

16 We have the following question: Question How about the behavior of population density for BRWRE? More precisely, does CLT hold for BRWRE? Answer Partially yes! d = 1, 2 Only the localization can occur d 3 Phase transition (CLT holds in one phase whereas the localization occurs in the other phase)

17 We have the following question: Question How about the behavior of population density for BRWRE? More precisely, does CLT hold for BRWRE? Answer Partially yes! d = 1, 2 Only the localization can occur d 3 Phase transition (CLT holds in one phase whereas the localization occurs in the other phase) In the rest of talk, we see the conditions for CLT and localization

18 Properties Notation P q : the probability measure describes branching random walk under fixed environment q = (q t,x ) (t,x) N Z d Q : the probability measure describes the iid offspring distributions assigned to each time-space location P( ) = P q ( )Q(dq)

19 Lemma 2 E[N t ] = m t, where m = Q[ k 1 kq t,x(k)] N t def = N t a non-negative martingale with respect to F t Therefore, there exists the limit m t = N t E[N t ] is N t N = lim t m t = lim N t t E[N t ] P-as and E[N ] = 1 or 0 Moreover, E[N ] = 1 N t are uniformly integrable F t is the filtration generated by branching, random walks, and environment up to time t

20 Fact It is known that 1 if d = 1 or 2 and Q(m t,x = m) 1, then N = 0 P-as, where m t,x = k 1 k q t,x(k) 2 if d 3, then a phase transition occurs, that is, { 1 if environment is not too random (regular growth) E[N ] = 0 if environment is random enough (slow growth) Regular growth N t E[N t ] Slow growth N t E[N t ]

21 Fact It is known that 1 if d = 1 or 2 and Q(m t,x = m) 1, then N = 0 P-as, where m t,x = k 1 k q t,x(k) 2 if d 3, then a phase transition occurs, that is, { 1 if environment is not too random (regular growth) E[N ] = 0 if environment is random enough (slow growth) Regular growth N t E[N t ] Slow growth N t E[N t ] We will look at the behavior of ρ t for each phase

22 Regular Growth Phase Theorem 3 (Heil-N-Yoshida 11) Suppose d 3 In the regular growth phase, we assume that m > 1, m (4) <, where m (4) = Q[ k 1 k4 q t,x (k)] Then, ρ t ( t ) ν in P( surv)-probability, where ν is a Gaussian measure with mean 0 and covariance 1 d I

23 Remark 1 We found from this theorem that central limit theorem holds when the process is in the regular growth phase 2 The sufficient conditions for regular growth phase are known Moreover, under some conditions, CLT is improved to almost sure convergence (N 11)

24 Slow Growth Phase In the slow growth phase, the behavior of ρ t has been completely changed Set Ψ = lim t Q[log Pq [N t ]] Theorem 4 (Heil-N 10) Suppose that Q[mt,x 1 ] <, Ψ > 0, and m (3) < If the process is in the slow growth phase, then there exists a constant c > 0 such that t 1 max ρ t (x) > c io on {surv} x This result is proved under non-extinction condition (Hu-Yoshida 09)

25 ρ t1 (x) ρ t2 (x) c

26 When Ψ > 0? It is difficult to check the assumptions in Theorem 4, since we cannot compute Ψ However, we have a sufficient conditions for Ψ > 0 as follows Q[ln(m t,x )] > 0 Ψ > 0 Also, we remark that P q [N t ] is the partition function of directed polymers in random environment

27 Thank you for your attention!