A REMARK ON LOCALIZATION FOR BRANCHING RANDOM WALKS IN RANDOM ENVIRONMENT

Branching Random Walks in Random Environment (BRWRE) are a model for the spread of particles on an inhomogeneous media, such as bacteria that move around and encounter food supply or environmental conditions variable in time and space. These environmental conditions have an impact on the reproduction rate of the particles. The randomness of the model occurs in two steps. The first step is the setting of the environment, which determines the offspring distribution at different times and places. In our case, these offspring distributions are to be i.i.d..


Informal descripion
Branching Random Walks in Random Environment (BRWRE) are a model for the spread of particles on an inhomogeneous media, such as bacteria that move around and encounter food supply or environmental conditions variable in time and space.These environmental conditions have an impact on the reproduction rate of the particles.The randomness of the model occurs in two steps.The first step is the setting of the environment, which determines the offspring distribution at different times and places.In our case, these offspring distributions are to be i.i.d..

Electronic Communications in Probability
The second step is the development of the population given the environment randomly generated in the first step.Starting with one particle at the origin, each particle generates offspring according to the offspring distribution associated with the time-space-location where it is born.It carries this offspring to adjacent sites in the manner of a simple random walk, and dies, leaving the new particles to start over, independently of each other.As it is possible that particles die without leaving any offspring, the whole population might die out.This phenomenon is described in the event of "extinction".In the present article, however, we are more interested in the long-term-behaviour of the population, and usually work on the complementary event, called "survival".All the notions will be thoroughly defined in Subsection 1.3.

Brief history
Branching random walks in random environment have been introduced in [Birk], and Birkner, Geiger and Kersting [BGK05] revealed a phase change of the model which was subsequently characterized as a dichotomy: [Nak11] revealed that this model exhibits a phase transition beween what is called slow and regular growth, respectively.The question of localization in this model, that is whether or not it is possible that in the long term, many particles may become concentrated on few sites, was answered positively for the slow growth phase by Hu and Yoshida [HY09] for environments that do not allow for extinction.A similar answer is given for the more general model of Linear Stochastic Evolution (LSE) in [Yos10].BRWRE's survival, together with growth rates for the population, are studied by Comets and Yoshida [CY].Uniting tools from the last three articles is what allows us to prove a localization result in a setting where extinction is possible.A central limit theorem for BRWRE in the regular growth phase is proved in [HNY].In that article, a more complete outline of the history of CLTs for BRW, BRWRE and related models can be found, and pictures of the BRWRE are given.

Thorough definition of the model
We define the random environment as i.i.d.offspring distributions (q t,x ) t∈N 0 ,x∈Z d under some (product-)measure Q on Ω q := (N 0 ) N 0 ×Z d , where (N 0 ) is the set of probability measures on N 0 , and may be equipped with the natural Borel-σ-field induced from that of [0, 1] N 0 .We call this product-σ-field q .
On a measurable space (Ω K , K ), to each fixed environment q = (q t,x ) t∈N 0 ,x∈Z d we associate a probability measure P q K such that the random variables K := (K ν t,x ) t∈N 0 ,x∈Z d ,ν∈N are independent in the number ν of the particle and the space-time point (t, x) while being distributed according to q t,x : These random variables K ν t,x describe the number of children born to the ν-th particle at timespace-location (t, x).The ν-th particle (ν ∈ N) at time t ∈ N 0 := {0, 1, . . .} =: N ∪ {0} and site x ∈ Z d moves (together with all of his offspring) to some site adjacent to his birthplace, determined by the Z d -valued random variable X ν t,x .The X := (X ν t,x ) t∈N 0 ,x∈Z d ,ν∈N , defined on a probability space (Ω X , X , P X ), are defined to be the one-step transitions of a simple random walk, and i.i.d. in all three time, space, and particles: At its time-space destination (t + 1, X ν t,x ), the said ν-th particle from (t, x) dies and leaves place to its children, and the procedure starts over for every child.Of course, we can combine the realization of X and K on one probability space (Ω X × Ω K , X ⊗ K , P q ), where P q := P X ⊗ P q K (1.3) and finally merge all our construction to (1.4) P q can be seen as the quenched measure and P as the annealed one of the model.Now we come to the population at time t and site x.We start at time 0 with one particle at the origin, and define inductively The filtration makes the process t → (N t,x ) x∈Z d adapted.The total population at time t can now be obtained by summation over all sites: Important quantities of this model are the averaged and local moments of the offpring distributions We also write m := m (1) .

The phase transition of the normalized population
It has been proven in [Nak11] that the total population exhibits a phase transition, where the one phase amounts to population growing as fast as its expectation, while the other phase means slower-than-the-expectation growth.
Sufficient conditions for both phases are given by the following two Propositions 1.4.3 and 1.4.4,which necessitate a bit of Notation 1.4.2.Given the simple symmetric random walk S t on Z d , we call π d the probability of the return event t≥1 {S t = 0}.Furthermore, we write Proposition 1.4.4.On the other hand, P(N ∞ = 0) = 1 is provided by any of the following three conditions: Remark 1.4.5.We would at this point recall the non-random environment case [AN72, Theorem 1, page 24], where In our case here, with the additional randomness of the environment, P(N ∞ = 0) = 1 can happen even if the K ν t,x are bounded (see Remark 1.6.3b) below).

Survival and the global growth estimate
Another dichotomy of this model is the one of survival and extinction.We define (1.12) The event of extinction is defined as the complement.
The following global growth estimate obtained in [CY, Theorem 2.1.1]characterizes the event of survival: Lemma 1.5.1.Suppose Q(m t,x + m −1 t,x ) < ∞ and let > 0.Then, for large t, N t ≤ e (Ψ+ )t , P-a.s., where the limit b) It is proved in [CY] as well that "Ψ > 0" is implied by The object we investigate is the population density It describes the distribution of the population in space.Related important objects are They are, respectively, the density at the most populated site and the probability that two particles picked randomly from the total population are at the same site at time t.We will call this latter value the "replica overlap".It is possible to relate the event of survival to this replica overlap. (1.17) The proof of this Theorem can be found in Section 2.2.While it is true that the opposite inclusion does hold under the stronger assumption m (3) < ∞, we do not state this formally here.The proof can be found in [HNY].

The main result
Hu and Yoshida, using the assumption that particles may not die, proved in [HY09, Theorem 1.3.2] the following Theorem 1.6.1.Suppose P(N ∞ = 0) = 1 and Then, there exists a non-random number c ∈ (0, 1) such that, In this setting, extinction (i.e. the event that at some time, the total population becomes 0) cannot occur.However, it is possible to drop this assumption with the help of a few additional tools.
Our main result is indeed that the last two hypotheses can be replaced by weaker ones.
Remark 1.6.3.a) The fact that Theorem 1.6.1 does not allow for dying particles has two implications, namely Ψ > 0 (rather trivially by (1.14)) and Our theorem shows that we can indeed content ourselves with these two weaker conditions themselves.b) The hypotheses P(N ∞ = 0) = 1 and Ψ > 0 are difficult to check in practice.Yet, it is possible to give an example that satisfies the easier (a1) − (a3) of Proposition 1.4.4 and (1.14), but not the hypotheses of Theorem 1.6.1.It is given by the following class of environments constituted only of two states: for n ∈ N,

Proofs
2.1 Tools for the proof of Theorem 1.5.3 The following Definition will be useful at several points.It provides notation for the thorough calculus of the fluctuation of the normalized population.

Proof of Theorem 1.5.3
We want to apply the abstract result that is Proposition 2.1.2to our setting.To get the notation right, we take X t := N t , and remark that the definition verifies (2.4); the U t,x are taken from Definition 2.1.1.As for the other hypothesis of the Proposition, we need not even to check it in order to find ∞ s=1 P (∆Y s ) 2 s−1 = ∞: if uniform boundedness does not hold, it is true anyway, and if uniform boundedness holds, we derive it from Proposition 2.1.2on the event {sur vi val} ∩ {N ∞ = 0}.Now, with (2.1), we see that s−1 shares its asymptotic behaviour with t s=1 s , so we conclude (1.17).

Tools for the proof of Theorem 1.6.2
One result that has not been taken into account in [HY09] and that helps us making the slight improvement of the hypotheses is the following improved version of the Borel-Cantelli-lemma, stated in [Yos10, Lemma 2.2.1]:Lemma 2.3.1.Let (R t ) t∈N be an integrable, adapted process defined on a probability space with measure E and a filtration ( t ) t∈N 0 .Define V 0 := 0 =: T 0 and s. for all t ∈ N. (2.7) Then, , and that there exists a constant C 2 ∈ (0, ∞) such that where Then, E-a.s., This Lemma admits in our setting, with a slight abuse of notation, for the following Corollary 2.3.2.On the event {lim t→∞ V t = ∞}, there exists a constant c 0 ∈ [1, ∞) such that holds for large t.
Proof.In fact, the hypotheses of both a) and b) of Lemma 2.3.1 are satisfied.Indeed, 0 ≤ t = x ρ 2 t,x ≤ 1 is square-integrable and adapted, and (2.7) is satisfied with C 1 = 2. Also, Hence, with a), {lim t→∞ V t = ∞} implies {sup t T t = ∞}.But T t is a sum over positive terms, so its supremum is equal to its limes, and we can readily apply part b).The statement is then trivial.
Lemma 2.3.3.Let (U i ) 1≤i≤n , n ≥ 2, be non-negative, independent and cube-integrable random variables on our general probability space with probability measure E such that for (2.9) Let furthermore X be a random variable such that 0 ≤ X ≤ U 2 1 a.s..Then, (2.10) (2.11) Proof.The first inequality is proved in [Yos10].We will prove the second one.Note that u −2 ≥ 3 − 2u for u ∈ (0, ∞).Thus, we have that At this point, we need some further notations.We denote by s (x, y) the probability that the simple random walk starting in x ∈ Z d goes to y ∈ Z d in exactly s ∈ N steps.We write r j := 2 j (x, x).Also, we can define the semigroup of the simple random walk by s f (x) := y s (x, y) f ( y).We write := 1 .
Remark 2.3.4.With the Cauchy-Schwarz-inequality, we have max (2.12) We now start estimates on the population density.The following result corresponds to the inequality (3.7) in [HY09, Lemma 3.1.4].
Lemma 2.3.5.Suppose (1.20).On the event of survival up to time s ∈ N, for any y 1 , y 2 ∈ Z d , we have where the c • are the same as in Definition 2.1.1.
Proof.We have (2.15) Now, we would like to estimate (2.14) and (2.15).We can rewrite these lines with the processes from Definition 2.1.1.These verify the hypotheses of Lemma 2.3.3.The estimates obtained by the application of this Lemma comprise second and third moments which we cannot provide explicitly.We therefore replace them by the estimates obtained in Definition 2.1.1;note that survival up to time s + 1 implies survival up to time s.
Lemma 2.3.6.Suppose (1.20).For all 1 ≤ j ≤ t − 1, Proof.If we apply the definition of the semigroup operator , we get Applying (2.13) gives where 2 by definition of the semigroup-operator; t− j t− j by Remark 2.3.4; In these computations, the symmetry of p(•, •) has been used at appropriate places.If we put together the pieces, we obtain the statement of the Lemma.
Later, in the proof of the main theorem, we are going to perform a division by t s=1 s at some point.The following Lemma helps showing that a certain term then vanishes asymptotically.We recall the definition of V t := t s=1 s from Corollary 2.3.2 and write V ∞ := lim t→∞ V t .Lemma 2.3.7.Assume (1.20), and fix some j ≥ 1.The martingale Z j (•) defined by satisfies the following law of large numbers: Proof of Lemma 2.3.7.The idea of the proof is to make use of the increasing process 〈Z j 〉 t associated with Z j (t) in order to monitor the growth of Z j (t) itself.With the previous Remark, it is indeed possible to estimate 〈Z j 〉 t by the sum of the conditional replica-overlap: (2.17) The rest is easy.Either 〈Z j 〉 ∞ < ∞, in which case Z j (t) converges and the statement is trivial anyway, or 〈Z j 〉 ∞ = ∞, in which case we can apply the law of large numbers for square-integrable martingales, see [Dur91, p. 253], which gives us As a final ingredient, we give a statement that compares parameters of the simple random walk with ones of the BRWRE-model.
Propositions 1.4.3 and 1.4.4 were obtained first in [BGK05, Theorem 4].Proposition 1.4.3 plays a crucial role in our proof as it allows us in the slow growth phase to conclude α > α * > 1/π d .

∼
ln n, and hence any dimension can be covered by n large enough.