www.imstat.org/aihp 2015, Vol. 51, No. 4, 1251–1289

DOI:10.1214/14-AIHP620

© Association des Publications de l’Institut Henri Poincaré, 2015

### Branching random walks in random environment and super-Brownian motion in random environment

### Makoto Nakashima

*Division of Mathematics, Graduate School of Pure and Applied Sciences,University of Tsukuba, 1-1-1 Ten-noudai, Tsukuba-shi, Ibaraki-ken,*
*Japan. E-mail:**nakamako@math.tsukuba.ac.jp*

Received 30 June 2013; revised 16 April 2014; accepted 16 April 2014

**Abstract.** We focus on the existence and characterization of the limit for a certain critical branching random walks in time–space
random environment in one dimension which was introduced by Birkner, Geiger and Kersting in (In*Interacting Stochastic Systems*
(2005) 269–291 Springer). Each particle performs simple random walk onZand branching mechanism depends on the time–space
site. The limit of this measure-valued processes is characterized as the unique solution to the non-trivial martingale problem and
called super-Brownian motion in a random environment by Mytnik in (Ann. Probab.**24**(1996) 1953–1978).

**Résumé.** Nous étudions l’existence et la caractérisation de la limite de marches branchantes critiques dans un environnement
spatio-temporel aléatoire en dimension 1 introduit par Birkner, Geiger and Kersting dans (In*Interacting Stochastic Systems*(2005)
269–291 Springer). Chaque particule effectue une marche aléatoire simple surZet le mécanisme de branchement dépend du site
indexé par l’espace et le temps. La limite de ce processus à valeur mesure est caractérisée comme l’unique solution d’un problème
de martingale non-trivial et correspond au super mouvement Brownien en environnement aléatoire par Mytnik dans (Ann. Probab.

**24**(1996) 1953–1978).

*MSC:*60H15; 60J68; 60J80; 60K37

*Keywords:*Superprocesses in a random environment; Branching random walks in a random environment; Stochastic heat equations; Uniqueness

We denote by*(Ω,F, P )*a probability space. LetN= {0,1,2, . . .},N^{∗}= {1,2,3, . . .}, andZ= {0,±1,±2, . . .}. Let
*C*_{x}_{1}_{,...,x}* _{p}*or

*C(x*

_{1}

*, . . . , x*

_{p}*)*be a non-random constant which depends only on the parameters

*x*

_{1}

*, . . . , x*

*.*

_{p}**1. Introduction**

Super-Brownian motion (SBM) is a measure-valued process which was introduced by Dawson and Watanabe inde- pendently [4,30] and is obtained as the limit of (asymptotically) critical branching Brownian motions (or branching random walks). There are many books for introduction of super-Brownian motion [6,10] and dealing with several aspects of it [8,9,15,25]. Also, super-Brownian motion appears in physics and population genetics.

An example of the construction is the following, where we always treat Euclidean space as the space,R* ^{d}* in this
paper. We assume that at time 0, there are

*N*particles in Z

*as the 0th generation particle. Each of*

^{d}*N*particles chooses independently of each other a nearest neighbor site uniformly, moves there at time 1, and then each particle independently of each other either dies or splits into two particles with probability 1/2 (1st generation). The newly produced particles in the

*nth generation perform in the same manner, that is each of them chooses independently of*each other a nearest neighbor site uniformly, moves there at time

*n*+1, and then each particle independently of each other either dies or splits into 2 particles with probability 1/2.

Let*X*^{(N )}_{t}*(*·*)*be the measure-valued Markov processes defined by
*X*^{(N )}_{t}*(B)*={particles in*B*√

*N* at*t N*th generation at time*t N*}

*N* *,*

where*B*∈*B(*R^{d}*)*are Borel sets inR* ^{d}* and

*B*√

*N*= {*x*=*y*√

*N* for*y*∈*B*}. Then, under some conditions, they con-
verge as*N*→ ∞to a measure-valued processes,*super-Brownian motion. In particular, the limit,X** _{t}*, is characterized
as the unique solution to the martingale problem:

⎧⎪

⎪⎨

⎪⎪

⎩

For all*φ*∈*C*_{b}^{2}*(*R*),*

*Z*_{t}*(φ)*:=*X*_{t}*(φ)*−*X*_{0}*(φ)*−_{t}

0 1

2d*X*_{s}*(φ)*ds
is an*F**t** ^{X}*-continuous square-integrable martingale

*Z*0

*(φ)*=0 and

*Z(φ)*

*t*=

*t*

0*X**s**(φ*^{2}*)*ds,

(1.1)

where*ν(φ)*=

*φ*dνfor any measure*ν.*

It is a well-known fact that one-dimensional super-Brownian motion is related to a stochastic heat equation [14,
26]. When*d*=1, super-Brownian motion*X*_{t}*(dx)*is almost surely absolutely continuous with respect to the Lebesgue
measure and its density*u(t, x)*satisfies the following stochastic heat equation:

*∂*

*∂tu*=1

2*u*+√

*uW (t, x),*˙

where*W (t, x)*˙ is time–space white noise. On the other hand, for*d*≥2,*X*_{t}*(*·*)*is almost surely singular with respect to
the Lebesgue measure [7,16,23,24].

In this paper, we consider super-Brownian motion in a random environment, which was introduced in [19]. Mytnik
showed the existence and uniqueness of the scaling limit*X*_{t}*(*·*)*for a certain critical branching diffusion in a random
environment with some conditions. It is characterized as the unique solution to the martingale problem:

⎧⎪

⎪⎪

⎨

⎪⎪

⎪⎩

For all*φ*∈*C*_{b}^{2}*(*R^{d}*),*

*Z*_{t}*(φ)*:=*X*_{t}*(φ)*−*X*_{0}*(φ)*−_{t}

0 1

2*X*_{s}*(φ)ds*

is an*F**t** ^{X}*-continuous square-integrable martingale and

*Z(φ)*

*t*=

*t*

0*X**s**(φ*^{2}*)*ds+*t*
0

R* ^{d}*×R

^{d}*g(x, y)φ(x)φ(y)X*

*s*

*(dx)X*

*s*

*(dy)*ds,

(1.2)

where*g(*·*,*·*)*is bounded continuous function in a certain class. In this paper, we construct a super-Brownian motion in
a random environment as the limit of scaled branching random walks in a random environment, which is a solution to
(1.2) for the case where*g(x, y)*is replaced by*δ**x,y*and*d*=1. The definition of such a martingale problem is formal.

The rigorous definition will be given later.

**2. Branching random walks in a random environment**

Before giving the system of the branching random walks in a random environment, we introduce the Ulam–Harris tree
*T* for labeling the particles. We set*T** _{k}*=

*(*N

^{∗}

*)*

^{k}^{+}

^{1}for

*k*≥1. Then, the Ulam–Harris tree

*T*is defined by

*T*=

*k*≥0*T** _{k}*.
We will give a name to each particle by using elements of

*T*.

(i) When there are*M*particles at the 0th generation, we label them as 1,2, . . . , M∈*T*0.

(ii) If the *nth generation particle* x=*(x*_{0}*, . . . , x*_{n}*)*∈ *T** _{n}* gives birth to

*k*

_{x}particles, then we name them as

*(x*

_{0}

*, . . . , x*

_{n}*,*1), . . . , (x0

*, . . . , x*

_{n}*, k*

_{x}

*)*∈

*T*

_{n}_{+}

_{1}.

Thus, every particle in the branching systems has its own name in*T*. We define|x|by its generation, that is ifxis an
element of*T** _{k}*, then|x| =

*k. For convenience, we denote by*|x∧y|the generation of the closest common ancestor of xandy. Ifxandy have no common ancestor, then we define|x∧y| = −∞. Also, we denote byy/xthe last digit ofy whenyis a child ofx, that is

y/x=

*k*_{y}*,* ifx=*(x*_{0}*, . . . , x*_{n}*)*∈*T** _{n}*,y=

*(x*

_{0}

*, . . . , x*

_{n}*, k*

_{y}

*)*∈

*T*

_{n}_{+}

_{1}, for some

*n*∈N,

∞*,* otherwise.

Now, we give the definition of branching random walks in a random environment. In our case, a particle moves on Zand the process evolves by the following rules:

(i) The initial particles are located at sites{*x** _{i}*∈2Z:

*i*=1, . . . , M

*N*}.

(ii) Each particle located at site*x*at time*n*chooses a nearest neighbor site independently of each others with proba-
bility^{1}_{2} and moves there at time*n*+1. Then, it is replaced by*k-children with probabilityq*_{n,x}^{(N )}*(k)*independently
of each others,

where{{*q*_{n,x}^{(N )}*(k)*}^{∞}_{k}_{=}_{0}: *(n, x)*∈N×Z}are the offspring distributions assigned to each time–space site*(n, x)*which are
i.i.d. in*(n, x). We denote byB*_{n}* ^{(N )}*and by

*B*

_{n,x}*the total number of particles at time*

^{(N )}*n*and the local number of particles at site

*x*at time

*n. Also, we denote bym*

^{(N,p)}*the*

_{n,x}*pth moment of offsprings for offspring distribution*{

*q*

_{n,x}

^{(N )}*(k)*}, that is

*m*^{(N,p)}* _{n,x}* =

^{∞}

*k*=0

*k*^{p}*q*_{n,x}^{(N )}*(k).*

This model is called branching random walks in a random environment (BRWRE) whose properties as measure- valued processes are studied well for “supercritical” case [12,13]. Also, the continuous counterpart, branching Brow- nian motions in a random environment was introduced by Shiozawa [28,29]. We know that the normalized random measure weakly converges to Gaussian measure in probability in one phase, whereas the localization has occurred in the other phase.

In this paper, we focus on the scaled measure-valued processes*X*_{t}* ^{(N )}*associated to this branching random walks:

*X*^{(N )}_{0} = 1
*N*

*M*_{N}*i*=0

*δ*_{x}

*i**/N*^{1/2}*,*
and

*X*^{(N )}* _{t}* = 1

*N*

*B*_{tN}^{(N )}

*i*=1

*δ*_{x}_{i}* _{(t )/N}*1/2

*,*for

*t*= 1

*N, . . . ,KN*

*N* for each*K >*0,

where*x*_{i}*(t)*is the position of the*ith particle att N*th generation. We remark that if we identify*B*_{tN,x}* ^{(N )}* as the measure

*B*

_{tN,x}

^{(N )}*δ*

*, then*

_{x}*X*

_{t}*is represented as*

^{(N )}*X*^{(N )}* _{t}* = 1

*N*

*x*∈Z

*B*_{tN,x}^{(N )}*δ** _{x/N}*1/2 for

*t*= 1

*N, . . . ,KN*
*N* *.*

Let*M**F**(*R*)*be the set of the finite measures onRwith the topology of weak convergence. For convenience, we extend
this model to the càdlàg paths in*M**F**(*R*)*by

*X*^{(N )}* _{t}* = 1

*N*

*x*∈Z

*B*_{t N,x}^{(N )}*δ** _{x/N}*1/2

*,*for

*t*≤

*t < t*+ 1

*N,*

where we define*t* for*t* and*N* by some positive number _{N}* ^{i}* for

*i*∈Nsatisfying

_{N}*≤*

^{i}*t <*

^{i}^{+}

_{N}^{1}. Then,

*X*

_{t}*∈*

^{(N )}*M*

*F*

*(*R

*)*for each

*t*∈ [0, K]. Let

*φ*∈

*B*

*b*

*(*R

*), where*

*B*

*b*

*(*R

*)*is the set of the bounded Borel measurable functions onR. We denote the product of

*ν*∈

*M*

*F*

*(*R

*)*and

*φ*∈

*B*

*b*

*(*R

*)*by

*ν(φ), that is*

*ν(φ)*=

R*φ(x)ν(dx).*

To describe the main theorem, we introduce the following assumption on the environment:

**Assumption A.**

*E*
*m*^{(N,1)}_{0,0}

=*E*
_{∞}

*k*=0

*kq*_{n,x}^{(N )}*(k)*

=1, lim

*N*→∞*E*

*m*^{(N,2)}_{0,0} −1

=*γ >*0,

sup

*N*≥1

*E*
*m*^{(N,4)}_{0,0}

*<*∞*,* lim

*N*→∞*N*^{1/2}*E*

*m*^{(N,1)}_{0,0} −12

=*β*^{2}*,*

sup

*N*≥1

*N*^{1/2}*E*

*m*^{(N,1)}_{0,0} −14

*<*∞*.*

**Example.***The simplest example satisfying Assumption*A*is the case where* *q*_{n,x}^{(N )}*(0)*=^{1}_{2}− ^{βξ(n,x)}_{2N}1/4 *, q*_{n,x}^{(N )}*(2)*=^{1}_{2} +

*βξ(n,x)*

2N^{1/4} *for i.i.d.random variables*{*ξ(n, x):* *(n, x)*∈N×Z}*such thatP (ξ(n, x)*=1)=*P (ξ(n, x)*= −1)=^{1}_{2}.
**Theorem 2.1.** *We suppose thatX*^{(N )}_{0} *(*·*)*⇒*X*_{0}*(*·*)inM**F**(*R*)and Assumption*A.*Then,the sequence of measure-valued*
*processes*{*X*^{(N )}_{·} : *N*∈N}*converges to a continuous measure-valued processX*_{·}∈*C(*[0,∞*),M**F**(*R*)).Moreover*,*for*
*anyt >*0,*X*_{t}*(dx)is almost surely absolutely continuous with respect to the Lebesgue measure and its densityu(t, x)*
*is the unique nonnegative solution to the following martingale problem:*

⎧⎪

⎪⎨

⎪⎪

⎩

*For allφ*∈*C*_{b}^{2}*(*R*),*
*Z*_{t}*(φ)*=

R*φ(x)u(t, x)*dx−

R*φ(x)X*_{0}*(dx)*−^{1}_{2}_{t}

0

R*φ(x)u(s, x)dx*ds
*is anF*_{t}^{X}*-continuous square-integrable martingale and*

*Z(φ)**t*=_{t}

0

R*φ*^{2}*(x)(γ u(s, x)*+2β^{2}*u(s, x)*^{2}*)*dxds.

(2.1)

**Remark 1.***We found in theExampleafter Assumption*A*that the fluctuation of the environment is mainly given by*
*(m*^{(N,1)}* _{n,x}* −1)

*and scaling factor isN*

^{−}

^{1/4}. (It appears clearly in the

*Exampleafter Assumption*A.)

*This scaling factor*

*is different from*

*N*

^{−}

^{1/2},

*the one in*[19].

*When the scaling factor isN*

^{−}

^{1/2},

*the limit is the usual super-Brownian*

*motion*(1.1).

*We roughly discuss how the scaling factor in our model is determined.For simplicity,we consider the model for*
*the case where the environment is the one given in theExample.*

*We scale the space byN*^{−}^{1/2}.*Then,the summation of the fluctuation of the first moment of offsprings in the segment*
{*k*} × [*x, y*]*is*

*z*∈[*xN*^{1/2}*,yN*^{1/2}]*βξ(k,z)*

*N*^{1/4} .*Since it is the summation of i.i.d.random variables of* ^{(y}^{−}^{x)N}_{2} ^{1/2},*the central*
*limit theorem holds and it weakly converges to a Gaussian random variable with distributionN (0,*^{β}^{2}^{(y}_{2}^{−}^{x)}*).Similar*
*argument holds for random variables other than Bernoulli random variables.*

**Remark 2.***The martingale problem*(2.1)*is the rigorous definition of the martingale problem wheng(x, y)* *is re-*
*placed byδ*_{x}_{−}_{y}*in*(1.2).*Also,the theorem implies the existence and the uniqueness of the nonnegative solution to the*
*stochastic heat equation*

*∂*

*∂tu*=1

2*u*+

*γ u*+2β^{2}*u*^{2}*W ,*˙ (2.2)

*and*lim*t*→+0*u(t, x)*dx=*X*_{0}*(dx)inM**F**(*R*),whereW*˙ *is time–space white noise.In*[18],*the existence of solutions*
*was proved for general SPDE containing*(2.2)*when the initial measureX*_{0}*(dx)has a continuous density with rapidly*
*decreasing at infinity.*

*Also,there are a lot of papers on uniqueness of the stochastic heat equation*_{∂t}^{∂}*u*=^{1}_{2}*u*+ |*u*|^{α}*W*˙.*It is known that*
*weak uniqueness for nonnegative solutions holds for* ^{1}_{2}≤*α*≤1 [20]*and pathwise uniqueness holds for* ^{3}_{4}*< α*≤1
[21].*However,uniqueness in law and pathwise uniqueness fails when solutions are allowed to take negative values*
*forα <*^{3}_{4}[17].

**3. Proof of Theorem2.1**

In this section, we will give a proof of Theorem2.1. The proof is divided into three steps:

(i) tightness,

(ii) identification of the limit point processes, (iii) weak uniqueness of the limit points.

In this section, we consider the following setting for simplicity.

**Assumption B.***The number of initial particles isN* *and all of them locates at the origin at time*0.*Also,q*_{n,x}^{(N )}*(0)*=

1

2−^{βξ(n,x)}_{2N}1/4 *, q**n,x*^{(N )}*(2)*=^{1}_{2}+^{βξ(n,x)}_{2N}1/4 *for i.i.d.random variables*{*ξ(n, x):* *(n, x)*∈N×Z}*such thatP (ξ(n, x)*=1)=
*P (ξ(n, x)*= −1)=^{1}_{2}.

To consider the general model, it is almost enough to replace ^{βξ(n,x)}* _{N}*1/4 by

*m*

^{(N,1)}*−1. We sometimes need to consider {{*

_{n,x}*q*

_{n,x}

^{(N )}*(k)*}

*k*≥0:

*(n, x)*∈N×Z}. Especially,

*γ*appears in the same situation as the construction of the usual super- Brownian motion, so the reader will not have any difficulties extending the proof to the more general case.

Before starting the proof, we will look at the*X*_{t}^{(N )}*(φ). SinceX*^{(N )}* _{t}* are constant in

*t*∈ [

*t , t*+

_{N}^{1}

*), it is enough to see*the difference between

*X*

_{t}*and*

^{(N )}*X*

_{t}

^{(N )}_{+}

_{1/N};

*X*^{(N )}_{t}_{+}_{1/N}*(φ)*−*X*_{t}^{(N )}*(φ)*= 1
*N* x∼*t*

*φ*

*Y*_{t N}^{x}_{+}_{1}
*N*^{1/2}

*V*^{x}−*φ*

*Y*_{t N}^{x}
*N*^{1/2}

*,*

wherex∼*t*means that the particlexis the*tN*th generation,*Y*_{t N}^{x} is the position of the particlexat time*tN*forx∼*t*,
*V*^{x}is the number of children ofxand for simplicity, we omit*N*. We define*Y*_{t N}^{x}_{+}_{1}=*Y*_{t N}^{y} _{+}_{1}fory which is a child
ofx.

Also, we divide this summation into four parts:

*(LHS)*

= 1
*N*_{x∼}

*t*

*φ*
*Y*_{t N}^{x}_{+}_{1}

*N*^{1/2}

*V*^{x}−1−*βξ(t N, Y*_{t N}^{x}*)*
*N*^{1/4}

+ 1
*N* _{x∼}

*t*

*φ*
*Y*_{t N}^{x}_{+}_{1}

*N*^{1/2}

*βξ(t N, Y*_{t N}^{x}*)*
*N*^{1/4}
+ 1

*N* _{x}_{∼}

*t*

*φ*

*Y*_{t N}^{x}_{+}_{1}
*N*^{1/2}

−*φ*
*Y*_{t N}^{x}

*N*^{1/2}

−

*φ*

*Y*_{t N}^{x} +1
*N*^{1/2}

+*φ*

*Y*_{t N}^{x} −1
*N*^{1/2}

−2φ
*Y*_{t N}^{x}

*N*^{1/2}
2

+ 1
*N* _{x}_{∼}

*t*

*φ*

*Y*_{t N}^{x} +1
*N*^{1/2}

+*φ*

*Y*_{t N}^{x} −1
*N*^{1/2}

−2φ
*Y*_{t N}^{x}

*N*^{1/2}
2

=*M*_{t}^{(b,N )}*(φ)*+*M*_{t}^{(e,N )}*(φ)*+*M*_{t}^{(s,N )}*(φ)*
+ 1

*N* _{x}_{∼}

*t*

*φ*

*Y*_{t N}^{x} +1
*N*^{1/2}

+*φ*

*Y*_{t N}^{x} −1
*N*^{1/2}

−2φ
*Y*_{t N}^{x}

*N*^{1/2}
2.

Thus, we have that
*X*^{(N )}_{t}*(φ)*−*X*_{0}^{(N )}*(φ)*=

*M*_{t}^{(b,N )}*(φ)*+*M*_{t}^{(e,N )}*(φ)*+*M*_{t}^{(s,N )}*(φ)*
+

_{t}

0

*X*_{s}^{(N )}*A*^{N}*φ*

ds, (3.1)

where

*M*_{t}^{(b,N )}*(φ)*= 1

*N* *s<t*x∼*s*

*φ*
*Y*_{sN}^{x}_{+}_{1}

*N*^{1/2}

*V*^{x}−1−*βξ(sN, Y*_{sN}^{x} *)*
*N*^{1/4}

*,*

*M*_{t}^{(e,N )}*(φ)*= 1

*N* *s<t*x∼*s*

*φ*
*Y*_{sN}^{x}_{+}_{1}

*N*^{1/2}

*βξ(sN, Y*_{sN}^{x}*)*
*N*^{1/4} *,*

*M*_{t}^{(s,N )}*(φ)*= 1

*N* *s<t*x∼*s*

*φ*

*Y*_{sN}^{x}_{+}_{1}
*N*^{1/2}

−*φ*
*Y*_{sN}^{x}

*N*^{1/2}

−

*φ*

*Y*_{sN}^{x} +1
*N*^{1/2}

+*φ*

*Y*_{sN}^{x} −1
*N*^{1/2}

−2φ
*Y*_{sN}^{x}

*N*^{1}^{2}
2

*,*

and*A** ^{N}*:

*B*

*b*

*(*R

*)*→

*B*

*b*

*(*R

*)*is the following operator;

*A*^{N}*φ(x)*=

*φ*

*x*+ 1
*N*^{1/2}

+*φ*

*x*− 1

*N*^{1/2}

−2φ(x) 2

*N.*
Actually, we have that

_{t}

0

*X*^{(N )}_{s}*A*^{N}*φ*

ds=

*s<t*x∼*s*

1
*NA*^{N}*φ*

*Y*_{sN}^{x}
*N*^{1/2}

*.*

Also, we remark that*M*_{t}^{(b,N )}*(φ),M*_{t}^{(e,N )}*(φ), andM*_{t}^{(s,N )}*(φ)*are*F*_{tN}* ^{(N )}*-martingales, where

*F*

*n*

*is the*

^{(N )}*σ*-algebra

*σ*

*V*^{x}*, Y*_{k}^{x}_{+}_{1}*, ξ(k, x):* |x| ≤*n*−1, k≤*n*−1, x∈Z
*,*

where*F*_{0}* ^{(N )}*= {∅

*, Ω*}. Indeed, since

*Y*

_{n}^{x}

_{+}

_{1}are independent of

*V*

^{x}and

*ξ(n, x),*

*E*

*M*_{t}^{(b,N )}*(φ)*−*M*_{t}^{(b,N )}_{−}_{1/N}*(φ)*|F_{t N}^{(N )}_{−}_{1}

= 1

*N* x∼*t*−1/N

*E*

*φ*
*Y*_{t N}^{x}

*N*^{1/2}

F_{t N}^{(N )}_{−}_{1}
*E*

*V*^{x}−1−*βξ(t N*−1, Y_{t N}^{x}_{−}_{1}*)*
*N*^{1/4}

F_{tN}^{(N )}_{−}_{1}

=0,
*E*

*M*_{t}^{(e,N )}*(φ)*−*M*_{t}^{(e,N )}_{−}_{1/N}*(φ)*|F_{t N}^{(N )}_{−}_{1}

= 1

*N* x∼*t*−1/N

*E*

*φ*
*Y*_{t N}^{x}

*N*^{1/2}

F*t N** ^{(N )}*−1

*E*

*βξ(t N*−1, Y_{t N}^{x}_{−}_{1}*)*
*N*^{1/4}

F_{tN}^{(N )}_{−}_{1}

=0, and

*E*

*M*_{t}^{(s,N )}*(φ)*−*M*_{t}^{(s,N )}_{−}_{1/N}*(φ)*|F_{t N}^{(N )}_{−}_{1}

=0, almost surely.

Moreover, the decomposition (3.1) is very useful since the martingales*M*_{t}^{(i,N )}*(φ)*(i=*b, e, s) are orthogonal to*
each others. Indeed, we have that

*E*

*M*_{t}^{(b,N )}*(φ)*

*M*_{t}^{(e,N )}*(φ)*

|F_{t N}^{(N )}_{−}_{1}

= 1
*N*^{2}

x,x^{}∼*t*−1/N

*E*

*φ*

*Y*_{t N}^{x}
*N*^{1/2}

*φ*

*Y*_{t N}^{x}^{}
*N*^{1/2}

F*tN** ^{(N )}*−1

×*E*

*E*

*V*^{x}−1−*βξ(t N*−1, Y_{tN}^{x}_{−}_{1}*)*
*N*^{1/4}

G_{t N}^{(N )}_{−}_{1}*βξ(t N*−1, Y_{t N}^{x}^{}_{−}_{1}*)*
*N*^{1/4}

F_{t N}^{(N )}_{−}_{1}

=0,

where *G**n** ^{(N )}* =

*F*

*n*

*∨*

^{(N )}*σ (ξ(n, x):*

*x*∈ Z

*)*almost surely. Also, we can obtain by similar arguments that

*E*[

*(M*

_{t}

^{(b,N )}*(φ))(M*

_{t}

^{(s,N )}*(φ))*|F

_{t N}

^{(N )}_{−}

_{1}] =

*E*[

*(M*

_{t}

^{(s,N )}*(φ))(M*

_{t}

^{(e,N )}*(φ))*|F

_{t N}

^{(N )}_{−}

_{1}] =0 almost surely.

3.1. *Tightness*

In this subsection, we will prove the following lemma.

**Lemma 3.1.** *The sequence*{*X** ^{(N )}*}

*is tight inD(*[0,∞

*),M*

*F*

*(*R

*)),and each limit process is continuous.*

To prove it, we will use the following theorem which reduces the problem to the tightness of real-valued process [25], Theorem II.4.1.

**Theorem 3.2.** *Assume thatEis a Polish space.LetD*0*be a separating class ofC**b**(E)containing*1.*A sequence of*
*càdlàgM**F**(E)-valued processes*{*X** ^{(N )}*:

*N*∈N}

*isC-relatively compact inD(*[0,∞

*),M*

*F*

*(E))if and only if*

(i) *for everyε, T >*0,*there is a compact setK*_{T ,ε}*inEsuch that*
sup

*N*

*P*

sup

*t*≤*T*

*X*_{t}^{(N )}*K*_{T ,ε}^{c}

*> ε*

*< ε,*

(ii) *and for allφ*∈*D*_{0},{*X*^{(N )}*(φ):* *N*∈N}*isC-relatively compact inD(*[0,∞*),*R*).*

**Assumption.***We chooseC*_{b}^{2}*(*R*)as* *D*0,*whereC*^{2}_{b}*(*R*)is the set of bounded continuous function on*R*with bounded*
*derivatives of order*1*and*2.

Hereafter, we will check the conditions (i) and (ii) of Theorem3.2for our case. In the beginning, we give the proof of (ii) by using the following lemmas:

**Lemma 3.3.** *Forφ*∈*C*_{b}^{2}*(*R*), sup*_{t}_{≤}* _{K}*|

*M*

_{t}

^{(s,N )}*(φ)*|→

^{L}^{2}0

*asN*→ ∞

*for allK >*0.

**Lemma 3.4 (See [25], Lemma II 4.5).** *Let(M*_{t}^{(N )}*,F*^{N}*t* *)be discrete time martingales withM*_{0}* ^{(N )}*=0.

*Let*

*M*

^{(N )}*t*=

0≤*s<t**E*[*(M*_{s}^{(N )}_{+}_{1/N}−*M*_{s}^{(N )}*)*^{2}|F^{N}*s* ],*and we extendM*_{·}^{(N )}*and* *M*^{(N )}_{·}*to*[0,∞*)as right continuous step functions.*

*If*{ *M*^{(N )}_{·}: *N*∈N}*isC-relatively compact inD(*[0,∞*),*R*)and*
sup

0≤*t*≤*K*

*M*_{t}^{(N )}_{+}_{1/N}−*M*_{t}* ^{(N )}*→

*0*

^{P}*asN*→ ∞

*for allK >*0, (3.2)

*thenM*_{·}^{(N )}*isC-relatively compact inD(*[0,∞*),*R*).*

*If,in addition,*
*M*_{t}* ^{(N )}*2

+
*M*^{(N )}

*t*: *N*∈N

*is uniformly integrable for allt,*

*then* *M*_{·}^{(N}^{k}* ^{)}* ⇒

^{w}*M*

_{·}

*in*

*D(*[0,∞

*),*R

*)*

*implies that*

*M*

*is a continuous*

*L*

^{2}

*-martingale and*

*(M*

_{·}

^{(N}

^{k}

^{)}*,*

*M*

^{(N}

^{k}*·*

^{)}*)*⇒

^{w}*(M*

_{·}

*,*

*M*

_{·}

*)inD(*[0,∞

*),*R

*)*

^{2}.

**Lemma 3.5.** *For anyφ*∈*C*_{b}^{2}*(*R*),the sequenceC*_{t}^{(N )}*(φ)*≡*t*

0*X*^{(N )}*s* *(A*^{N}*φ)*ds*isC-relatively compact inD(*[0,∞*),*R*).*

When we can verify the conditions of Lemma3.4for*M*_{·}^{(b,N )}*(φ), andM*_{·}^{(e,N )}*(φ), the sequence*{*X*^{(N )}_{·} *(φ):* *N*∈N}

is*C*-relatively compact in*D(*[0,∞*),*R*). Moreover, if we check the condition of (i) in Theorem*3.2, then the tightness
of{*X*_{·}* ^{(N )}*:

*N*∈N}follows immediately.

Before starting the proof of the above lemmas, we prepare the following lemma. It tells us the mean of the measure
*X*_{t}* ^{(N )}*is the same as the distribution of the scaled simple random walk.

**Lemma 3.6.** *We define historical process by*

*H*_{t}* ^{(N )}*= 1

*N*

_{x∼}

*t*

*δ*_{Y}^{x}

*(·∧t)N**/N*^{1/2}∈*M**F*

*D*

[0,∞*),*R
*,*

*whereY*_{s}^{x}=*Y*_{s}^{y} *for*0≤*s <*|x∧y| +1,*that isY*_{s}^{x}*is the position of thesN-generation’s ancestor of*x.

*Ifψ*:*D(*[0,∞*),*R*)*→R_{≥}0*is Borel,then for anyt*≥0
*E*

*H*_{t}^{(N )}*(ψ )*

=*E*_{Y}

*ψ*

*Y*_{(}_{·∧}_{t )N}*N*^{1/2}

*,* (3.3)

*whereY*_{·}*is the trajectory of simple random walk on*Z.*In particular*,*for allφ*∈*B*_{+}*(*R*),*
*E*

*X*_{t}^{(N )}*(φ)*

=*E*_{Y}

*φ*
*Y*_{t N}

*N*^{1/2}

*.* (3.4)

*Moreover*,*for allx*,*K >*0,*we have that*
*P*

sup

*t*≤*K*

*X*^{(N )}_{t}*(1)*≥*x*

≤*x*^{−}^{1}*.* (3.5)

To prove this lemma, we introduce some notation. For*x(*·*), y(*·*)*∈*D(*[0,∞*),*R*)*such that*y(0)*=0,
*(x/s/y)(t )*=

*x(t)* if 0≤*t < s,*
*x(s)*+*y(t*−*s)* if*t*≥*s.*

Then,*(x/s/y)(*·*)*∈*D(*[0,∞*),*R*).*

**Proof.** (3.3) follows from the Markov property. Indeed, we have
*E*

*H*_{t}^{(N )}*(ψ )*

=*E*
1

*N* _{y}_{∼}

*t*

*ψ*

*Y*_{(}^{y}_{·∧}_{t )N}*N*^{1/2}

=*E*
1

*N*x∼*t*−1/N

*ψ*

*Y*_{(}^{x}_{·∧}_{t )N}*N*^{1/2}

*E*

*V*^{x}|F_{t N}^{(N )}_{−}_{1}

=*E*

*E*_{Z}_{1}
1

*N* x∼*t*−1/N

*ψ*

*(Y*_{(}^{x}_{·∧}_{(t}_{−}_{1/N ))N}*/tN/Z*_{1}*)((*· ∧*t )N )*
*N*^{1/2}

*,*

where*Z*_{1}*(*·*)*is a random function independent of*Y*_{·}^{x}* _{N}* such that

*Z*

_{1}

*(s)*=0 for 0≤

*s <*1,

*P (Z*

_{1}

*(s)*=1 for

*s*≥1)=

*P (Z*1

*(s)*= −1 for ≥1)=

^{1}

_{2}. Iterating this,

*E*

*H*_{t}^{(N )}*(ψ )*

=*E*

*E**Z*_{1}*,Z*_{2}

1

*N* x∼*t*−2/N

*ψ*

*((Y*_{(}^{x}_{·∧}_{t}_{−}_{2/N )N}*/t N*−2/Z2*)/t N*−1/Z1*)((*· ∧*t )N )*
*N*^{1/2}

=*E*_{Y}

*ψ*

*Y**((*·∧*t )N )*

*N*^{1/2}

*,*

where*Z*_{2}is independent copy of*Z*_{1}and*Y (*·*)*is the trajectory of simple random walk. Also, (3.5) follows from the
fact that*X*^{(N )}_{t}*(1)*is an*F*_{t N}* ^{(N )}*-martingale, from the

*L*

^{1}inequality for nonnegative submartingales, and from (3.4).

**Proof of Lemma3.5.** We know*X*^{(N )}_{0} *(φ)*=*φ(0). Also, we have that for anyK >*0
C_{t}^{(N )}*(φ)*−*C*_{s}^{(N )}*(φ)*≤

_{t}

*s*

X^{(N )}_{u}

*A*^{N}*φ*du

≤ sup

*u*≤*K*

*C(φ)X*^{(N )}_{u}*(1)*|*t*−*s*|*,* (3.6)

where we have used that the boundedness of *A*^{N}*φ. We can use the Arzela–Ascoli Theorem by (3.5) and (3.6) so*
that{*C*_{·}^{(N )}*(φ):* *N*∈N}are*C-relatively compact sequences inD(*[0,∞*),*R*). (See Corollary 3.7.3, Remark 3.7.4, and*

Theorem 3.10.2 in [11].)

**Proof of Lemma3.3.** Let*h**N**(y)*=*E** ^{y}*[

*(φ(*

^{Y}^{1}

*N*^{1/2}*)*−*φ(* ^{Y}^{0}

*N*^{1/2}*))*^{2}]. First, we remark that
*φ*

*Y*_{sN}^{x}_{+}_{1}
*N*^{1/2}

−*φ*
*Y*_{sN}^{x}

*N*^{1/2}

− 1
*NA*^{N}*φ*

*Y*_{sN}^{x}
*N*^{1/2}

are orthogonal forx=x^{}∼*s. SinceM*_{t}^{(s,N )}*(φ)*is a martingale, we have that
*E*

*M*_{K}^{(s,N )}*(φ)*2

=

*s<K*

*E*

*M*_{s}^{(s,N )}*(φ)*2

= 1
*N*^{2}

*s<K*

*E*

x∼*s*

*E*

*φ*
*Y*_{sN}^{x}_{+}_{1}

*N*^{1/2}

−*φ*
*Y*_{sN}^{x}

*N*^{1/2}

− 1
*NA*^{N}*φ*

*Y*_{sN}^{x}
*N*^{1/2}

2F_{sN}^{(N )}

≤ 2
*N* *s<K*

*E*
1

*N* _{x}_{∼}

*s*

*h*_{N}

*Y*_{sN}^{x}
+ 1

*N*^{2} A^{N}*φ *^{2}

≤2E
_{K}

0

*X*_{s}^{(N )}*(h*_{N}*)*+ A^{N}*φ *^{2}_{∞}*N*^{−}^{2}*X*_{s}^{(N )}*(1)*
ds

≤2

*E**Y*

*K*
0

*φ*

*Y*_{sN}_{+}_{1}
*N*^{1/2}

−*φ*
*Y*_{sN}

*N*^{1/2}
2

ds

+ *K*
*N*^{2}sup

*N*

*A*^{N}*φ* ^{2}

∞*X*^{(N )}_{0} *(1)*

→0,

where we have used Lemma3.6and the fact that sup_{N}*A*^{N}*φ*∞*<*∞for*φ*∈*C*_{b}^{2}*(*R*)*and{*X*^{(N )}_{t}*(1): 0*≤*t*≤*K*}is a

martingale with respect to*F*_{tN}* ^{(N )}*in the last line.

Next, we will check the conditions in Lemma3.4for*M*_{·}^{(b,N )}*(φ)*and*M*_{·}^{(e,N )}*(φ), that is,*
(1) { *M*^{(b,N )}*(φ)*_{·}+ *M*^{(e,N )}*(φ)*_{·}: *N*∈N}is*C*-relatively compact in*D(*[0,∞*),*R*),*

(2) sup_{0}_{≤}_{t}_{≤}* _{K}*|

*M*

_{t}

^{(b,N )}_{+}

_{1/N}

*(φ)*−

*M*

_{t}

^{(b,N )}*(φ)*+

*M*

_{t}

^{(e,N )}_{+}

_{1/N}

*(φ)*−

*M*

_{t}

^{(e,N )}*(φ)*|→

*0 as*

^{P}*N*→ ∞for all

*K >*0,

(3) {*(M*_{t}^{(b,N )}*(φ))*^{2}+*(M*_{t}^{(e,N )}*(φ))*^{2}+ *M*^{(b,N )}*(φ)**t*+ *M*^{(e,N )}*(φ)**t*: *N*∈N}is uniformly integrable for all*t*.
As we verified that*M*^{(b,N )}*(φ)*and*M*^{(e,N )}*(φ)*are orthogonal, we have that

*M*^{(b,N )}*(φ)*+*M*^{(e,N )}*(φ)*

·=

*M*^{(b,N )}*(φ)*

·+

*M*^{(e,N )}*(φ)*

·*.*

Moreover, since under fixed environment{*ξ(n, x):* *(n, x)*∈N×Z},*V*^{x} and*V*^{y} are independent for x=y, we
have that

*M*^{(b,N )}*(φ)*

*t*

=

*s<t*

*E*

*M*_{s}^{(b,N )}_{+}_{1/N}*(φ)*−*M*_{s}^{(b,N )}*(φ)*2

|F_{sN}^{(N )}

= 1
*N*^{2}

*s<t*x∼*s*

*E*

*φ*
*Y*_{sN}^{x}_{+}_{1}

*N*^{1/2}

2F_{sN}^{(N )}

*E*

*V*^{x}−1−*βξ(sN, Y*_{sN}^{x}*)*
*N*^{1/4}

2F_{sN}^{(N )}

= 1
*N*^{2}

*s<t*x∼*s*

*E*

*φ*
*Y*_{sN}^{x}_{+}_{1}

*N*^{1/2}

−*φ*
*Y*_{sN}^{x}

*N*^{1/2}

+*φ*
*Y*_{sN}^{x}

*N*^{1/2}

2F_{sN}^{(N )}

1− *β*^{2}
*N*^{1/2}

= 1
*N* *s<t*

*X*_{s}^{(N )}*φ*^{2}

1− *β*^{2}
*N*^{1/2}

+ 1

*N*^{1/2}
*O(1)*

*N* *s<t*

*X*^{(N )}_{s}*(1)*

1− *β*^{2}
*N*^{1/2}

=
_{t}

0

*X*^{(N )}_{s}

*φ*^{2}+*O(1)*
*N*^{1/2}

ds, and

*M*^{(e,N )}*(φ)*

*t*

=

*s<t*

*E*

*M*_{s}^{(e,N )}_{+}_{1/N}*(φ)*−*M*_{s}^{(e,N )}*(φ)*2

|F_{sN}^{(N )}

= *β*^{2}
*N*^{2}

*s<t*x,x˜∼*t*

*E*

*φ*
*Y*_{sN}^{x}_{+}_{1}

*N*^{1/2}

*φ*
*Y*_{sN}^{x}^{˜} _{+}_{1}

*N*^{1/2}

F*sN*^{(N )}

**1{***Y*_{sN}^{x} =*Y*_{sN}^{x}^{˜} }
*N*^{1/2}

= *β*^{2}
*N*^{2}

*s<t*x,x∼˜ *t*

*φ*

*Y*_{sN}^{x}
*N*^{1/2}

2

+*O(1)*
*N*^{1/2}

**1{***Y*_{sN}^{x} =*Y*_{sN}^{x}^{˜} }
*N*^{1/2}

= 1
*Nβ*^{2}

*s<t**x*∈Z

*φ*

*x*
*N*^{1/2}

2

+*O(1)*
*N*^{1/2}

*(B*_{sN,x}^{(N )}*)*^{2}
*N*^{3/2}

=*β*^{2}
*t*

0 *x*∈Z

*φ*

*x*
*N*^{1/2}

2

+*O(1)*
*N*^{1/2}

*(B*_{sN,x}^{(N )}*)*^{2}
*N*^{3/2} ds,
where|O*(1)*| ≤*C** _{φ}*for a constant

*C*

*that depends only on*

_{φ}*φ.*

Therefore, we have that
*M*^{(b,N )}*(φ)*

*t*+

*M*^{(e,N )}*(φ)*

*t*−

*M*^{(b,N )}*(φ)*

*s*−

*M*^{(e,N )}*(φ)*

*s*

≤*C**φ*

*M*^{(b,N )}*(1)*

*t*+

*M*^{(e,N )}*(1)*

*t*−

*M*^{(b,N )}*(1)*

*s*−

*M*^{(e,N )}*(1)*

*s*

=*C*

*X*^{(N )}*(1)*

*t*−

*X*^{(N )}*(1)*

*s*

*,* (3.7)

where we remark that{*X*_{t}^{(N )}*(1): 0*≤*t*}is a martingale with respect to*F*_{tN}* ^{(N )}*and

*M*

_{t}

^{(s,N )}*(1)*=0 for any 0≤

*t <*∞.

We will prove*C*-relative compactness of (3.7) by showing the following lemma.