getdoc28f9. 373KB Jun 04 2011 12:04:11 AM

El e c t ro n ic

Jo ur

n a l o

f P

r o

b a b i_{l i t y} Vol. 16 (2011), Paper no. 22, pages 618–657.

Journal URL

http://www.math.washington.edu/~ejpecp/

Convergence of Rescaled Competing Species Processes

to a Class of SPDEs

Sandra Kliem

12 Department of Mathematics, UBC,

1984 Mathematics Road, Vancouver, BC V6T1Z2, Canada University of British Columbia

e-mail: kliem@eurandom.tue.nl Abstract

One can construct a sequence of rescaled perturbations of voter processes in dimensiond=1 whose approximate densities are tight. By combining both long-range models and fixed ker-nel models in the perturbations and considering the critical long-range case, results of Cox and Perkins (2005) are refined. As a special case we are able to consider rescaled Lotka-Volterra models with long-range dispersal and short-range competition. In the case of long-range inter-actions only, the approximate densities converge to continuous space time densities which solve a class of SPDEs (stochastic partial differential equations), namely the heat equation with a class of drifts, driven by Fisher-Wright noise. If the initial condition of the limiting SPDE is integrable, weak uniqueness of the limits follows. The results obtained extend the results of Mueller and Tribe (1995) for the voter model by including perturbations. In particular, spatial versions of the Lotka-Volterra model as introduced in Neuhauser and Pacala (1999) are covered for parameters approaching one. Their model incorporates a fecundity parameter and models both intra- and interspecific competition.

Key words: Voter model, Lotka-Volterra model, spatial competition, stochastic partial differen-tial equations, long-range limits.

AMS 2000 Subject Classification:Primary 60F05, 60K35; Secondary: 60H15. Submitted to EJP on November 13, 2009, final version accepted February 27, 2011.

1_{Present address: Technische Universiteit Eindhoven, EURANDOM, P.O.Box 513, 5600 MB,} Eindhoven, The Netherlands.

2_{The work of the author was supported by a “St John’s College Sir Quo-Wei Lee Fellowship”,} a “University of BC Graduate Fellowship” and an NSERC Discovery Grant.

1

Introduction

We define a sequence of rescaled competing species modelsξN_t in dimensiond =1, which can be described as perturbations of voter models. In theNth-model the sites are indexed byx _∈N−1Z_.

We label the state of sitex at time t byξN_t (x)where ξN_t(x) =0 if the site is occupied at time t by type 0 andξN

t (x) =1 if it is occupied by type 1.

In what follows we shall write x _∼ y if and only if 0< |x ₋ y_{| ≤} N−1/2, i.e. if and only if x is a neighbour of y. Observe that eachx has 2c(N)N1/2,c(N)N_→→∞1 neighbours.

The rates of change incorporate both long-range models and fixed kernel models with finite range. The long-range interaction takes into account the densities of the neighbours ofx at long-range, i.e.

f_i(N)(x,ξ)_≡ 1 2c(N)pN

X

0<|y−x|≤1/pN, y∈Z_/N

1(ξN(y) =i), i=0, 1

and the fixed kernel interaction considers

g_i(N)(x,ξ)_≡ X y∈Z_/N

p(N(x₋y))1(ξN(y) =i), i=0, 1, (1)

where p(x) is a random walk kernel on Z _{of finite range, i.e. 0 ≤ p(x) ≤ 1,} P

x∈Zp(x) = 1

and p(x) = 0 for all |x| ≥C_p. In what follows we shall often abbreviate f_i(N)(x,ξ) by f_i(N) and

g_i(N)(x,ξ)by g_i(N)if the context is clear. Note in particular that 0_≤f_i(N),g_i(N)_≤1 andf₀(N)+f₁(N)=

g₀(N)+g₁(N)=1.

Now define the rates of change of our configurations. At site x in configurationξN ∈ {0, 1_}Z/N the coordinateξN_{(x)makes transitions}

0→1 at rateN f₁(N)+f₁(N) n

g₀(N)G(₀N)

f₁(N)

+g(₁N)H(₀N)

f₁(N) o

, (2)

1_→0 at rateN f₀(N)+f₀(N) n

g₀(N)G(₁N)

f₀(N)

+g(₁N)H(₁N)

f₀(N) o

,

whereG(_iN),H_i(N),i=0, 1 are functions on[0, 1]. Every configurationξN

t can be rewritten in terms of its corresponding measure. Indeed, introduce the following notation.

Notation 1.1. For f,g:N−1Z_→R, we set_〈f,g_〉= _N1 P_x f(x)g(x). Letν be a measure onN−1Z. Then we set_〈ν,f〉=R f dν.

Now we can rewrite every configurationξN

t in terms of its corresponding measure as follows. Let

ν_tN≡ 1

N X

x

δx1(ξNt (x) =1), (3)

We next define approximate densitiesA(ξN_t )for the configurationsξN_t via

A(ξN_t)(x) = 1 2c(N)N1/2

X

y∼x

ξN_t(y), x_∈N−1Z ₍₄₎

and note thatA(ξN_t)(x) = f₁(N)€x,ξN_t Š. By linearly interpolating between sites we obtain approxi-mate densitiesA(ξN_t)(x)for allx∈R.

Notation 1.2. Set_{C1 ≡ {}f :R_→[0, 1]continuous_}and let_C1 be equipped with the topology of uniform convergence on compact sets.

We obtain thatt_7→A(ξN

t)is cadlagC1-valued, where we used that 0≤A(ξN_t)(x)_≤1 for allx ∈N−1Z_.

Definition 1.3. LetS be a Polish space and let D(S)denote the space of cadlag paths from R₊to

S with the Skorokhod topology. Following Definition VI.3.25 in Jacod and Shiryaev[8], we shall say that a collection of processes with paths in D(S)isC-tight if and only if it is tight in D(S)and all weak limit points are a.s. continuous. Recall that for Polish spaces, tightness and weak relative compactness are equivalent.

In what follows we shall investigate tightness of_{A(ξN_· ):N_≥1_}inD(_C1)and tightness of_{ν_tN :N_≥

1_}in D(_M(R)), where_M(R)is the space of Radon measures equipped with the vague topology (M(R)is indeed Polish, see Kallenberg[9], Theorem A2.3(i)). We next impose certain assumptions on the functionsG_i(N)andH_i(N)in (2). These assumptions are rather technical and only become clear later in the proper context.

Definition 1.4. Let_P~₀ be the class of sequences (f(N):N_∈N_{)of real-valued functions on[0, 1],}

that can be expressed as power series f(N)(x)_≡ P∞_m=0γ(m+1,N)xm, x _∈[0, 1]withγ(m+1,N) ∈R,

m≥0 such that there existsN0∈Nsuch that

sup N≥N0

∞

X

m=0 n€

γ(m+1,N)Š++ (m+1)€γ(m+1,N)Š−o<∞, (5)

wherea+=max(a, 0)anda−=max(₋a, 0)fora_∈R. Theorem 1.5. Suppose that A(ξN

0) →u0 in C1. Let the transition rates of ξN(x) be as in (2) with (G_i(N):N ∈N),(H_i(N):N ∈N)_∈P~₀,i=0, 1. Then€A(ξN

t):t≥0

Š

are C-tight as cadlagC1-valued

processes and the€νN

t :t ≥0

Š

are C-tight as cadlag Radon measure valued processes with the vague topology. If AξNk

t

,νNk

t

t≥0 converges to(ut,νt)t≥0, thenνt(d x) =ut(x)d x for all t ≥0.

From now on we consider the special case of no fixed kernel interaction (also to be called no short-range competition in what follows, for reasons that become clear later) and investigate the limits of our tight sequences. Recall thatg₀(N)+g₁(N)=1. Hence, the special case can be obtained by choosing

Definition 1.6. Let _P~_{1 ⊂P}~₀ be the class of sequences (f(N) :N _∈N) of real-valued functions on [0, 1]such that f(N)(x) =P∞_m=0γ(m+1,N)xmwith

γ(m+1,N)N→→∞γ(m+1)for allm≥0 (6) and

lim N→∞

∞

X

m=0 n€

γ(m+1,N)Š++ (m+1)€γ(m+1,N)Š−o=

∞

X

m=0 n€

γ(m+1)Š++ (m+1)€γ(m+1)Š−o. (7)

Remark 1.7. For(f(N):N∈N)_∈P~₁ let f(x)_≡lim_N→∞ f(N)(x). Then we have

f(x) =

∞

X

m=0

γ(m+1)xm,x ∈[0, 1].

Indeed, this holds by (7) and Royden[16], Proposition 11.18.

Theorem 1.8. Consider the special case with no short-range competition. Under the assumptions of Theorem 1.5 we have for(G(_iN):N _∈N)_∈P~₁,i= 0, 1that the limit points of A(ξN_t) are continuous

C1-valued processes ut which solve

∂u ∂t =

∆u

6 + (1−u)u

G₀(u)₋G₁(1₋u) +p2u(1₋u)W˙ (8)

with initial condition u0. If we assume additionally 〈u0, 1〉<∞, then ut is the unique in law[0, 1]

-valued solution to the above SPDE.

1.1

Literature review

In[13], Mueller and Tribe show that the approximate densities of type 1 of rescaled biased voter processes converge to continuous space time densities which solve the heat equation with drift, driven by Fisher-Wright noise. This model and result are covered by our following example.

Example 1.9. Choose G₀(N)(x) =H₀(N)(x)_≡ θ ∈R _{and G}(N)

1 (x) =H

(N)

1 (x)≡ 0in (2). Using that

g₀(N)+g₁(N)=1by definition, we obtain a sequence of rescaled biased voter models with rates of change

0_→1at rate c(x,ξ) =N

1+ θ

N

f₁(N)(x,ξ),

1→0at rate c(x,ξ) =N f₀(N)(x,ξ).

Forθ >0we thus have a slight favour for type1and forθ <0we have a slight favour for type0. Theorem 1.5 and Theorem 1.8 yield that the sequence of approximate densities A(ξN_t )is tight and every limit point solves (8) with G0(u) =θ, G1(1−u) =0and initial condition u0. Uniqueness in law holds

for initial conditions of finite mass.

Forθ ≥0, the above result, except for the part on weak uniqueness, coincides (up to scaling) with Theorem 2 of[13].

Example 1.10. For i=0, 1choose

a(_i(N₁)

−i)≡1+

θ_i(N) N withθ

(N)

i

N→∞

→ θi. (9)

Let G_i(N)(x) =H_i(N)(x)_≡θ_i(₍N₁₋)_i)x and observe that G_i(N)(x) =H_i(N)(x)N_→→∞θ_ix_≡G_i(x) =H_i(x). We obtain a sequence of rescaled Lotka-Volterra models with rates of change

0→1at rate N f₁(N)+θ₀(N)f₁(N) 2

(10)

=N f₁(N)+N(a₀₁(N)₋1)f₁(N)2=N f₁(N)f₀(N)+a(₀₁N)f₁(N),

1_→0at rate N f₀(N)+θ₁(N)

f₀(N)

2

=N f₀(N)

f₁(N)+a₁₀(N)f₀(N)

instead, where we used that f₀(N)+ f₁(N) = 1 by definition. The interpretation of a₀₁(N) and a₁₀(N) will become clear once we introduce spatial versions of the Lotka-Volterra model with finite range in (12) (chooseλ=1). Observe in particular that if we choose a₀₁(N),a₁₀(N)close to1as above, the Lotka-Volterra model can be seen as a small perturbation of the voter model.

Theorem 1.5 and Theorem 1.8 yield that the sequence of approximate densities A(ξN

t )is tight and every

limit point solves

∂u ∂t =

∆u

6 + (1−u)u

θ0u−θ1(1−u) + p

2u(1₋u)W˙ (11)

with initial condition u0. Uniqueness in law holds if<u0, 1><∞.

If we chooseθ0 =−θ1 >0 in (11) we obtain the Kolmogorov-Petrovskii-Piscuinov (KPP) equation driven by Fisher-Wright noise. This SPDE has already been investigated in Mueller and Sowers[12]

in detail, where the existence of travelling waves was shown forθ₀ big enough.

In Cox and Perkins[5]it was shown that stochastic spatial Lotka-Volterra models as in (10), satis-fying (9) and suitably rescaled in space and time, converge weakly to super-Brownian motion with linear drift. [5]extended the main results of Cox, Durrett and Perkins [4], which proved similar results for long-range voter models. Both papers treat the low density regime, i.e. where only a finite number of individuals of type 1 is present. Instead of investigating limits for approximate densities as we do, both papers define measure-valued processesXN_t by

X_tN= 1

N′

X

x∈Z_/(MN

p

N)

ξN_t(x)δx,

i.e. they assign mass 1/N′, N′=N′(N)to each individual of type 1 and consider weak limits in the space of finite Borel measures onR. In particular, they establish the tightness of the sequence of measures and the uniqueness of the martingale problem, solved by any limit point.

Note that both papers use a different scaling in comparison to[13]. Using the notation in[4], for

d = 1 they take N′ = N and the space is scaled by MN p

N with MN/ p

N → ∞ (see for instance Theorem 1.1 of [4] for d = 1) in the long-range setup. According to this notation, [13] used

M_N =pN, which is at the threshold of the results in[4], but not included. By letting M_N =pN

in our setup non-linear terms arise in our limiting SPDE. Also note the brief discussion of the case whereMN/

p

Next recall spatial versions of the Lotka-Volterra model with finite range as introduced in Neuhauser and Pacala[14](they consideredξ(x)_{∈ {}1, 2_}instead of_{0, 1_}). They use transition rates

0→1 at ratec(x,ξ) = λf1(x,ξ)

λf₁(x,ξ) + f₀(x,ξ) f0(x,ξ) +α01f1(x,ξ)

, (12)

1_→0 at ratec(x,ξ) = f0(x,ξ)

λf1(x,ξ) + f0(x,ξ)

f₁(x,ξ) +α10f0(x,ξ)

,

where α01,α10 ≥ 0,λ > 0. Here fi(x,ξ) = _{|N |}1

P

y∈x+N 1(ξ(y) = i), i = 0, 1 with the set of

neighbours of 0 being_N =_{y : 0<|y_{| ≤}R_}withR_≥1.

We can think ofRas the finite interaction range of the model.[14]use this model to obtain results on the parameter regions for coexistence, founder control and spatial segregation of types 0 and 1 in the context of a model that incorporates short-range interactions and dispersal. As a conclusion they obtain that the short-range interactions alter the predictions of the mean-field model.

Following[14]we can interpret the rates as follows. The second multiplicative factor of the rate governs the density-dependent mortality of a particle, the first factor represents the strength of the instantaneous replacement by a particle of opposite type. The mortality of type 0 consists of two parts, f₀ describes the effect of intraspecific competition, α01f1 the effect of interspecific competi-tion.[14]assume that the intraspecific competition is the same for both species. The replacement of a particle of opposite type is regulated by the fecundity parameterλ. The first factors of both rates of change added together yield 1. Thus they can be seen as weighted densities of the two species. If

λ >1, species 1 has a higher fecundity than species 0.

Example 1.11. Choose the competition and fecundity parameters near one and consider the long-range case. Namely, the model at hand exhibits the following transition rates:

0_→1at rate N 



λ(N)f₁(N) λ(N)_f(N)

1 + f

(N)

0

f₀(N)+a(₀₁N)f₁(N)



,

1_→0at rate N 



f₀(N) λ(N)_f(N)

1 + f

(N)

0

f₁(N)+a(₁₀N)f₀(N)



.

We suppose that

λ(N)≡1+λ

′

N, a

(N)

01 ≡1+

a₀₁ N , a

(N)

10 ≡1+

a₁₀ N . Using f₀(N)+f₁(N)=1we can therefore rewrite the rates as

0→1at rate N+λ′ f₁(N)

 1+ a01

N f

(N)

1

‹ X

n≥0 ‚

−λ

′

N f

(N)

1 Œn

, (13)

1_→0at rate N f₀(N) 

1+ a10

N f

(N)

0

‹ X

n≥0 ‚

−λ

′

N f

(N)

1 Œn

=N f₀(N) 

1+a10

N f

(N)

0

‹ X

k≥0

f₀(N) k

‚

λ′ N

Œk X

n≥k

n k

‚ −λ

′

N Œn−k

Here we used that f

(N)

i

≤1,i=0, 1and that

λ′

N

→0for N → ∞. We can use the explicit calculations for a geometric series, in particular that we haveP_n≥0n_|q_|n<∞andP_n≥k|q_|n−k n_k

= ₍₁ 1

−|q|)k+1 for

|q_|<1to check that(G_i(N):N_∈N),(H(_iN):N_∈N)_∈P~₁,i=0, 1. Using Theorem 1.8 we obtain that the limit points of A(ξN

t )are continuousC1-valued processes ut which solve

∂u ∂t =

∆u

6 + (1−u)u

λ′+u a01−λ′

− −λ′+ (1₋u) a10+λ′ + p

2u(1₋u)W˙

=∆u

6 + (1−u)u

λ′−a₁₀+u a₀₁+a₁₀ +p2u(1₋u)W˙ (14)

by rewriting the above rates (13) in the form (2) and taking the limit for N_{→ ∞}. For_〈u₀, 1_〉<∞, u_t is the unique weak[0, 1]-valued solution to the above SPDE.

Additionally, [4] and [5] consider fixed kernel models in dimensions d ≥ 2 respectively d ≥ 3. They set g(_iN)(x,ξ) = P_y∈Zd_/(M

N

p

N)p(x − y)1(ξN(y) = i), i = 0, 1 (compare this to (1)) and

choose M_N =1 and a fixed random walk kernel qsatisfying some additional conditions such that

p(x) =q(pN x)on x _∈Zd_/(M_NpN). In Cox and Perkins[6], the results of[5]ford_≥3 are used to relate the limiting super-Brownian motions to questions of coexistence and survival of a rare type in the original Lotka-Volterra model.

Example 1.12. Consider rescaled Lotka-Volterra models with long-range dispersal and short-range competition, i.e. where (10) gets generalized to

0→1at rate N f₁(N)

g(₀N)+a₀₁(N)g₁(N)

,

1→0at rate N f₀(N)

g(₁N)+a₁₀(N)g₀(N)

.

Here f_i(N),i=0, 1is the density corresponding to a long-range kernel pLand g(_iN),i=0, 1is the density corresponding to a fixed kernel pF (also recall the interpretation of both multiplicative factors following equation (12)).

We obtain for a_i(₍N₁)_−i),i=0, 1as in (9) that H₀(N)(x)_≡θ₀(N), G₁(N)(x)_≡θ₁(N)and G₀(N)(x) =H₁(N)(x)_≡ 0in (2). Under the assumption that the initial approximate densities A(ξN

0)converge inC1, Theorem

1.5 yields tightness of the sequence of approximate densities A(ξN t ).

1.2

Discussion of results and future challenges

In the present paper we first prove tightness of the local densities for scaling limits of more general particle systems. The generalization includes two features.

Firstly, we extend the model in[13]to limits of small perturbations of the long-range voter model, including negative perturbations (recall Example 1.9 and that[13]assumedθ ≥0) and the setup from[14](cf. Example 1.11). As the rates in[14](see (12)) include taking ratios, we extend our perturbations to a set of power series (for extensions to polynomials of degree 2 recall (10)), thereby including certain analytic functions. Recall in particular from (9) that we shall allow the coefficients of the power series to depend onN.

Secondly, we combine both long-range interaction and fixed kernel interaction for the perturbations. As we see, the tightness results carry over (cf. Example 1.12).

Finally, in the case of long-range interactions only we show that the limit points are solutions of a SPDE similar to[13]but with a drift depending on the choice of our perturbation and small changes in constants due to simple differences in scale factors. Hence, we obtain a class of SPDEs that can be characterized as the limit of perturbations of the long-range voter model.

Example 1.13. Let Gi(x) =

P_∞

m=0γ

(m+1)

i x

m_{,i= 0, 1be two arbitrary power series with coefficients}

satisfying

X

i=0,1

∞

X

m=0 §

γ(_im+1)++ (m+1)γ(_im+1)−

ª

<∞.

Set G(_iN)(x)_≡G_i(x)for all N ∈N_{. Then}(G(_iN):N∈N)_∈P~₁and Theorem 1.8 yields that a solution to (8) with u_{0 ∈ C1} can be obtained as the limit point of a sequence of approximate densities A(ξN_t ), where(ξN_t :N ∈N)is a sequence of rescaled competing species models with rates of change

0→1at rate N f₁(N)+f₁(N)G0

f₁(N)

,

1_→0at rate N f₀(N)+f₀(N)G1

f₀(N)

(recall (2) with G(_iN)_≡H_i(N)) and initial configurations satisfying A(ξN₀)_→u₀ in_C1. This includes in particular the case where Gi(x),i=0, 1are polynomials.

If the limiting initial condition u₀ satisfies Ru₀(x)d x < _∞, we can show the weak uniqueness of solutions to the limiting SPDE and therefore show weak convergence of the rescaled particle densities to this unique law.

To include more general perturbation functions we resolved to rewrite the transition rates (2) so that they involve non-negative contributions only (cf. (18)). A representation for the evolution in time ofξN_t(x)is given in (23) and used to obtain a Green’s function representation (37) (compare this to (2.9) in[13]). The right choice of Poisson processes in the graphical construction (cf. (22)) turned out to be crucial. For an example of how the proper choice of rates in (22) yields terms involving approximate densities A(ξN_t) in the approximate semimartingale decomposition, see the third equality in (34). Also see (36) for an example of how error bounds can be obtained and why the additional fixed kernel interaction does not impact the result.

While dealing with non-constant functionsG_i(N),H_i(N),i=0, 1, certain cancelation tricks from[13]

were not available anymore. Instead, techniques of[13]had to be modified and refined. See for instance the calculations (51) to (52), where only the leading term of the perturbation part of the transition rates in (18) effects the error bounds. Here we can also see best why the additional perturbations do not change the tightness result.

Finally, as a further extension to[13], we include results on weak uniqueness.

It would be of interest to see if the results can be extended to more general functionsG_i(N),H_i(N),i= 0, 1 in (2). The techniques utilized in the present paper require the functions to be power series with coefficients satisfying (5). In short, we model each non-negative contribution to the rewritten transition rates (18) of the approximating processes ξN via independent families of i.i.d. Poisson processes (for more details see Subsection 3.2) and as a result obtain a graphical construction of

ξN in (23). The non-negativity assumption makes it necessary to rewrite negative contributions in terms of positive contributions, which results in assumption (5).

When there exists a fixed kernel, the question of uniqueness of all limit points and of identifying the limit remains an open problem. Also, when we consider long-range interactions only with R

u0(x)d x=∞the proof of weak uniqueness of solutions to the limiting SPDE remains open. In Example 1.11 we apply our results to characterize the limits of spatial versions of the Lotka-Volterra model with competition and fecundity parameter near one (see (9)) in the case of long-range interactions only. We obtain a class of parameter-dependent SPDEs in the limit (see (14)). This opens up the possibility to interpret the limiting SPDEs and their behaviour via their approxi-mating long-range particle systems and vice versa. For instance, a future challenge would be to use properties of the SPDE to obtain results on the approximating particle systems, following the ideas of[5]and[6].

A major question is how the change in the parameter-dependent drift, in particular, possible addi-tional zeros, impacts the long-time behaviour of the solutions and if there exist phase transitions. The author conjectures that there are parameter regions that yield survival and others that yield extinction. Aronson and Weinberger[1]showed that the corresponding class of deterministic PDEs exhibits a diverse limiting behaviour.

1.3

Outline of the rest of the paper

In Section 2 we prove Theorem 1.5 and Theorem 1.8. The first part of the proof consists in rewriting the transition rates (2) of the rescaled modelsξN

t . We then present results comparable to the results of Theorems 1.5 and 1.8 for a class of sequences of rescaled models with transition rates that include the transition rates of the rewritten system. The advantage of the new over the old model is, that the new model can be approached by the methods used in[13]. The proof of the results for the new model is given in Section 3.

2

Proof of Theorem 1.5 and Theorem 1.8

We prove both theorems together. Let(G(_iN):N _∈N),(H_i(N):N _∈N)_∈P~₀,i=0, 1. Then we can write

G(_iN)(x)_≡

∞

X

m=0

α(_im+1,N)xmandH_i(N)(x)_≡

∞

X

m=0

β_i(m+1,N)xm, x ∈[0, 1] (15)

withi=0, 1 andα(_jm,N),β_j(m,N)_∈Rfor all j=0, 1,m_∈N, satisfying (5). We can now rewrite the rates of change (2) ofξN_(x)as

0→1 at rateN f₁(N)+g(₀N)

∞

X

m=1

α(₀m,N)f₁(N) m

+g₁(N)

∞

X

m=1

β₀(m,N)f₁(N) m

, (16)

1_→0 at rateN f₀(N)+g(₀N)

∞

X

m=1

α(₁m,N)f₀(N) m

+g₁(N)

∞

X

m=1

β₁(m,N)f₀(N) m

.

Remark 2.1. The above rates of change determine indeed a unique, _{0, 1_}Z/N-valued Markov process ξN_t for N ≥N0 with N0as in Definition 1.4. For a proof, see Appendix A.

Following[13], we would like to model each term in (16) via independent families of i.i.d. Pois-son processes. This technique is only applicable to non-negative contributions. As we allow the

α(_im,N),β_i(m,N)to be negative, too, the first part of the proof consists in rewriting (16) with the help of f₀(N)+f₁(N)=1 andg₀(N)+g(₁N)=1 in a form, where all resulting coefficients are non-negative. Lemma 2.2. We can rewrite our transitions as follows.

0→1at rate (17)

€

N₋θ(N)Š

f₁(N)+ f₁(N) 



 X

i=0,1

a(_iN)g(_iN)+ X m≥2,i,j=0,1

q_{i j}(0,m,N)g_i(N)f_j(N)

f₁(N) m−2





 ,

1_→0at rate €

N₋θ(N)Šf₀(N)+ f₀(N) 



 X

i=0,1

b(_iN)g_i(N)+ X m≥2,i,j=0,1

q_{i j}(1,m,N)g_i(N)f_j(N)

f₀(N) m−2





 ,

with correspondingθ(N)_,a(N)

i ,b

(N)

i ,q

(k,m,N)

i j ∈R

+_{,i,j,k=0, 1,m}

≥2.

Proof. We shall drop the superscripts of f_i(N),g_i(N),i = 0, 1 in what follows to simplify notation. Suppose for instanceα(₀m,N)<0 for somem≥1 in (16). Using that

−xm= (1₋x) m−1 X

l=1

xl₋x

and recalling that 1₋f₁= f₀we obtain

g0α

(m,N)

0 f m 1 =g0

(

−α(₀m,N)f0 m−1 X

l=1

f₁l+α(₀m,N)f1 )

.

Finally, we can use g₀=1₋g₁to obtain

g₀α(₀m,N)f₁m=g₀

−α(₀m,N)f₀

m−1 X

l=1

f₁l+g₁

−α(₀m,N)f₁+α(₀m,N)f₁.

All terms on the r.h.s. but the last can be accommodated into an existing representation (17) as follows:

q(₀₀0,n,N)_→q₀₀(0,n,N)+

−α(₀m,N)

for 2_≤n_≤m,

a₁(N)_→a(₁N)+

−α(₀m,N).

Finally, we can assimilate the last term into the first part of the rate 0→1, i.e. we replace

As we use the representation (17), a change inθ(N) also impacts the rate 1_→0 in its first term. Therefore we have to fix the rate 1 _→ 0 by adding a term of (₋α(₀m,N))f₀ = g₀f₀(₋α(₀m,N)) +

g₁f₀(₋α(₀m,N))to the second and third term of the rate, i.e. by replacing

b(₀N)_→b(₀N)+

−α(₀m,N), b₁(N)_→b(₁N)+

−α(₀m,N).

The general case with multiple negativeα′sand/orβ′sfollows inductively. Remark 2.3. The above construction yields the following non-negative coefficients:

q(₀₀0,m,N)_≡

∞

X

n=m

α(₀n,N)−, q₁₀(0,m,N)_≡

∞

X

n=m

β₀(n,N)−,

q(₀₁0,m,N)_≡

α(₀m,N)

+

, q(₁₁0,m,N)_≡

β₀(m,N)

+ ,

θ(N)≡ X j=0,1

∞

X

n=1

α(_jn,N)−+β(_jn,N)−,

a₀(N)≡α(₀1,N)++

∞

X

n=1

β₀(n,N)−+

∞

X

n=1

α(₁n,N)−+

∞

X

n=1

β₁(n,N)−,

a₁(N)_≡

β₀(1,N)++

∞

X

n=1

α(₀n,N)−+

∞

X

n=1

α(₁n,N)−+

∞

X

n=1

β₁(n,N)−,

q(₀₀1,m,N)≡α(₁m,N)+, q(₁₀1,m,N)≡β₁(m,N)+

q(₀₁1,m,N)≡

∞

X

n=m

α(₁n,N)−, q₁₁(1,m,N)≡

∞

X

n=m

β₁(n,N)−,

b₀(N)_≡

α(₁1,N)++

∞

X

n=1

β₁(n,N)−+

∞

X

n=1

α(₀n,N)−+

∞

X

n=1

β₀(n,N)−,

b₁(N)≡β₁(1,N)++

∞

X

n=1

α(₁n,N)−+

∞

X

n=1

α(₀n,N)−+

∞

X

n=1

β₀(n,N)−.

For(G(_iN):N _∈N_),(H(N)

i :N ∈N)∈P~0,i=0, 1, this implies in particular that there exists N0∈N

such that

sup N≥N0

X

i,j,k=0,1 X

m≥2

q(_{i j}k,m,N)<∞.

Remark 2.4. Observe that we can rewrite the transition rates in (17) such that a(_iN) = b(_iN) = 0, i=0, 1, i.e.

0_→1at rate €N₋θ(N)Šf₁(N)+f₁(N) X

m≥2,i,j=0,1

q_{i j}(0,m,N)g_i(N)f_j(N)

f₁(N) m−2

, (18)

1_→0at rate €N₋θ(N)Š

f₀(N)+f₀(N) X

m≥2,i,j=0,1

q_{i j}(1,m,N)g_i(N)f_j(N)

f₀(N) m−2

Indeed, using that f₀(N)+f₁(N)=1, we can change for instance

a₀(N)g₀(N)+q₀₀(0,2,N)g₀(N)f₀(N)

f₁(N) 0

+q(₀₁0,2,N)g₀(N)f₁(N)

f₁(N) 0

with a₀(N),q₀₀(0,2,N),q₀₁(0,2,N)nonnegative into

a(₀N)+q(₀₀0,2,N)

g₀(N)f₀(N)

f₁(N) 0

+a(₀N)+q(₀₁0,2,N)

g₀(N)f₁(N)

f₁(N) 0

,

where the new coefficients are nonnegative again.

In the second part of this proof we shall now present results for rescaled competing species models

ξN with transition rates as in (18). Tightness results for such models then immediately yield tight-ness results for the former models with rates of change as in (16). A bit more work is needed to translate convergence results of the latter model to obtain convergence results for the former model. The relevant part of the proof is given at the end of this section.

Moving on to models with transition rates as in (18), we introduce hypotheses directly on theq(_{i j}k,m,N)

as the primary variables. Observe in particular that they will be assumed to be non-negative.

Hypothesis 2.5. Assume that there exist non-negative q(_{i j}k,m,N), i,j,k=0, 1and m≥2such that

sup N≥N0

X

i,j,k=0,1 X

m≥2

q(_{i j}k,m,N)<∞

for some N0∈N.

Remark 2.6. Recall the end of Remark 2.3. We can use the above condition as in Remark 2.1 to show that the rewritten transition rates can be used to determine a{0, 1}Z_/N

-valued Markov processξN t for

N_≥N₀.

Hypothesis 2.7. In the special case with no short-range competition, i.e. where we consider

q(₀₀k,m,N)=q(₁₀k,m,N)and q(₀₁k,m,N)=q(₁₁k,m,N) (19)

in (18), we assume additionally to Hypothesis 2.5 that θ(N)N→→∞θ,

q₀(k_j,m,N)N_→→∞q₀(k_j,m)for all j,k=0, 1and m_≥2

and

lim N→∞

X

j,k=0,1 X

m≥2

q(₀k_j,m,N)= X j,k=0,1

X

m≥2

q(₀k_j,m). (20)

Remark 2.8. In the special case with no short-range competition, observe that if we assume that the q(₀k_j,m,N),j,k=0, 1,m_≥2were obtained fromα(_jm,N),j=0, 1,m_≥1as described earlier in Remark 2.3 and Remark 2.4, then G(_iN) = H_i(N),i = 0, 1 implies (19). The additional assumption (G(_iN) : N ∈

N_),(H(N)

i :N ∈N)∈P~1,i= 0, 1implies Hypothesis 2.7. Indeed, use for instance[16], Proposition

Notation 2.9. Fork=0, 1 anda_∈Rwe let

Fk(a) =

(

1−a, k=0,

a, k=1.

We give the proof of the following result for rescaled competing species models ξN _{with rates of} change as in (18) in Section 3.

Theorem 2.10. Suppose that A(ξN₀)_→u₀ in_C1. Let the transition rates ofξN(x)be as in (18) and q(_{i j}k,m,N)satisfying Hypothesis 2.5. Then the€A(ξN_t ):t_≥0Šare C-tight as cadlag_C1-valued processes and the€νN

t :t≥0

Š

are C-tight as cadlag Radon measure valued processes with the vague topology. If

A

ξNk

t

,νNk

t

t≥0 converges to(ut,νt)t≥0, thenνt(d x) =ut(x)d x for all t ≥0.

For the special case with no short-range competition we further have that if Hypothesis 2.7 holds, then the limit points of A(ξN

t )are continuousC1-valued processes ut which solve

∂u ∂t =

∆u

6 + X

k=0,1

(1₋2k) X m≥2,j=0,1

q(₀k_j,m)Fj(u) F1−k(u)

m−1

Fk(u) +

p

2u(1₋u)_W˙ ₍₂₁₎

with initial condition u0. If we assume additionally 〈u0, 1〉<∞, then ut is the unique in law[0, 1]

-valued solution to the above SPDE.

The claim of Theorem 1.5 now follows from the first part of Theorem 2.10. Indeed, rewrite (16) in the form (18) and use Remark 2.3.

Assume additionally that there is no short-range competition and recall Remark 2.8. The second part of Theorem 2.10 yields that the limit points ofA(ξN

t ), withξNt being the system with transition rates as in (16), are continuous_C1-valued processesu_t which solve (21). To obtain the claim of Theorem 1.8, it remains to show that every solution to (21) can be rewritten as a solution to (8). This follows from Corollary 2.11 below. Uniqueness in law of the former solution then implies uniqueness in law of the latter.

Corollary 2.11. Under the assumptions of Theorem 1.8, the SPDE (21) may be rewritten as ut=

∆u

6 + (1−u)u

∞

X

m=0

α(₀m+1)um−u(1₋u)

∞

X

m=0

α(₁m+1)(1₋u)m+p2u(1₋u)W˙.

Proof. First recall Remark 1.7. Next use the definition ofFk(a)and collect terms appropriately. Then recall how we rewrote the transition rates in Lemma 2.2 and Remark 2.4 to obtain (18) from (16). Now, analogously, rewrite (8) as (21).

3

Proof of Theorem 2.10

3.1

Overview of the proof

The proofs in Subsections 3.2-3.7 are generalizations of the proofs in [13]. In [13], limits are considered for both the long-range contact process and the long-range voter process. Full details

are given for the contact process. For the voter process, once the approximate martingale problem is derived, almost all of the remaining steps are left to the reader. Many arguments of our proof are similar to[13]but as they did not provide details for the long-range voter model and as additions and adaptations are needed due to our broader setup, we shall not omit the details.

In Subsection 3.2 we shall introduce a graphical construction for each approximating modelξN. This allows us to write out the time-evolution of our models. By integrating it against a test function and summing over x _∈Z_/N we finally obtain an approximate martingale problem for theNth-process

in Subsection 3.4. We defined the approximate densityA(ξN_t )(x)as the average density of particles of type 1 on Z_{/N in an interval centered at x of length 2/}p_{N (recall (4)). By choosing a specific}

test function, the properties of which are under investigation at the beginning of Subsection 3.5, an approximate Green’s function representation for the approximate densities A(ξN_t )(_·) is derived towards the end of Subsection 3.5 and bounds on error-terms appearing in it are given. Making use of the Green’s function representation, tightness ofA(ξN

t )(·)is proven in Subsection 3.6. Here the main part of the proof consists in finding estimates on pth-moment differences. In Subsection 3.7

the tightness of the approximate densities is used to show tightness of the measure corresponding to the sequence of configurationsξN_t . Finally, in the special case with no short-range competition, every limit is shown to solve a certain SPDE.

In Subsection 3.8 we additionally prove that this SPDE has a unique weak solution ifRu0(x)d x < ∞. In this case, weak uniqueness of the limits of the sequence of approximate densities follows.

3.2

Graphical construction

Recall that the rates of change of the approximating processesξN_t that we consider in Theorem 2.10 are given in (18). Note in particular that by Hypothesis 2.5 the coefficientsq(_{i j}k,m,N)are non-negative. We shall first derive a graphical construction and evolution in time of our approximating processes

ξN_t . The graphical construction uses independent families of i.i.d. Poisson processes: €

Pt(x;y):x,y ∈N−1Z

Š

i.i.d. Poisson processes of rate N−θ

(N)

2c(N)N1/2, (22) and form≥2,i,j,k=0, 1,

Qm_t,i,j,k(x;y1, . . . ,ym;z):x,y1, . . . ,ym,z∈N−1Z

i.i.d. Poisson processes of rate

q(_{i j}k,m,N)

(2c(N))m_Nm/2p(N(x−z)).

Note that we suppress the dependence onN in the family of Poisson processesP_t(x;y)and

Qm_t,i,j,k(x;y₁, . . . ,y_m;z).

At a jump of P_t(x;y) the voter at x adopts the opinion of the voter at y provided that y is a neighbour ofx with opposite opinion.

At a jump ofQm_t,i,j,k(x;y₁, . . . ,y_m;z)the voter atx adopts the opinion 1₋kprovided that y₁, . . . ,y_m

This yields the following stochastic integral equation to describe the evolution in time of our ap-proximating processesξN_t:

ξN_t (x) =ξN₀(x) +X y∼x

Z t

0 ¦

1€ξN_s−(x) =0Š1€ξ_sN₋(y) =1Š−1€ξ_sN₋(x) =1Š1€ξN_s−(y) =0Š©d Ps(x;y) (23)

+ X

k=0,1

(1₋2k) X m≥2,i,j=0,1

X

y1,...,ym∼x

X

z

Z t

0 1€ξN

s−(x) =k

Š 1€ξN

s−(y1) = j Š

× m

Y

l=2 1€ξN

s−(yl) =1−k

Š 1€ξN

s−(z) =i

Š

dQm_s,i,j,k(x;y₁, . . . ,y_m;z)

for allx ∈N−1Z_.

An explanation of why (23) has a unique solution can be found at the beginning of Section 4.3 of the author’s thesis, Kliem[10]. There it is further shown that the solution is the spin-flip system with ratesc(x,ξN)given by (18). In what follows we shall often drop the superscripts w.r.t. N to simplify notation.

3.3

Preliminary notation

In what follows we shall considere_λ(x) =exp(λ|x_|)forλ∈Rand we let

C =_{f :R_{→[0,∞)continuous with|f(x)eλ(x)| →0 as|x| → ∞for allλ <0}}

be the set of non-negative continuous functions with slower than exponential growth. Define

kfkλ=sup

x |

f(x)e_λ(x)_|

and give_C the topology generated by the norms(_k·kλ:λ <0).

Remark 3.1. We work on the spaceC instead of C1 because in Subsection 3.5 we shall introduce

functions0_≤ψz_t(x)_≤C N1/2and shall show in Lemma 3.9(b) that they converge in_C to the Brownian transition density p€₃t,z₋xŠ. Finally, in Subsection 3.6 we shall derive estimates on pth-moment differences ofAˆ(ξt)(z)≡A(ξt)(z)− 〈ξ0,ψzt〉, where A(ξ0)→u0 inC to finally establish the tightness

claim for the sequence of approximate densities A(ξN_)(x).

Notation 3.2. Forx ∈N−1Z_{,f :N}−1Z_→R_{andδ >0 we shall write}

D(f,δ)(x) =sup_{|f(y)₋f(x)_|:_|y₋x_{| ≤}δ,y_∈N−1Z_}, (24)

∆(f)(x) = N−θ

(N)

2c(N)N1/2 X

y∼x

(f(y)₋ f(x)),

3.4

An approximate martingale problem

We now derive the approximate martingale problem. In short, the idea is to express the integral ofξ_t against time-dependent test functions as the sum of a martingale, average (drift) terms and fluctuation (error) terms.

Take a test function φ : [0,_∞)_×N−1Z _→ R with t _7→ φ_t(x) continuously differentiable and satisfying

Z T

0

〈|φs|+φs2+|∂sφs|, 1〉ds<∞ (25)

(this ensures that the following integration and summation are well-defined). We apply integration by parts toξt(x)φt(x), sum over x and multiply by _N1, to obtain for t ≤T (recall the definition of

νt from (3) and that〈ξt,φ〉=〈νt,φ〉) 〈ν_t,φ_t〉=_〈ν₀,φ_0〉+

Z t

0

〈ν_s,∂_sφ_s〉ds (26) + 1

N X

x

X

y∼x

Z t

0

ξs−(y) φs(x)−φs(y)

d P_s(x;y) (27)

+ 1

N X

x

X

y∼x

Z t

0

ξs−(x)φs(x) d Ps(y;x)−d Ps(x;y)

(28)

+ X

k=0,1

(1₋2k) X m≥2,i,j=0,1

1

N X

x

X

y1,...,ym∼x

X

z

Z t

0

1 ξ_s−(x) =k

1 ξ_s−(y₁) = j

× m

Y

l=2

1 ξ_s−(y_l) =1₋k

1 ξ_s−(z) =i

φ_s(x)dQ_sm,i,j,k(x;y₁, . . . ,y_m;z). (29)

The main ideas for analyzing terms (27) and (28) will become clear once we analyze term (29) in detail. The latter is the only term where calculations changed seriously compared to[13]. Hence, we shall only summarize the results for terms (27) and (28) in what follows.

We break term (27) into two parts, an average term and a fluctuation term and after proceeding as for term (3.1) in[13]we obtain

(27) = Z t

0

〈ν_s−,∆ φ_s

〉ds+E(_t1)(φ),

where

E(_t1)(φ)_≡ 1

N X

x

X

y∼x

Z t

0

ξs−(y) φs(x)−φs(y)

d P_s(x;y)₋d_〈P(x;y)_〉s .

We have suppressed the dependence on N in E(_t1)(φ). E(_t1)(φ) is a martingale (recall that if N _∼

Pois(λ), thenN_t−λt is a martingale with quadratic variation_〈N_〉t=λt) with predictable brackets process given by

d

E(1)(φ)

t≤

D

φt, 1 p

N

2

λ

Alternatively we also obtain the bound

d

E(1)(φ)

t≤4kφtk0〈 φ_t

, 1〉d t (31)

withkφtk0=supx|φt(x)|.

The second term (28) is a martingale which we shall denote byM_t(N)(φ)(in what follows we shall drop the superscripts w.r.t. N and write M_t(φ)). It can be analyzed similarly as the martingale

Zt(φ)of (3.3) in[13]. We obtain in particular that 〈M(φ)_〉t=2N−θ

(N)

N

¨Z t

0

〈ξs−,φs2〉ds−

Z t

0

〈A ξs−φs

,ξs−φs〉ds

«

. (32)

Using that

A ξ_s−φ_s

(x)≡ 1 2c(N)N1/2

X

y∼x

ξs−(y)φs(y)

≤sup y∼x|

φs(y)| we can further dominate_〈M(φ)_〉t by

〈M(φ)_〉t≤C(λ) Z t

0 €

kφsk2_λ〈1,e−2λ〉

Š

∧ kφsk0〈ξs−,|φs|〉

ds. (33)

We break the third term (29) into two parts, an average term and a fluctuation term. Recall Notation 2.9 and observe that if we only considera_{∈ {}0, 1_}we have F_k(a) =1(a=k). We can now rewrite (29) to

X

k=0,1

(1₋2k) X m≥2,i,j=0,1

1

N X

x

X

y1,...,ym∼x

X

z

Z t

0

1 ξs−(x) =k

1 ξs−(y1) = j

(34)

× m

Y

l=2

1 ξ_s−(y_l) =1₋k

1 ξ_s−(z) =i

φ_s(x)

q(_{i j}k,m,N)

(2c(N))m_Nm/2p(N(x−z))ds+E

(3)

t (φ)

= X

k=0,1

(1₋2k) X m≥2,i,j=0,1

q_{i j}(k,m,N) Z t 0 1 N X x 1 2c(N)N1/2

X

y1∼x

1 ξs−(y1) = j

!

× m

Y

l=2

1 2c(N)N1/2

X

yl∼x

1 ξs−(yl) =1−k

!

X

z

p(N(x−z))1 ξs−(z) =i

!

×1 ξs−(x) =k

φs(x)ds+E

(3)

t (φ)

= X

k=0,1

(1₋2k) X m≥2,i,j=0,1

q_{i j}(k,m,N) Z t 0 1 N X x

F_j(A(ξ_s−)(x)) × F1−k(A(ξs−)(x))

m−1

F_i((pN∗ξ_s−)(x))1 ξ_s−(x) =k

φ_s(x)ds+E(_t3)(φ)

= X

k=0,1

(1₋2k) X m≥2,i,j=0,1

q_{i j}(k,m,N) Z t

0

〈€Fj◦A(ξs−)

Š

× F1−k◦A(ξs−)

m−1€

Fi◦(pN∗ξs−)

Š

1 ξs−(·) =k

,φs〉ds+E

(3)

where for x_∈Z_/N we set €

pN_∗fŠ(x)_≡ X z∈Z_/N

p(N(x₋z))f(z) (35)

and

E(_t3)(φ)_≡ X k=0,1

(1₋2k) X m≥2,i,j=0,1

1

N X

x

X

y1,...,ym∼x

X

z

Z t

0

1 ξs−(x) =k

×1 ξs−(y1) = j

m

Y

l=2

1 ξs−(yl) =1−k

1 ξs−(z) =i

φs(x)

× 

 dQ

m,i,j,k

s (x;y1, . . . ,ym;z)−

q_{i j}(k,m,N)

(2c(N))m_Nm/2p(N(x−z))ds 

 .

We have suppressed the dependence onN inE(_t3)(φ). Here,E(_t3)(φ)is a martingale with predictable brackets process given by

E(3)(φ)

t≤

X

m≥2,i,j,k=0,1

q(_{i j}k,m,N) 1 N2

X

x m

Y

l=0 X

yl∼x

1 2c(N)N1/2

! X

z

p(N(x₋z)) !_{Z t}

0

φ_s2(x)ds

(36)

≤ 1

N

X

m≥2,i,j,k=0,1

q(_{i j}k,m,N) Z t

0

kφsk2λ〈e−2λ, 1〉ds.

Taking the above together we obtain the following approximate semimartingale decomposition from (26).

〈ν_t,φ_t〉=_〈ν₀,φ_0〉+ Z t

0

〈ν_s,∂_sφ_s〉ds+ Z t

0

〈ν_s−,∆ φ_s

〉ds+E(_t1)(φ) +M_t(φ) +E_t(3)(φ) (37)

+ X

k=0,1

(1₋2k) X m≥2,i,j=0,1

q(_{i j}k,m,N) Z t

0

〈€F_j◦A(ξs−)

Š

× F_1−k◦A(ξs−)

m−1€

F_i◦(pN_∗ξs−)

Š

1 ξs−(·) =k

,φs〉ds.

Remark 3.3. Note that this approximate semimartingale decomposition provides the link between our approximate densities and the limiting SPDE in (21) for the case with no short-range competition. Indeed, uniqueness of the limit u_t of A(ξN

t ) will be derived by proving that ut solves the martingale

problem associated with the SPDE (21).

3.5

Green’s function representation

Analogous to[13], define a test function

then

dPu dPv

Ft

=R_t for all t>0, (64)

whereFt is the canonical filtration of the process v(t,x).

Proof. We proceed analogously to the proof of Theorem IV.1.6(a),(b) in[15]. Observe in particular that we take

T_n=inf

(

t_≥0 :

Z t

0

Z

(1₋u_s(x)) Q(u_s(x))2

2 us(x)d x+1ds≥n

)

.

Lemma C.1 shows that underPu

Z t

0

Z

(1₋u_s(x)) Q(u_s(x))2

2 us(x)d x ds≤

€

C_QŠ2 2

Z t

0

〈us, 1〉ds<∞

for allt>0Pu₋a.s. and soT_{n↑ ∞}Pu-a.s. As in Theorem IV.1.6(a) of[15]this gives uniqueness of the lawPu _{of a solution to (62). As in Theorem IV.1.6(b) of}[15]the fact that Tn↑ ∞Pv-a.s. (from

Lemma C.1) shows that (64) defines a probabilityPu_{which satisfies (62).}

A

Proof of Remark 2.1

In what follows we shall prove the claim of Remark 2.1. See for instance Theorem B3, p.3 in Liggett[11]and note the uniform boundedness assumption on the rates from p.1 of[11]. Following the notation in[11], letc(x,ξN_{)denote the rate at which the coordinateξ}N_{(x)flips from 0 to 1 or}

from 1 to 0 when the system is in stateξN_{. Then using(G}(N)

i :N∈N),(H

(N)

i :N∈N)∈P0~,i=0, 1,

(15), (16) and 0_≤ f_i(N),g_i(N)_≤1,i=0, 1 yield

sup

x∈Z_/N

sup

ξN_∈{0,1}Z/N

c€x,ξNŠ≤N+ ∞

X

m=0

α

(m+1,N) 0 + β

(m+1,N) 0 ! ∨ ∞ X

m=0

α

(m+1,N) 1 + β

(m+1,N) 1

!

≡N+C₀(N)<∞ and

sup

x∈Z_/N

X

u∈Z_/N

sup

ξN_∈{0,1}Z_/N

c

€

x,ξNŠ₋c€x,ξ_uNŠ

≤ sup

x∈Z_/N

X

u∼x

sup

ξN_∈{0,1}Z_/N

c

€

x,ξNŠ₋c€x,ξN_uŠ

+ sup

x∈Z_/N

X

u∈Z_/N

sup

ξN_∈{0,1}Z/N

X

i=0,1

g

(N)

i €

x,ξNŠ−g_i(N)€x,ξN_uŠ

C0(N)

≤2c(N)N1/22(N+C0(N)) + sup

x∈Z_/N

X

u∈Z_/N

2p(N(x−u))C₀(N)

where

ξN_u(v) =

(

ξN_{(v), v6=u,}

1₋ξN_{(v), v=u.}

Following[11], the two conditions are sufficient to ensure the claim holds.

B

Auxiliary results to prove Lemma 3.4

The following lemma and corollary are necessary to prove Lemma 3.4.

Lemma B.1. There exists N0<∞such that for all N≥N0,k≥1 (a)

ρ

k_{(t)−exp−c}

2(N)kt

2 6N ≤C 1 kexp

−c2(N)kt

2

12N

for t≤ÆN₃, (b) |ρ(t)_{| ≤}exp₋C₁₂t2_N

for t≤_c6N

2(N)

1/2

,

(c) There existsδ >0such that_|ρ(t)_{| ≤}1₋δfor t_∈

6N c2(N)

1/2

,πN

.

Proof. The proof mainly follows along the lines of the proof of Lemma 8 in [13]. Some small changes ensued due to the different setup. Recall the definition ofρ(t)from equation (42).

For (b), we could not find the reference mentioned in[13]but the following reasoning in[13]based on applying Taylor’s theorem at t=0 works well without it.

For (a), first observe thatρk(t) =E”ei tSk—_{and use Bhattacharya and Rao[2], (8.11), (8.13) and}

[2], Theorem 8.5. as suggested in[13]. We used thatE[Y₁] =E[Y₁3] =0.

It remains to prove (c). We have to change the proof of[13], Lemma 8(c) slightly, as we usedx _6∼x. We get

|ρ(t)_|=

1 2c(N)N1/2

X

0<j≤c(N)pN

2Re

h

ei tNj

i = 1 c(N)N1/2Re



 

ei t1N −ei t c(N)pN+1

N 1−ei tN1

   ≤ 1

c(N)N1/2

2 2 sin€₂t_NŠ

.

For 1+ε

c(N)N1/2 ≤ t

2N ≤ π

2 withε >0 fixed we get as an upper bound 1

c(N)N1/2

1 sin

1+ε c(N)N1/2

≤ 1

1+ε <1,

Corollary B.2. For N_≥N₀, y_∈N−1Zwe have

NP(S_k= y)₋p

c₂(N) k 3N,y

≤

C₁¦Nexp

−kC₂ +N1/2k−3/2©, where C₁,C₂>0are some positive constants.

Proof. This result corresponds to Corollary 9 in [13]. The proof works similarly. Instead of the reference given at the beginning of the proof of Corollary 9 in[13], we used Durrett[7], p. 95, Ex. 3.2(ii) and[7], Thm. (3.3).

Note in particular that the result of Lemma B.1(c) can be extended tot ∈hÆN₃,πN

i

if we choose

δ >0 small enough. Indeed, using Lemma B.1(b) we obtain |ρ(t)_{| ≤}e−C t

2 12N ≤e−C

N/3

12N ≤(1−δ) as claimed.

C

Auxiliary results to prove Lemma 3.10

The following lemma is used to prove Lemma 3.10. LetuandQ(u)be as in (62) andv as in (63). Also recall the definitions ofPu_,Pv _{andCQ in between (62) and (63).}

Lemma C.1. Given u0=v0satisfying〈u0, 1〉<∞, we havePu-a.s.

Rt

0〈us, 1〉ds<∞for all t≥0and Pv_-a.s.Rt

0〈vs, 1〉ds<∞for all t≥0.

Proof. We shall prove the claim forPu_{. The other claim then follows by considering the special case} Q _≡ 0. As a first step we shall use a generalization of the weak form of (62) to functions in two variables. In the proof of Theorem 2.1 on p. 430 of[18]it is shown that for everyψ∈ D2r ap(T)and

0<t<T we have

〈u_t,ψ_t〉=_〈u₀,ψ_0〉+

Z t

0 〈u_s,

∂ ∂s+

∆ 6

ψ_s〉ds+

Z t

0

〈u_s(1₋u_s)Q(u_s),ψ_s〉ds (65)

+

Z t

0

Z p

2u_s(x)(1₋us(x))ψs(x)dW(x,s).

Here we have forT>0, C(R) =

f :R_→Rcontinuous , Cr ap=

f _{∈ C}(R_{)such that sup}

x

eλ|x|_|f(x)_|<_∞for allλ >0

,

Cr ap2 = ¦

ψ∈ C2(R)such thatψ,ψ′,ψ′′∈ Cr ap ©

,

Dr ap2 (T) = n

ψ∈ C1,2([0,T)_×R)such thatψ(t,_·)is_{Cr ap}2 -valued continuous and

∂ ψ

∂t (t,·)isCr ap-valued continuous in 0≤t<T

Also observe that the condition (2.2) of [18] is satisfied as we have 0 _≤u_s(x)_≤ 1 and therefore |Q(u_s(x))_{| ≤}CQ.

Now recall that the Brownian transition density is p(s,x) = p1 2πse

−x22s. Let P

sφ

(x) =

R

p€₃s,y−xŠφ(y)d ywithφ∈ Cc∞,φ≥0 and let

ψ(s,x) =ψs(x) =eCQ(T−s) PT−sφ

(x)and thusψ∈ D2r ap(T).

Note that _∂∂_s PT−sφ

(x) =₋∆

6 PT−sφ

(x), where we used that ∂

∂sp(s,x) =

1

2∆p(s,x). We obtain for the drift term in (65) that

〈u_s,

_∂

∂s+ ∆

6

ψs〉+〈us(1−us)Q(us),ψs〉

=_〈u_s,₋C_Qψ_s−∆

6ψs+ ∆

6ψs〉+〈us(1−us)Q(us),ψs〉 ≤0

using thatψ(s,x)_≥0 forφ≥0. Additionally, the local martingale in (65) is a true martingale as

DZ ·

0

Z p

2u_s(x)(1₋u_s(x))ψ_s(x)dW(x,s)

E

t

=

Z t

0

〈2u_s(1₋u_s),ψ2_s〉ds_≤2e2CQT

Z t

0

〈1, P_T−sφ2

〉ds

≤2e2CQT _kφk0

Z t

0

〈1,PT−sφ〉ds=2e2CQT kφk0〈1,φ〉t

<∞.

Hence we obtain from (65) for all 0<t<T after taking expectations E

〈u_t,ψ_t〉

≤ 〈u₀,ψ_0〉, i.e.

eCQ(T−t)_E〈u

t,(PT−tφ)〉

≤eCQT_〈u

0,(PTφ)〉.

Now choose an increasing sequence of non-negative functionsφn_{∈ C}∞

c such thatφn↑1 forn→ ∞.

Using the monotone convergence theorem, we obtain from the above

eCQ(T−t)_E〈u

t, 1〉

= lim

n→∞e

CQ(T−t)_E〈u

t,(PT−tφn)〉

≤ lim

n→∞e

CQT_〈u

0,(PTφn)〉=eCQT〈u0, 1〉. Hence by the Fubini-Tonelli theorem,

E

–Z t

0

〈us, 1〉ds ™

≤ 〈u0, 1〉

Z t

0

eCQs_ds<∞ for allt_≥0, which proves the claim.

Acknowledgement

The author wishes to thank Ted Cox for suggesting to extend the results of[13]for the voter model to perturbations of the voter model. Further thanks go to my Ph.D. supervisor Ed Perkins for giving helpful feedback after reading my manuscript. Finally, the author wishes to thank two referees for a careful reading of my paper and for many suggestions that helped to improve the exposition of this paper.

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