Fluctuations of the front in a stochastic combustion model

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Fluctuations of the front in a stochastic combustion model

Francis Comets

^a,^∗^,1,2

, Jeremy Quastel

^b,3

, Alejandro F. Ramírez

^c,2,4

aLaboratoire de Probabilités et Modèles Aléatoires, Université Paris 7-Denis Diderot, 2, Place Jussieu, 75251 Paris cedex 05, France bDepartments of Mathematics and Statistics, University of Toronto, 40 St. George Street, Toronto, Ontario M5S 1L2, Canada

cFacultad de Matemáticas, Pontificia Universidad Católica de Chile, Vicuña Mackenna 4860, Macul, Santiago, Chile

Received 3 November 2005; accepted 3 January 2006 Available online 5 October 2006

Abstract

We consider an interacting particle system on the one-dimensional latticeZmodeling combustion. The process depends on two integer parameters 2aM <∞. Particles move independently as continuous time simple symmetric random walks except that (i) when a particle jumps to a site which has not been previously visited by any particle, it branches intoaparticles, (ii) when a particle jumps to a site withMparticles, it is annihilated. We start from a configuration where all sites to the left of the origin have been previously visited and study the law of large numbers and central limit theorem forrt, the rightmost visited site at timet.

The proofs are based on the construction of a renewal structure leading to a definition of regeneration times for which good tail estimates can be performed.

Résumé

On considère un système de particules en interaction surZmodélisant les particules incandescentes d’un mécanisme de combustion. Le processus dépend de deux paramètres entiers 2aM <∞. Les particules se déplacent indépendamment selon des promenades aléatoires simples symétriques à temps continu, mises à part les interactions suivantes : 1-quand une particule saute vers un site qui n’a jamais encore été visité, elle branche et fait place àa particules ; 2-quand une particule saute vers un site abritantMparticules, elle disparait. On démarre d’une configuration où seuls les sites à gauche de l’origine ont déja été visités et on étudie la loi des grands nombres et le théorème de la limite centrale pourr_t, la position du site le plus à droite visité à l’instantt.

Les preuves reposent sur la construction d’une structure de renouvellement associée à des temps de régénération dont les queues peuvent être convenablement estimées.

MSC:primary 82C22, 82B41; secondary 82B24, 60K35, 60G99

Keywords:Regeneration times; Interacting particle systems; Random walks in random environment

* Corresponding author.

E-mail addresses:comets@math.jussieu.fr (F. Comets), quastel@math.toronto.edu (J. Quastel), aramirez@mat.puc.cl (A.F. Ramírez).

1 Partially supported by CNRS, UMR 7599.

2 Partially supported by ECOS-Conycit grant CO5EO2.

3 Partially supported by NSERC, Canada.

4 Partially supported by Fondo Nacional de Desarrollo Científico y Tecnológico grant 1020686.

doi:10.1016/j.anihpb.2006.01.005

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1. Introduction

The method of regeneration times has been very successfully applied to problems of random walk in random environment (for example see [16,17,6]); in a one-dimensional setting it was already used via a renewal structure in [9].

In this article, we extend the method of regeneration times to the study of front propagation in an interacting particle system. The system we consider is a one-dimensional stochastic model of combustion. Heat particles move by symmetric nearest neighbor random walks on sites{x∈Z: xr}of the integer lattice, with rrepresenting the position of a flamefront. To the right of the flame front is a propellant. The first heat particle which reaches the propellant atr+1 immediately branches intoa2 particles, and the front moves one step to the right. We also include an upper boundM on the number of particles at each site, so that if a particle tries to jump to a site with Mparticles, it is immediately killed. We show that the front in the model moves ballistically to the right, or, more precisely, prove a law of large numbers forr_t. The next question one might ask is of the fluctuations about the law of large numbers. We prove here that these are Gaussian, with a central limit theorem fort⁻^1/2(rt−vt ).

ForM= ∞a discrete time version of the system we are considering has appeared in the literature under the enigmatic name “frog model”, and laws of large numbers for the position of the front were proved, also in higher dimensions, using methods based on the sub-additive ergodic theorem [1–4,8,11,12]. The continuous time case under the namestochastic combustion processwas treated in [13,14]. It appears that the sub-additivity is completely lost whenM <∞. Our main purpose here is to develop new methods to study such models, and to go beyond methods based on sub-additivity. Also, we are particularly interested here in the fluctuations of the fronts. IfN_t represents the number of particles at the frontrt at timet, then rt moves to the right at rateNt, and hence rt −_t

0Nsds is a martingale. The standard approach to the law of large numbers would then be to show that, as observed from the front, the system has an ergodic invariant measure μ, and t⁻¹t

0Nsds→Eμ[N]. The central limit theorem would be proved by showing that the time correlations ofN_s decay fast enough. However, very few methods exist for proving uniqueness, or ergodicity, of invariant measures of such systems. Furthermore, although one can guess that the correlations ofN_s decay quickly – likely exponentially fast – it is not at all apparent how to prove it. The method we use to prove the law of large numbers and central limit theorem for the front,r_t, is that of regeneration times. Because the front moves ballistically while the particles move diffusively, one expects there are regeneration times after which the trajectory of the front is decoupled from that of the particles behind the front. Note that a concept similar to that of regeneration times, known ascluster structure, has been well know for many years in the context of mechanically interacting one-dimensional dynamical systems of particles [5] (also see [15] and references therein).

The combustion process that we consider is related to deterministic reaction–diffusion equations of the form∂_tu=

∂_x²u+f (u). The two key differences are the discreteness of the variable – we have a number of particles as opposed to a continuum variableu– and the stochasticity. The effect of discreteness and/or stochasticity on the traveling waves of reaction–diffusion equations is a question that has not received the attention it deserves, as these effects are usually present in real systems. Therefore, we believe it is crucial to develop methods to study such systems. In the literature of reaction–diffusion equations one separates several cases according to the behavior off near zero. Anf which vanishes on[0, θ]for someθ >0 with f (u) >0 for u > θ is said to have a combustion nonlinearity with ignition temperature cutoffθ(see, for example [7]). Note that the discreteness of the particle models of the type we consider makes them analogous to the combustion nonlinearity with the ignition temperature cutoff corresponding to a single particle.

In the second section of this article, we define the combustion process and state the main result (Theorem 1). In Section 3, we define an auxiliary and a labeled process, which will be needed to define the renewal structure leading to the regeneration times. The labeled process can be understood as a combustion process where particles are labeled so that if at a given site a particle has to be killed, it is the one with the smallest label. The auxiliary process moves ballistically to the right and is coupled to the labeled combustion process in such a way that it is always to the left of the right-most visited siter_t. In Section 4, these processes are used to define the renewal structure, and the corresponding regeneration times. Then in Section 5, it is proved that the regeneration times have finite moments of order 2, under appropriate assumption on the thresholdM. In Section 6 we complete the proof of Theorem 1.

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2. Combustion process

We define a stochastic process describing the dynamics of particles on the latticeZwhich move as rate 2 continuous time simple symmetric random walks and branch and are killed at rates depending on the configuration of neighboring particles. The branching and killing depend on natural number parameters 2a M. Each particle performs a continuous time symmetric simple random walk independent of the others, with two twists: There is a positionr∈Z which is therightmost visited site. When a particle jumps right from this site, it branches intoa particles, with the result that there area particles at a new rightmost visited site,r+1. In addition, we allow at mostMparticles at any site, and maintain this requirement by killing any particle that attempts to jump to a site withMparticles.

The state space of our system is Ω:=

(r, η): r∈Z, η∈ {0, . . . , M}^{^...,r⁻^1,r^} .

Hereη(x)denotes the number of particles atxrand we will writeηt(x)orη(t, x)to denote the same quantity at timet.

The process is Markov with infinitesimal generator

Lf (r, η)=

xr,yr,|x−y|=1

η(x)

f (r, T_xyη)−f (r, η) +η(r)

f (r+1, η−δ_r+aδ_r₊₁)−f (r, η)

, (1)

whereδ_xdenotes the configuration with one particle atxand T_xyη=η−δ_x+δ_y1

η(y) < M

. (2)

In the following we will assume for technical reasons that

8< M <∞. (3)

We can now state the main results.

Theorem 1. Suppose the process is started with initial conditions r =0, η(0,0)∈ {1, . . . , M} and η(0, x)∈ {0, . . . , M}arbitrary forx <0.

(i) (Law of large numbers). There exists a nonrandomv∈(0,∞), which does not depend on the initial condition {η(0, x): x0}, such that almost surely,

tlim→∞

r_t

t =v. (4)

(ii) (Central limit theorem). There exists a nonrandomσ∈(0,∞), which does not depend on the initial condition {η(0, x): x0}, such that

^1/2

r₋1t−⁻¹vt

, t0, (5)

converges in law as→0to Brownian motion with varianceσ².

To prove this theorem we will define a renewal structure for the right-most visited siter_t, in terms of regeneration times, and then will show that such times have finite second moments.

3. Auxiliary and labeled processes

In this section, we will define an auxiliary process and a labeled process, both coupled to the combustion process.

The auxiliary process{˜r_t: t0}, will take values onZand will have two fundamental properties: Under appropriate initial conditions on the stochastic combustion process it will always be to the left of the right-most visited site {r_t:t0}, and its dynamics will be independent of all particles appearing to the left ofr₀in the stochastic combustion process. These properties will help us define a renewal structure for the right-most visited site{r_t:t0}and will give a lower bound for the limiting speedvofr_t. This is the content of Lemma 4.

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The labeled process has a state space larger than the combustion process, with particles carrying labels. An explicit rule for killing is given in which when a particle is killed, it is the one with the smallest label. We also introduce a variant of the labeled process, called the enlarged process, in which we keep track of the killed particles as well.

We now begin the constructions. Start with a set of independent continuous time rate 2 simple symmetric random walksY_x,i, with label(x, i), wherex∈Zandi∈ {1, . . . , a−1}. Each random walkY_x,i,i∈ {1, . . . , a−1}starts at sitex. We order the labels by

(x, i) < (x, i) ifx < x orx=xandi < i. (6)

3.1. Auxiliary process

Letr∈Zand for eachz∈Z, defineBz as the set ofMlargest labels not exceeding(z, a−1)in the order (6).

DefineAr,zto be the labels inBzwithxr.

We define a sequence of waiting times. Letν₀:=0 and define ν₁ as the first time one of the random walks {Y_z,i: (z, i)∈A_r,r}, hits the siter+1. Next, defineν₂as the first time one of the random walks{Y_z,i:(z, i)∈A_r,r₊₁}, hits the siter+2. In general, fork3, we defineν_kas the first time one of the random walks{Y_z,i: (z, i)∈A_r,r₊_k₋₁}, visits the siter+k. Finally, forn∈N, let

˜

r_t^r:=r+n, if n

k=0

νkt <

n+1

k=0

νk, (7)

where the superscriptrinr˜_t^r indicates thatr˜₀^r=r.

Lemma 1.There exists anα >0which does not depend onrsuch that a.s.

tlim→∞

˜ r_t^r

t =α. (8)

Proof. Once one notes that for every j 0 the random variables {νk+j: k1} are independent whenever > M/(a−1), the proof is a simple exercise. 2

3.2. Labeled process

We enlarge the state space of the stochastic combustion process so that particles carry labels which tell us where they originated.

We will want to keep track of where particles came from, even after restarting at stopping times. Hence each particle will have a starting positionz∈Zand label(x, i),x∈Z,i∈ {1, . . . , a−1}describing its birthplace. We will allow the possibility thatz=x.

At time 0, we have anr∈Zrepresenting the rightmost visited site, and a subset I(0)of the labels(x, i) with xr, representing the set of labels of live particles at time 0. To each one of these labels is assigned a position z=Zx,i(0)rwhich is the position at timet=0 of that particle. The position at timet isZx,i(t )=Yx,i(t )+z−x.

To keep track of the killing in this process, we make a rule that whenever a particle jumps to a site withMparticles, the particle at that site with the smallest label is killed and the corresponding label is removed from the setIof labels of live particles.

The first time a particle jumps to siter+1, the labels{(r+1,1), . . . , (r+1, a−1)}corresponding to particles with initial positionsr+1, are added to the set of labels of live particles. The time this happens will be denotedρ₁. These particles then have trajectoriesZ_r₊_1,i(t )which is equal toY_r₊_1,i(t−ρ₁)fortρ₁.

Similarly, fork2,ρ1+ · · · +ρk will be the first time a particle jumps tor+kand at that time{(r+k,1), . . . , (r+k, a−1)}are added toI, with trajectoriesZr+k,i(t )=Yr+k,i(t−ρ1− · · · −ρk)fortρ1+ · · · +ρkuntil such time as the particles are killed and their label removed from the set of labels of live particles.

We denote byI(t )the set of labels of live particles at timet and byZ(t )= {Z_x,i(t ): (x, i)∈I(t )}the positions of the corresponding random walks.

To avoid pathologies it is useful to insist that initially the set of labels of live particles includes at least one with x=r. The rightmost visited siter_t is the supremum of thex over the collection of labels,r_t=sup{x: (x, i)∈I(t )}.

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Let us formalize the above discussion. CallLthe set of triples(r,I,Z)formed by an integerr∈Z, a set of labels I⊂ {(x, i): xr,1ia−1}and position functionZ:I→ {. . . , r−2, r−1, r}taking values in the integers smaller than or equal tor. We now define

S:=

(r,I,Z)⊂L: max

(x,i)∈Ix=r,max

zr

(x,i)∈I

1_Z(x,i)(z)M , which will be the state space of our process.

Thelabeled processstarting fromw=(r,I(0),Z(0)), is now defined as the triplew_t = {(rt,I(t ),Z(t )): t0}, defining a strong Markov process taking values onS, and with a law given by a probability measurePw defined on the Skorohod spaceD([0,∞);S).

The combustion process described in Section 1 is the particle count η(t ):= {η(t, y): y r_t}where η(t, y)=

(x,i)∈I(t )1(Zx,i(t )=y). We will occasionally use the more explicit notationη_w(t )andη_w(t, y)instead ofη(t )and η(t, y)respectively, with the understanding thatη_w(0)is the particle count of the initial conditionw.

We already definedρ₁to be the first time that one of these particles hitsr+1.

Lemma 2.Suppose that(r,1), . . . , (r, a−1)∈I(0), and that all particles with these labels are initially atr. Then ρ₁ν₁.

Proof. The dynamics of thea−1 particles which start atris the same in the labeled process as in the ones we look at in the auxiliary process. Indeed, these are the only particles considered in the auxiliary process up to timeν1, while the labeled process may have many others, each of which has a chance to be the first to hitr+1. 2

Recall thatρ_k+ · · · +ρ₁is the first time one of the labeled particles in the labeled process hitsr+k. Note that the hitting could be done by one of theanew particles created atr+iat timeρ_i,i < k.

Lemma 3.Suppose that(r,1), . . . , (r, a−1)∈I(0), and that all particles with these labels are initially atr. Then ρ_kν_k.

Proof. Before giving a general proof, we describe the special case ofM=a=k=2 where the idea is more trans- parent.

Case M=a=k=2: Note first of all that ifa=2, we havei=1 always and hence we can drop thei in the labels. The labeled process starts with one particle with labelratr, possibly one other particle at siter, with a label smaller than r, and other particles with labels smaller thanr at arbitrary positions to the left ofr. At timeρ₁we have one particle labeledr+1 atr+1, which, after that time, has trajectoryZ_r₊₁(t )=Y_r₊₁(t−ρ₁), and another particle, labeledrat some sitexr+1, which, after that time, has trajectoryZ_r(t )=Y_r(t ). Note that neither particle can be killed before timeρ₁+ρ₂because they are the two with the highest labels until that time. There could also be other particles with other labels at positionsxr+1. Denote byτ_r₊₁=inf{tρ₁: Z_r₊₁(t )=r+2}, byτ_r = inf{tρ₁: Z_r(t )=r+2}and byτ_othersthe first time one of the others hitsr+2. Thenρ₂=min{τ_r₊₁, τ_r, τ_others} − ρ₁min{τ_r₊₁, τ_r} −ρ₁.

On the other hand,ν2=min{σr, σr+1}whereσr =inf{t0: Yr(t )=r+2}andσr+1=inf{t0: Yr+1(t )= r+2}. Now

τ_r₊₁=inf

tρ₁: Z_r₊₁(t )=r+2

=inf

tρ₁: Y_r₊₁(t−ρ₁)=r+2

=inf

t0: Y_r₊1(t )=r+2

+ρ1=σ_r₊1+ρ1 (9)

and

τ_r=inf

tρ₁: Z_r(t )=r+2

=inf

tρ₁: Y_r(t )=r+2 inf

t0:Y_r(t )=r+2

+ρ₁=σ_r+ρ₁. The result follows.

General case: Let us examine the configuration in the labeled process at timeρ₁+ · · · +ρ_k₋₁. It hasa−1 particles at siter+k−1 with labels(r+k−1,1), . . . , (r+k−1, a−1)plus one additional particle with an unknown label, the

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one which hitr+k−1. An additionalM−a+1 particles (which might include the additional particle of unknown label which hitr+k−1) with the next highest labels after the firsta−1 have some unknown positions, and there may in addition be any number of other particles in the configuration. In the time interval[0, ρ1+ · · · +ρ_k₋₁]none of theM−a+1 particles has hitr+k. Their trajectories in the time interval[ρ₁+ · · · +ρ_k₋₁, ρ₁+ · · · +ρ_k]are Y_(r₊_k₋2,a−1)(t−ρ1+ · · · +ρk−2), . . .. The definition ofνk, on the other hand, involves the particles with theseM leading indices, but without the time shift. Hence the first time that any of the firsta−1 particles hits is identical to that of the auxiliary process, the first time that any of the nextM−a+1 particles hits is greater in the auxiliary process, and one of the other possible particles in the labeled process could be the one which hit first, making the time shorter still. 2

3.3. Enlarged process

We have seen that the labeled process is defined in terms of a set of labelsI(t ), for t0, with the property rt =sup{x: (x, i)∈I(t )}<∞. Whenever a particle with a label in I(t )is killed, this label is removed. In the enlarged process we keep track of the killed particles as well. Define for eacht0, the set of labels of all activated particles up to timetin the labeled process,

I(t )=

0st

I(s).

Consider the set Z(t )= {Zx,i(t ): (x, i) ∈ I(t )} of all the corresponding random walks at time t. The en- larged process is defined as the triple w_t = {(r_t,I(t ),Z(t )): t 0}. Observe that the particle count (t, y)→

(x,i)∈ I(t )1(Zx,i(t )=y)of the enlarged process, is the combustion process without threshold (M= ∞).

Consider now the set of labelsR(t ), obtained after removing fromI(t )all labels(x, i)withx < r=sup{y: (y, i)∈ I(0)}. We define foryr_tthe particle count

ζ (y, t )=

(x,i)∈R(t )

1

Z_x,i(t )=y

. (10)

Also letL(t )be the set of labels obtained after removing fromI(t )all labels(x, i) xr. We define foryr_t the particle count

φ(y, t)=

(x,i)∈L(t )

1

Zx,i(t )=y

. (11)

We similarly defineL(t )as the set of labels obtained after removing fromI¯(t )all labels(x, i)with positionsxr, and the corresponding particle count foryr_t as

φ(y, t )¯ =

(x,i)∈ L(t )

1

Z_x,i(t )=y

. (12)

Sometimes we will writeζ_w,φ_wandφ¯_w to emphasize the dependence on the initial conditionw∈S.

Lemma 4.

(i) For every initial conditionw∈Sand everyt0andxr_t

φ_w(x, t )φ¯_w(x, t ). (13)

(ii) For everyw=(r,I(0),Z(0))∈Swith labels(r,1), . . . , (r, a−1)at siter and corresponding initial positions Z_r,i(0)=rfor1ia−1,

˜

r_t^rr_t. (14)

(iii) For everyw=(r,I(0),Z(0))∈S, the processesφ¯_wandr˜_t^r are independent.

Proof. Part (i) follows directly from the definitions. Part (ii) is a consequence of Lemma 3. Part (iii) follows from the observation thatφ¯andr˜_t^r are defined in terms of the random walks{Y_x,i: x < r}and{Y_x,i: xr}respectively, which are independent. 2

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4. The renewal structure

At a heuristic level, regeneration occurs every time the front moves forward and the particles behind never catch it up later on. After such a time, the trajectory of the front depends only on thea−1 newly created particles sitting on the front, and the new particles they will create in the future.

In order to estimate the tails of regeneration times, it is useful to decouple particles initially on the front from particles to the left of it. We require that, from such a time on, thea−1 new particles on the front, and the ones they will create, move fast enough, and particles initially to the left do not move fast. The precise definition is given now in a sequence of definitions leading to (18).

Consider the enlarged processw_t and its natural filtrationFt, with an initial conditionw₀ having particles with labels (r,1), . . . , (r, a−1)at site r, and any allowable configuration of particles with labels to the left of r. Let α=limt→∞r˜_t/tbe as in (8) and choose any 0< α< α. Define

U:=inf

t0: r˜_t−r <αt+1/2

wherexdenotes the greatest integer less than or equal tox.

DefineV as the first time one of the particles initially strictly to the left ofrhits{x: x >αt +r}, V :=inf

t0: sup

x>αt+r

φ¯_w₀(x, t ) >0

.

U andV in[0,∞]are stopping times with respect to{ Ft: t0}. Let

D:=min{U, V}. (15)

Note that when{D= ∞}, sincer_t r˜_t, the jumps in the frontr_t are never produced by particles initially strictly to the left ofr. For eachy∈Z, let

T_y:=inf{t0:r_ty}

denote the hitting time ofy by the rightmost visited site in the labeled processw_t. We will also need the first timeUandV happen after times0,

U◦θ_s:=inf

t0: r˜_t^r^s−r_s<αt+1/2

, (16)

V ◦θ_s:=inf

t0: sup

x>αt+rs

¯

φ_w_s(x, t ) >0

, (17)

andD◦θ_s:=min{U◦θ_s, V ◦θ_s}.

Choose an integerL:=M and define sequences{S_k: k0}and{D_k: k1}of Ft-stopping times as follows.

Start withS₀=0 andR₀=r. Then define

S₁:=T_R₀₊_L, D₁:=D◦θ_S₁+S₁, R₁:=r_D₁, and, fork1,

S_k₊₁:=T_R_k₊_L, D_k₊₁:=D◦θ_S_k₊₁+S_k₊₁, R_k₊₁:=r_D_k₊₁.

Note that these times are not necessarily finite and we make the convention thatr_∞= ∞. Similarly, we defineU_k:=

U◦θ_S_k+S_kandV_k:=V ◦θ_S_k+S_kfork1. Let K:=inf

k1: S_k<∞, D_k= ∞ , and define theregeneration time,

κ:=S_K. (18)

Note thatκ isnota stopping time with respect toFt.

Denote byGthe information up to timeκ, defined as the completion with respect toPwof the smallestσ-algebra containing all sets of the form{κt} ∩A,A∈ Ft.

In Section 5, we will show thatEw[κ²]<∞and hence, in particular

Pw[κ <∞] =1. (19)

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Newly created particles (and the ones they will further create) start afresh from regeneration times, ignoring all other active particles.

Proposition 1.LetAbe a Borel subset ofD([0,∞);Ω). Then, Pw

τ₋_r_κζ (κ+ ·)∈A|G

=P(a−1)δ₀

η(·)∈A|U= ∞ ,

where(a−1)δ0denotes a configuration witha−1particles at0and none anywhere else.

Proof. We have to show that for anyB∈G, Pw

B,

τ₋_r_κζ (κ+ ·)∈A

=Pw[B]P(a−1)δ0

η(·)∈A|U= ∞

. (20)

Now, using (19), Pw

B,

τ₋_r_κζ (κ+ ·)∈A

=Pw

{κ <∞}, B,

τ₋_r_κζ (κ+ ·)∈A

=^∞

k=1

Pw

{S_k<∞, D_k= ∞}, B, τ₋_r_κζ (κ+ ·)∈A

= ∞ k=1

x∈Z

Pw

rS_k=x, Sk<∞, Dk= ∞, B, τ₋xζ (Sk+ ·)∈A

. (21)

From the definition ofGthere is an eventBk∈ FS_ksuch thatBk=Bonκ=Sk. Therefore, we can continue developing (21) to obtain,

= ∞ k=0

x∈Z

Pw

rS_k=x, Sk<∞, Dk= ∞, Bk, τ₋xζ (Sk+ ·)∈A

=

k,x

Ew

1r_Sk=x,Sk<∞,BkPw

D_k= ∞, τ₋_xζ (S_k+ ·)∈A| FSk

,

whereEw is the expectation defined byPw. But on the eventsS_k<∞andr_S_k =x, we have by parts (i) and (ii) of Lemma 4, that

ζ_w(S_k+ ·)=η_(a₋_1)δ_x(·) (22)

whenUk=Vk= ∞, and thatη(a−1)δ_x(·)is independent of the configuration of particles initially to the left ofx. Here, (a−1)δx, is the configuration with(a−1)particles at sitex with labels(x,1), . . . , (x, a−1)and none elsewhere.

Indeed, part (i) of Lemma 4, and the eventV_k= ∞, imply that the particles with initial positionszto the left ofx, are never at the right ofαt +x. And part (ii) of Lemma 4, and the eventU_k= ∞, imply that the frontr_t is never to the left ofαt+1/2 +x. Hence, there is no effect of the particles initially to the left ofx on the frontr_t, so thatζ_w(S_k+ ·)=η_(a₋_1)δ_x(·). Then, (22) combined with the independence ofU_k andV_k givenFSk by part (iii) of Lemma 4, the translation invariance, and the strong Markov property imply that on the eventsS_k<∞andr_S_k=x,

Pw

Uk= ∞, Vk= ∞, τ₋xζ (Sk+ ·)∈A| FS_k

=Pw

Uk= ∞, τ₋xη(a−1)δ_x(·)∈A| FS_k

Pw[Vk= ∞| FS_k]

=P(a−1)δ0

U= ∞, η(·)∈A

Pw[V_k= ∞| FSk]. Summarizing, we have,

Pw[κ <∞]Pw

B, τ₋r_κζ (κ+ ·)∈A

=P(a−1)δ₀

U= ∞, η(·)∈A

k,x

Pw[V_k= ∞, r_S_k =x, S_k<∞, B_k]. (23) LettingA=Ωgives

Pw[κ <∞]Pw[B] =P(a−1)δ0[U= ∞]

k,x

Pw[V_k= ∞, r_S_k =x, S_k<∞, B_k]. (24) (23) and (24) together imply (20). 2

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Now defineκ1κ2· · ·byκ1:=κ and forn1

κn+1:=κn+κ(wκ_n+·). (25)

whereκ(w_κ_n_+·)is the regeneration time starting fromw_κ_n_+·and we setκ_n₊₁= ∞onκ_n= ∞forn1. We will call κ₁thefirst regeneration timeandκ_nthenth regeneration time.

For eachn1 theσ-algebra,Gn will be the completion with respect toPwof the smallestσ-algebra containing all sets of the form{κ₁t₁} ∩ · · · ∩ {κ_nt_n} ∩A,A∈Ftn. ClearlyG1=G.

Lemma 5.{U= ∞} ∈G1.

Proof. Note that{κ₁= ∞}is a null event forPwand hence, sinceG1is complete, it is enough to show that{U <∞}∩

{κ₁<∞} ∈G1.

For notational convenience, we writer˜_·^kinstead ofr˜_·^r^Sk. Note that wheneverU <∞,S_k<∞andr˜_U > r_S_k happen, for some k1, then necessarily κ₁> S_k. Indeed, from the observation thatr˜_S_k_+·= ˜r_·^k, we see that if the events U <∞,S_k<∞andr˜_U r_S_k happen, then we must have thatU_k<∞, and henceD_k<∞. By summing over the intersection with{κ₁=S_k}we see that it follows that{U <∞} ∩ {κ₁<∞} ∩ {˜r_U > r_κ₁} = ∅. So

{U <∞} ∩ {κ₁<∞} = {˜r_Ur_κ₁} ∩ {κ₁<∞}. (26)

Sincer˜Urt implies thatU <∞, it follows that {˜r_U> r_κ₁} ∩ {˜r_Ur_t} ∩ {κ1t}

is empty. Hence

{˜r_Ur_t} ∩ {κ₁< t} = {˜r_Ur_κ₁} ∩ {˜r_Ur_t} ∩ {κ₁< t} = {˜r_Ur_κ₁} ∩ {κ₁< t}. Thus

{˜r_Ur_κ₁} ∩ {κ₁<∞} =^∞

n=1

{˜r_Ur_n} ∩ {κ₁< n}. (27)

The result then follows from the fact that{˜rUrt} ∈ Ftfor eacht >0 which is a direct consequence of the construction of the processes. 2

Proposition 2.LetAbe a Borel subset ofD([0,∞);Ω). Then, Pw

τ₋_r_κnζ (κ_n+ ·)∈A|Gn

=P(a−1)δ0

η(·)∈A|U= ∞ . Proof. Letψ:D[[0,∞);S)→D[[0,∞);S)be the map given by,

ψ (w)(·)=τ₋r_κ₁w κ1

w(·) + ·

. (28)

Then note that, Gk+1 is generated byG1 andψ⁻¹(Gk), and that theσ-algebrasG1 andψ⁻¹(Gk)are independent.

The proof of this theorem now follows from the above observations, induction onn∈Z⁺ using Proposition 1, and Lemma 5. 2

We can now describe the full renewal structure.

Proposition 3. Let w∈S.(i) Under Pw,κ1, κ2−κ1, κ3−κ2, . . . are independent, and κ2−κ1, κ3−κ2, . . . are identically distributed with law identical to that ofκ1 underP(a−1)δ₀[ · |U= ∞].(ii)UnderPw,r_·∧κ₁, r(κ₁+·)∧κ₂− rκ₁, r(κ₂+·)∧κ₃−rκ₂, . . .are independent, andr(κ₁+·)∧κ₂−rκ₁, r(κ₂+·)∧κ₃−rκ₂, . . .are identically distributed with law identical to that ofrκ₁underP(a−1)δ₀[ · |U= ∞].

Proof. This follows directly from Proposition 2. 2

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5. Expectations and variances of the regeneration times 5.1. Estimates for the auxiliary process

In this subsection we obtain some estimates for the auxiliary process. The process will always start from(a−1)δ0

and we denote the corresponding measure on the trajectories of the auxiliary process byP, and expectations byE.

Lemma 6.For1p < M/2andjM, there exists a constantC=C(p) <∞such that E

|νj|^p

< C. (29)

Proof. ForjMthe auxiliary process has its full complement ofMparticles attempting to hitj+1.ν_j is then the minimum ofγ₁, . . . , γ_M which are the hitting times of 1 ofMrandom walks starting in{−M/ax0}. Standard estimates for random walks giveP[γ_i> t]Ct⁻^1/2for someC <∞. HenceP[ν_j> t]C^Mt⁻^M/2. 2

Lemma 7.For1p < M/2, there exists a constantC=C(p) <∞such that for all initial conditionsw, and all t >0

P[t < U <∞]Ct⁻^(p/2). (30)

Proof. By translation invariance we can assumer=0. Lett1be such thatαt1+1/2 =M. Then, whentt1, we have that

P[t < U <∞]P

˜ r_t⁰

1M,

s>t

˜ r_s⁰<

αs+1

2 . (31)

But, ifr˜_s⁰<αs+1/2for s > t, thenαs+1/2

j=1 ν_j> s, which in turn implies that, ¹_nn

j=1ν_j _α¹(1−_2n¹)for somenαt+1/2. Similarly,r˜_t⁰

1> M, implies that_M

j=1ν_j (M+1)/α. Therefore, wheneverαt+1/2 M+1, we have

P[t < U <∞]P _M

j=1

ν_jM+1 α ,

∞ n=αt+1/2

1 n

n

j=1

ν_j 1 α

1− 1

2n

P

_∞

n=αt+1/2

1 n

n

j=M+1

ν_j 1 α

1−M+1/2 n

. (32)

Now remark that forj M, the random variablesν_j are identically distributed and have finite moments of order p < M/2 by Lemma 6. Each has expected value 1/α. Defineγ_j=ν_j−1/α. Chooset₂as any real number such that

β= 1 α

1−M+1/2 αt2

−1 α>0

andαt₂+1/2M+1. Then, whenevertt₂, ifN= αt+1/2, we have that, P[t < U <∞]P

sup

nN

1 n

n

j=M+1

γ_jβ

. (33)

Recall that for each 0i < ,= M/(a−1) +1, the random variables{ν_k₊_i: k1}are independent. Observe from (33) that, ifβ=βa/M,

P[t < U <∞]

−1

i=0

P

sup

nN

1 n

n

j=M+1, j=k+i

γ_jβ

. (34)

SupposeX₁, X₂, . . .are independent and identically distributed random variables with mean 0. By Kolmogorov’s inequality,

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P

sup

nN

1 n

n

i=1

Xi

∞ k=0

P

sup

2^kNn2^k+1N

1 2^kN

n

i=1

Xi

^∞

k=0

2^kN ₋p

E

2^k⁺¹N i=1

X_i

p

. (35)

Now, forp2, if E[|Xi|^p]<∞ thenE[|2^k⁺¹N

i=1 Xi|^p]C(2^k⁺¹N )^p/2 for someC <∞(see item 16, page 60 of [10]), and hence for anotherC <∞,

P

sup

nN

1 n

n

i=1

X_i

C⁻^pN⁻^p/2. (36)

Applying (36) to (34), by Lemma 6 we obtain (30). 2 5.2. Estimates for the enlarged process

We start with a few standard estimates for hitting times of random walks.

Lemma 8.Let{X_t: t0}be a continuous time simple symmetric random walk onZ, of total jump rate2, starting fromx0,c >0, and

τ :=inf

t0: Xt>ct

. (37)

Then

P[τ= ∞]

1−exp{xθ_c}, x−1, exp{−2/c}(1−exp{−θc}), x=0,

whereθ_c>0is the nonzero solution ofcθ−2(coshθ−1)=0, and there exist0< C, C<∞such that P[t < τ <∞]< Cexp{−Ct}

exp

−θc|x+1|

2 +exp

−I

|x+1|

2t , (38)

whereI (u)=2+usinh⁻¹(u/2)−√

4+u²is the rate function for the random walk.

Proof. Forθ∈R, exp{θ X_t−2(coshθ−1)t}is a martingale. By the optional stopping time theorem, E

exp

θ Xτ∧n−2(coshθ−1)τ∧n

=exp{θ x}. (39)

Now,Xτ∧ncτ ∧n +1, henceθ Xτ∧n−2(coshθ−1)τ∧n(cθ−2(coshθ−1))τ∧n+θ. It follows that if θθc, we can apply the bounded convergence theorem in equality (39) taking the limit whenn→ ∞, to conclude that,

E

1(τ <∞)exp

θ X_τ−2(coshθ−1)τ

=exp

θ (x−1)

. (40)

Whenτ <∞, we haveX_τ= cτ +1 and thereforeθ X_τ−2(coshθ−1)τ(cθ−2(coshθ−1))τ. Lettingθ↓θ_c in (40) and applying the bounded convergence theorem, we obtain,

P[τ <∞]exp{xθ_c}. (41)

To get the bound for x=0, note that the probability that the random walk does not move before timet=1/c is exp{−2/c}. Then use the Markov property and the result forx= −1.

To prove (38):

P[t < τ <∞]P[Xt> B] +P[t < τ <∞, XtB]. (42) Now, wheneverB−x >0,

P[X_t> B]exp

−t I

B+ |x| t

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and by the strong Markov property, translation invariance and inequality (41), P[t < τ <∞, X_tB]P₋₍_ct₋_B)[τ <∞]exp

−

ct −B θ_c

.

ChoosingB=(ct +x)/2, on the above inequalities, using the inequalityt I (u+v)I (v)+(t−1)I (u)valid for u, v0, and substituting back in (42), gives (38). 2

Lemma 9.For eachα< α, there existsC <∞such that fort1, and allw∈S,

Pw[t < V <∞]Cexp{−Ct}. (43)

Proof. Note that in the worst case in which there areMrandom walks at the origin and at each site to the left of the origin, we get the bound,

Pw[t < V <∞]M

−∞

x=0

P_x[t < τ <∞],

whereτ is defined in display (37). On the other hand, it is true that_∞

k=1e⁻^{I (k/(n}⁺¹⁾⁾(n+1)_∞

k=0e⁻^{I (k)}. This estimate, the previous inequality and inequality (38) of Lemma 8 give us the result. 2

From Lemmas 7 and 9 we have

Corollary 1.For each 0p < M/2 there is a C <∞ depending only onp, M and α such that for all initial conditionsw,

Pw[t < D <∞]Ct⁻^p/2. Lemma 10. There is aδ1>0such that,

Pw[V <∞]<1−δ1.

Proof. Without loss of generality, r=0. Now, take the worst case scenario wherew hasM particles at each site x0. By Lemma 8,

Pw[V = ∞]e⁻^2M/α

1−e⁻^θ^cM∞ n=1

1−e⁻^nθ^cM

=δ₁>0. 2

Lemma 11. Suppose thatM >4. There is aδ₂>0such that for all initial conditionswwith at leasta−1particles at the rightmost visited siter

Pw[U <∞]<1−δ2.

Proof. We can also assume thatr=0. To estimatePw[U= ∞]below, note that Pw[U= ∞] =P

_∞

k=1

_k

j=1

νj2k−1 2α

. (44)

Letn∈Nand 0< <1/(2α)and defineGto be the event that each random walkY_(x,i) with 0xn, moves M+1 steps to the right before time. WhenG happens,νk<1/(2α)and hencek

j=1νj < k/(2α)for allk n+ M/a :=n. Note thatG∩_∞

k=1{_k

j=1ν_j< (2k−1)/α} ⊃G∩H where

H:= ^∞

k=n+1

_k

j=n+1

ν_j2k−1 2α −n

. (45)