Spatial Epidemics and Local Times for Critical Branching Random Walks in Dimensions 2 and 3

0.29 MB·English

by  Steven P. Lalley

SPATIAL EPIDEMICS AND LOCAL TIMES FOR CRITICAL BRANCHING RANDOM
WALKS IN DIMENSIONS 2 AND 3
9
0
STEVENP.LALLEYANDXINGHUAZHENG
0
2

n
a
J ABSTRACT. ThebehavioratcriticalityofspatialSIR(susceptible/infected/recovered)epidemicmod-

2 elsindimensionstwoandthreeisinvestigated.Inthesemodels,finitepopulationsofsizeN aresitu-
atedattheverticesoftheintegerlattice,andinfectiouscontactsarelimitedtoindividualsatthesame

] oratneighboringsites.Susceptibleindividuals,onceinfected,remaincontagiousforoneunitoftime
R
and then recover, after which they are immune to further infection. It isshown that the measure-
P
valued processes associated with these epidemics, suitably scaled, converge, in the large-N limit,
.
h eithertoastandardDawson-Watanabeprocess(super-Brownianmotion)ortoaDawson-Watanabe
t processwithlocation-dependentkilling,dependingonthesizeofthetheinitiallyinfectedset.Akey
a
elementoftheargumentisaproofofAdler’s1993conjecturethatthelocaltimeprocessesassociated
m
withbranchingrandomwalksconvergetothelocaltimedensityprocessassociatedwiththelimiting
[ super-Brownianmotion.

1
v 1. INTRODUCTION
6
4 1.1. Spatial SIR epidemics. Simple spatial models of epidemics are known to exhibit critical
2
thresholdsinonedimension: Roughly,whenthedensityoftheinitiallyinfectedsetexceedsacertain
0
. level, theepidemic evolves inamarkedly different fashion thanitsbranching envelope. SeeLalley
1
(2007)foraprecisestatement,andAldous(1997),Martin-Löf(1998),andDolgoarshinnykh andLalley
0
9 (2006)for analogous results in the simpler setting of mean-field models. Themain purpose of this
0 article is to show that spatial SIR epidemics (SIR stands for susceptible/infected/recovered) in di-
:
v mensionstwoandthreealsoexhibitcriticalthresholds.
i Theepidemic models studied here take place inpopulations ofsize N located at thesites ofthe
X
integer lattice Zd in d dimensions. Each of the N individuals at a site x Zd may at any time be
r ∈
a either susceptible, infected, orrecovered. Infected individuals remaininfected foroneunitoftime,
and then recover, after which they are immune to further infection. The rules of infections are as
follows: ateachtimet = 0,1,2,...,foreachpair(i ,s )ofaninfectedindividuallocatedatxand
x y
asusceptibleindividualaty,i infectss withprobabilityp (x,y).Weshallonlyconsiderthecase
x y N
where the transmission probabilities p (x,y) are spatially homogeneous, nearest-neighbor, and
N
symmetric, and scale with the village size N in such away that the expected number ofinfections
byacontagious individual inanotherwise healthy population is1(sothattheepidemic iscritical),
thatis,
Assumption1. p (x;y) = 1/[(2d+1)N]if y x 1;and=0otherwise.
N
| − |≤
Ourmain result, Theorem 2below, asserts that under suitable hypotheses on the initial configu-
rationsofinfectedindividuals, thecriticalspatialSIR depidemiccanberescaledsoastoconverge
−
Date:January5,2009.
1
2 STEVENP.LALLEYANDXINGHUAZHENG
to aDawson-Watanabe measure-valued diffusion in both d = 2and d = 3. Depending on the size
oftheinitially infected set, thelimiting Dawson-Watanabe process haseitherapositive killing rate
or no killing at all. The analogous result for d = 1 was proved in Lalley (2007), using the fact
that one-dimensional super-Brownian motion (the Dawson-Watanabe process with no killing) has
samplepathsinthespaceofabsolutelycontinuousmeasures. Inhigherdimensionsthisisnolonger
true,soadifferent strategyisneeded.
1.2. Branchingenvelopeofaspatialepidemic. ThespatialSIRepidemicinddimensionsisnatu-
rallycoupledwithanearestneighborbranching randomwalkontheintegerlatticeZd;thisbranch-
ing random walk is often referred to as the branching envelope of the epidemic. Particles of this
branching random walk represent infection attempts in the coupled epidemic, some of which may
fail to be realized in the epidemic because the targets of the attempts are either recovered or are
targets of other simultaneous infection attempts. The branching envelope evolves as follows: Any
particle located at site x at time t lives for one unit of time and then reproduces, placing random
numbers ξ of offspring at the sites y such that y x 1. The random variables ξ are i.i.d.,
y y
| − | ≤
withBinomial(N,1/[(2d +1)N]) distributions. Denote this reproduction rule by , and denote
N
R
by thecorresponding offspring lawinwhichtheBinomialdistribution isreplaced bythePois-
R∞
son distribution with mean 1/(2d + 1). Since offspring are placed independently at each of the
2d+1nearestneighbors, theexpectedtotalnumberofoffspringofaparticleis1,i.e.thebranching
random walk is critical. Moreover, under either reproduction rule or , particle motion is
N
governed bythe law ofthe simple nearest neighbor random walk1 oRn Zd (wRit∞h holding probability
1/(2d + 1)): In particular, given that a particle at site x has k offspring, each of these offspring
independently choosesaneighboring sitey according tothelaw
(1) P (x,y) = 1/(2d+1) for y x 1.
1
| − |≤
Note that the covariance matrix of the increment has determinant σ2d, where σ2 is the variance
parameter ofthejumpdistribution, definedby
2
(2) σ2 := .
2d+1
(cid:18) (cid:19)
ThespatialSIR-depidemiccanbeconstructedtogetherwithitsbranchingenvelopeonacommon
probabilityspaceinsuchawaythatthebranchingenvelopedominatestheepidemic,thatis,foreach
time n and each site x the number of infected individuals at site x at time n is no larger than the
number ofparticles in the branching envelope. Theconstruction, inbrief, is asfollows (see Lalley
(2007)): Particles of the branching random walk will be colored either red or blue according to
whether or not they represent infections that actually take place, with red particles representing
actual infections. Initially, all particles are red. At each time t = 0,1,2,..., particles produce
offspring at the same or neighboring sites according to the law described above. Offspring
N
R
of blue particles are always blue, but offspring of red particles may be either red or blue, with the
choices made according to the following procedure: All offspring of red particles at a location y
choose numbers j [N] := 1,2,...,N at random, independently of all other particles. If a
∈ { }
particle chooses anumber j thatwaspreviously chosen byaparticle ofanearlier generation atthe
samesitey,thenitisassignedcolorblue. Ifk > 1offspringofredparticleschoosethesamenumber
1Throughoutthepaper,thetermsimplerandomwalkwillmeansimplerandomwalkwithholdingprobability1/(2d+
1).
BRWLOCALTIME 3
jatthesametime,andifjwasnotchoseninanearliergeneration,then1oftheparticlesisassigned
color red, whiletheremaining k 1areassigned color blue. Underthis rule, thesubpopulation of
−
redparticles evolvesasanSIR depidemic.
−
Itisapparentthatwhenthenumbersofinfectedandrecoveredindividualsatasiteanditsnearest
neighbors are small compared to N, then blue particles will be produced only infrequently, and
so the epidemic process will closely track its branching envelope. Only when the sizes of the
recovered and infected sets reach certain critical thresholds will blue particles start to be produced
inlargenumbers,atwhichpointtheepidemicwillbegintodivergesignificantlyfromthebranching
envelope. Our main result, Theorem 2 below, implies that the critical threshold for the number of
initially infectedindividuals isontheorderN1/(3 d/2).
−
TheSIR-depidemicisrelatedtoitsbranchingenvelopeinasecond—andforourpurposesmore
important –way. Thelawoftheepidemic, asaprobability measureonthespace ofpossible popu-
lation trajectories, is absolutely continuous relative to the law of its branching envelope. The like-
lihood ratio can be expressed as aproduct over time and space, with each site/neighbor/generation
contributing afactor (see §3.3below). Eachsuchfactor involves thetotal occupation timeRN(x)
n
of the site, that is, the sum of the number of particles at site x over all times prior to n. Thus, the
asymptoticbehavioroftheoccupationtimestatisticsforbranchingrandomwalkswillplayacentral
roleintheanalysis ofthelarge-N behavior oftheSIR-depidemic.
1.3. Watanabe’s Theorem. A fundamental theorem of Watanabe (1968) asserts that, under suit-
able rescaling (the Feller scaling) the measure-valued processes naturally associated with critical
branching random walks converge to a limit, the standard Dawson-Watanabe process, also known
assuper-Brownian motion.
Definition 1. The Feller-Watanabe scaling operator scales mass by 1/k and space by 1/√k,
k
thatis,foranyfiniteBorelmeasureµ(dx)onRd andaFnytestfunction ψ C (Rd),
∈ c∞
(3) ψ, µ = k 1 ψ(√kx)µ(dx).
k −
h F i
Z
Watanabe’s Theorem . Fix N, and for each k = 1,2,..., let Xk be a branching random walk
t
withoffspringdistribution andinitialparticleconfiguration Xk. (Inparticular, Xk(x)denotes
RN 0 t
the number of particles at site x Zd in generation [t], and Xk is the corresponding counting
∈ t
measure.) Iftheinitialmassdistributions converge, afterrescaling, ask ,thatis,if
→ ∞
(4) Xk µ = X
Fk 0 ⇒ 0
for some finite Borel measure µ on Rd, then the rescaled measure-valued processes ( Xk)
k kt
F
converge inlawask :
→ ∞
( Xk) X ,
k kt t
F ⇒
where represents theweakconvergence relative totheSkorokhod topology on
D([0, ⇒);M (Rd)). The limit is the standard Dawson-Watanabe process X (super-Brownian
F t
∞
motion) with variance parameter σ2 (equivalently, standard super-Brownian motion run at speed
σ).
SeeEtheridge(2000)foranin-depthstudyoftheDawson-Watanabeprocessandadetailedproof
ofWatanabe’sTheorem. BecausetheprocessX hascontinuous samplepathsinthespaceoffinite
t
4 STEVENP.LALLEYANDXINGHUAZHENG
Borel measures, it follows routinely from Watanabe’s theorem that the occupation measures for
branching random walksconverge tothoseofsuper-Brownian motion:
Lemma1. Thefollowing jointconvergence holds:
t t
(5) ( Xk) , ( Xk) ds X , X ds ,
k kt k ks t s
F F ⇒
(cid:18) (cid:18)Z0 (cid:19)(cid:19) (cid:18) Z0 (cid:19)
where representsweakconvergencerelativetotheSkorokhodtopologyonD([0, );M (Rd))2.
F
⇒ ∞
Proof. TheDawson-WatanabeprocessX hascontinuoussamplepathsinD([0, );M (Rd)),see,
t F
∞
t
e.g.,Proposition2.15inEtheridge(2000). Thefunctional(X ) ( X ds)iscontinuousrelative
t 7→ 0 s
to the Skorokhod topology on the subspace of continuous measure-valued processes, so the result
R
followsfromWatanabe’stheorem andthecontinuous mappingprinciple. (cid:3)
1.4. Local times of critical branching random walks. In dimension d = 1 the super-Brownian
motion has sample paths in the space of absolutely continuous measures, that is, for each t > 0
the random measure X is absolutely continuous relative to Lebesgue measure (KonnoandShiga
t
(1988),Reimers(1989)). Moreover,theRadon-NikodymderivativeX(t,x)isjointlycontinuousin
t,x (for t > 0). It is shown in Lalley (2007) that if a sequence of branching random walks satisfy
theassumptionsinWatanabe’sTheorem,thenthedensityprocessesassociatedwiththosebranching
random walks, under some smoothness assumptions on the initial configurations and after suitable
scaling, converge tothedensityprocessofthelimitingsuper-Brownian motion.
In dimensions d 2 the measure X is almost surely singular (DawsonandHochberg (1979)).
t
≥
Therefore, one cannot expect the convergence of density processes as in Lalley (2007). We shall
prove, however, that the occupation measures of critical branching random walks have discrete
densities thatconverge weakly—seeTheorem1below. Thelimitprocessisthelocaltimeprocess
associated withtheoccupation measure
t
L := X ds
t s
Z0
of the super-Brownian motion. Indimensions d = 2,3, the random measure L is, for each t > 0,
t
absolutely continuous, despite the factthat X issingular —see Sugitani (1989), Iscoe (1986)and
t
Fleischmann (1988). Moreover, under suitable hypotheses on the initial condition X , the density
0
processL (x)isjointlycontinuousfort > 0andx Rd: ThisisthecontentofSugitani’s theorem.
t
For the reader’s convenience, we state Sugitani’s T∈heorem precisely here. For t > 0 and x Rd,
∈
set
t e x2/2t
−| |
q (x) = φ (x)ds, where φ (x) =
t s t
(2πt)d/2
Z0
istheusualheatkernel.
Sugitani’s Theorem . Assume that d = 2 or 3, and that the initial configuration µ := X of the
0
super-Brownian motionX issuchthattheconvolution
t
(6) (q µ)(x)isjointlycontinuous int 0andx Rd.
t
∗ ≥ ∈
Then for each t 0 the occupation measure L is absolutely continuous, and there is a jointly
t
≥
continuous versionL (x)ofthedensity process.
t
BRWLOCALTIME 5
Wecall(L (x)) thelocaltimedensityprocess associated withthesuper-Brownian mo-
t t 0,x Rd
tion. In view of W≥atan∈abe’s and Sugitani’s theorems, it is natural to conjecture (see Remark 1
below) that the local time density processes of branching random walks, suitably scaled, converge
to the local time density process of the super-Brownian motion. Theorem 1 below asserts that this
conjecture istrue. LetXk beasequence ofbranching randomwalksonZd. Write
(7) Xk(x) := #particlesat xattimei, and
i
Rk(x) := Xk(x).
n i
i<n
X
(We use the notation Rk instead of Lk because in the corresponding spatial epidemic model, the
n n
quantity Rk(x)represents thenumberofrecovered individuals atsitexandtimen.) Denoteby
n
(8) Pn = (Pn(x,y))x,y∈Zd = (Pn(y−x))x,y∈Zd
thetransition probability kernelofthesimplerandomwalkonZd,thatis,Pn = P Pn 1 isthenth
−
∗
convolution power of the one-step transition probability kernel given by (1). Let G (x,y) be the
n
associated Green’sfunction:
G (x) := P (x).
n i
i<n
X
ForanyfinitemeasureµonZd withfinitesupport, set
(µG )(x) := (µ G )(x) = µ(y)G (x y),
n n n
∗ −
y
X
anddenotebyµG (y)thecontinuous extension to[0, ) Rd bylinearinterpolation.
t
∞ ×
Theorem 1. Assume that d = 2 or d = 3. For each k = 1,2,..., let Xk be a branching random
t
walk whose offspring distribution is Poisson with mean 1. Assume that the initial configurations
µk := Xk satisfy hypothesis (4) of Watanabe’s theorem, where the limit measure µ has compact
0
support andsatisfiesthehypothesis (6)ofSugitani’s theorem. Assumefurtherthat
µkG (√kx)
(9) k2ktd/2 =⇒ [(qσ2t∗µ)/σ2](x),
−
where indicates weakconvergence inthetopology ofD([0, ),C (Rd)). Thenask ,
b
⇒ ∞ → ∞
Rk (√kx)
(10) kt = L (x),
t
k2 d/2 ⇒
−
whereL (x)isthelocaltimedensityprocessassociatedwiththesuper-Brownian motionwithvari-
t
anceparameterσ2 startedintheinitialconfiguration X = µ.
0
Theorem1willbeprovedin§2.
Remark1. TheanalogousresultforcriticalbranchingBrownianmotionswasconjectured byAdler
(1993),whoprovedthemarginal convergence foranyfixedtandx.
Remark 2. The assumption that the offspring distribution is Poisson with mean 1 can be relaxed.
All that is really needed is that the offspring distribution has an exponentially decaying tail. See
Remark7.
6 STEVENP.LALLEYANDXINGHUAZHENG
Remark 3. Thehypothesis (4)does notbyitself imply(9),evenifthelimitmeasure µsatisfies the
hypothesis (6)ofSugitani’stheorem. Sufficientconditions for(9)aregiveninProposition 1below.
In particular, in dimension 2, if (4) holds and the maximal number of particles on a single site is
bounded ink,then(9)issatisfied.
Remark 4. Let X be super-Brownian motion in dimension d = 2. For each t > 0 the random
t
measure X is singular, so by Fubini’s theorem, for almost every point x R2 the set of times
t
∈
t > 0suchthatxisapointofdensity ofX hasLebesguemeasure0. Underhypothesis (9)wecan
t
makeananalogousquantitativestatementforbranchingrandomwalk: Foranyfixedx Z2andall
∈
t > 0,
[kt]
(11) E I = O(k/logk).
Xk(x)>0
m
m=1
X
Proof. ByProposition 35 in LalleyandZheng (2007), there exists δ > 0such that for all k and m
sufficiently large,
E Xk(x)Xk(x) > 0 δlogm.
m | m ≥
Buthypothesis (9)impliesthat h i
ERk (x) = (µkG )(x) = O(k),
kt kt
and
ERk (x) = EXk(x) = E Xk(x)Xk(x)> 0 P Xk(x) > 0 .
kt m m | m · m
mX<kt mX<kt h i h i
(cid:3)
1.5. Scalinglimitofspatial SIRepidemic. Beforestating ourresult, wefirstrecall thedefinition
ofDawson-Watanabe processes withvariable-rate killing. TheDawson-Watanabe process X with
t
killingrateθ = θ(x,t,ω)(assumedtobeprogressively measurableandjointlycontinuousin(t,x))
and variance parameter σ2 can be characterized by a martingale problem (DawsonandPerkins
(1999),§6.2): Foranytestfunction ψ C2(Rd),
∈ c
σd t t
X ,ψ X ,ψ X ,∆ψ ds+ X ,θ(,s)ψ ds
t 0 s s
h i−h i− 2 h i h · i
Z0 Z0
is a martingale with the same quadratic variation as for super-Brownian motion with variance pa-
rameter σ2. The Dawson-Watanabe process with killing rate 0 (which we sometimes refer to as
the standard Dawson-Watanabe process) is super-Brownian motion. Existence and distributional
uniqueness of Dawson-Watanabe processes in general is asserted in DawsonandPerkins (1999)
and proved, in various cases, in Dawson(1978) and EvansandPerkins (1995). Itis also proved in
thesearticlesthatthelawofaDawson-Watanabeprocesswithkillingonafinitetimeintervalisab-
solutelycontinuouswithrespecttothatofastandardDawson-Watanabeprocesswiththesamevari-
ance parameter, and that the likelihood ratio (Radon-Nikodym derivative) is (DawsonandPerkins
(1999))
1
(12) exp θ(t,x)dM(t,x) X ,θ(t, )2 dt ,
t
− − 2 h · i
(cid:26) Z Z (cid:27)
BRWLOCALTIME 7
where dM(t,x) is the orthogonal martingale measure attached to the standard Dawson-Watanabe
process (see Walsh (1986)). Absolute continuity implies that sample path properties are inherited:
In particular, when d = 2,3, if X is a Dawson-Watanabe process with killing, then almost surely
t
its occupation time process L is absolutely continuous, with local time density L (x) jointly con-
t t
tinuousinxandt.
ItisshowninLalley(2007)thatfortheSIR-1epidemicinZwithvillagesizeN,theparticleden-
sityprocesses, suitablyrescaled,convergeasN tothedensityprocessofastandardDawson-
→ ∞
Watanabe process or a Dawson-Watanabe process with location-dependent killing, depending on
whether the total number of initial infections is below a critical threshold or not. In dimensions
d 2, one cannot expect such a result to hold, because the Dawson-Watanabe process is almost
≥
surely singular with respect to the Lebesgue measure and therefore has no associated density pro-
cess. However,asmeasure-valuedprocesses,theSIR-d(d= 2,3)epidemics,undersuitablescaling,
doconverge, asthenexttheoremasserts. FortheSIR-dmodelwithvillagesizeN,define
(13) XN(x) : = #infected particlesat xattime i;
i
RN(x) : = #recoveredparticles at xattimen= XN(x).
n i
i<n
X
Theorem 2. Assume that d = 2 or 3, and suppose that for some α 1/(3 d/2) the initial
≤ −
configurations µN := XN aresuchthat
0
(14) µN µwithcompactsupport, and
Nα
F ⇒
(15) ((µNGNαt)(√Nαx))/Nα(2−d/2) [(qσ2t µ)/σ2](x) C(R1+d)
⇒ ∗ ∈
wherethesecond convergence isinD([0, );C (Rd)). Then
b
∞
(16) ( XN) = X
Nα Nαt t
F ⇒
where the limit process X is a Dawson-Watanabe process with initial configuration X = µ,
t 0
variance parameterσ2,andkillingrateθ. Thekillingratedepends onthevalueofαasfollows:
(i) ifα < 1/(3 d/2),thenθ 0; and
− ≡
(ii) ifα = 1/(3 d/2),thenθ = L (x),
t
−
whereL (x)isthelocaltimedensityoftheprocessX . Theconvergence in(16)isweakconver-
t t
gencerelativetotheSkorokhod topology onD([0, );M (Rd)). ⇒
F
∞
Theorem2willbeprovedin§3.
Remark5. Theorem2assertsthatthereisacriticalthresholdfortheSIR depidemicindimensions
−
d = 2,3: Below the threshold (when the sizes of the initially infected populations are Nα∗,
≪
whereα = 1/(3 d/2)isthecritical exponent) theeffectoffinitepopulation sizeisnotfelt,and
∗ −
theepidemiclooksmuchlikeitsbranching envelope. Atthecriticalthreshold, thefinite-population
effects begin to show, and the epidemic now looks like a branching random walk with location-
dependent killing.
Remark6. ThecriticalbehavioroftheSIR-depidemicsindimensionsd 4isconsiderablysimpler.
≥
Itcanbeshownthatinalldimensionsd 5,thecriticalthresholdisα = 1;inthefourdimensional
≥
case, the critical threshold is still α = 1 but there will be logarithmic corrections – we thank Ed
Perkinsforpointing thisouttous.
8 STEVENP.LALLEYANDXINGHUAZHENG
1.6. Notational conventions. Since the proof of Theorem 2 is based on likelihood ratio calcula-
tions,weshall,attheriskofminorconfusion, usethesamelettersX andR,withsubscripts and/or
superscripts, todenote particlecounts andoccupation counts forbothbranching randomwalksand
theSIR-depidemicprocesses(seeequations(12)and(13))andfortheircontinuouslimits. Through-
out the paper, weusethe notation f g to meanthat the ratio f/g remains bounded awayfrom 0
≍
and . Also,C,C ,etc.denote generic constants whosevaluesmaychange fromlinetoline. The
1
∞
notation δ (y) is reserved for the Kronecker delta function. The notation Y = o (f(n)) means
x n P
thatY /f(n) 0inprobability; andY = O (f(n))meansthatthesequence Y /f(n)istight.
n n P n
→ | |
Finally,weusea“localscopingrule”fornotation: Anynotationintroducedinaproofislocaltothe
proof,unlessotherwise indicated.
2. LOCAL TIME FOR BRANCHING RANDOM WALK IN d= 2,3
2.1. Estimatesontransitionprobabilities. RecallthatPn = (P (x y))isthen steptransition
n
probability kernel for the simple random walk on Zd (with holding−parameter 1/−(2d + 1)). For
critical branching random walk, P (x,y) is the expected number of particles at site y at time n
n
giventhatthebranchingrandomwalkisinitiatedbyasingleparticleatsitex. Forthisreason,sharp
estimates on these transition probabilities will be ofcrucial importance in the proof of Theorem 1.
Wecollectseveralusefulestimateshere. Astheproofsaresomewhattechnical,werelegatethemto
theAppendix(section 4below). Write
Φ (x,y) = φ (x)+φ (y) where
n n n
1 x2
φ (x) = exp | |
n
(2πn)d/2 − 2n
(cid:18) (cid:19)
istheGausskernelinRd. Thefirsttworesultsrelatetransition probabilities totheGausskernel.
Lemma 2. For all sufficiently small β > 0 there exists constant C = C(β) > 0 such that for all
integers m,n 1andallx Zd,
≥ ∈
(17) P (x) Cφ (βx) and
n n
≤
(18) (P φ )(βx) Cφ (βx/2).
m n m+n
∗ ≤
Furthermore, for each A > 0 and each T > 0 there exists C = C(A,T) > 0 such that for all k
sufficiently largeandall x A√k,
| | ≤
(19) φ (βx) C P (x).
n n
≤
n kT n kT
X≤ X≤
Lemma 3. For all sufficiently small β > 0 there exists constant C = C(β) > 0 such that for all
integers n 1andallx,y Zd,
≥ ∈
x y
(20) P (x) P (y) C | − | 1 Φ (βx,βy).
n n n
| − | ≤ √n ∧ ·
(cid:18) (cid:19)
Inparticular, forallγ 1,
≤
γ
x y
(21) P (x) P (y) C | − | Φ (βx,βy).
n n n
| − | ≤ √n ·
(cid:18) (cid:19)
Ourarguments willalsorequirethefollowingestimatesonthediscretized Greenkernel.
BRWLOCALTIME 9
Lemma4. Foreachγ (0,2 d/2),β > 0,h 1,n N,andx,y Zd,define
∈ − ≥ ∈ ∈
1
(22) F (x,y;β) = F (x,y;β) = Φ (β(x+ρ),β(y +ρ)),
n,h n,h;γ lγ/2 l
ρ<hl<n
|X| X
wherethefirstsummationisoverρ Zdwith ρ < h. ThenthereexistsC =C(γ,β,d) < such
thatforalln N,h ,h 1,anda∈llx,y Z|d|,thefollowinginequalities hold: ∞
1 2
∈ ≥ ∈
(23) F (x,y;β) F (x,y;β) Cn2 (d+γ)/2F (x,y;β),
n,h1 · n,h2 ≤ − n,h1+h2−1
and
P (z) [F (x z,y z;β) F (x z,y z;β)]
i · n−i,h1 − − · n−i,h2 − −
i<n z
XX
(24) Cn2 (d+2γ)/2 Φ (β(x+ρ)/2,β(y +ρ)/2)
− l
≤
|ρ|<hX1+h2−1Xl<n
Cn2 (d+γ)/2F (x,y;β/2).
≤ − n,h1+h2−1
(Notethatinthelasttermtheβ parameterischangedtoβ/2.)
Lemma5. Foreachβ > 0,h 1,m,n N,andx Zd,define
≥ ∈ ∈
(25) J (x;β) = φ (β(x+ρ)),
m,n,h l
ρ<hm l<m+n
|X| ≤X
wherethefirst summation isover ρ Zd with ρ < h. Thenthere exists C = C(β) > 0such that
forallm,n N,h ,h 1,andall∈x Zd,th|e|followinginequalities holds:
1 2
∈ ≥ ∈
(26) J (x;β) J (x;β) Cn2 d/2J (x;β),
m,n,h1 · m,n,h2 ≤ − m,n,h1+h2−1
and
(27)
P (z) [J (x z;β) J (x z;β)] Cn2 d/2J (x;β/2).
i · m,n−i,h1 − · m,n−i,h2 − ≤ − m,n,h1+h2−1
i<n z
XX
2.2. Proof of Theorem 1. Fornotational ease, we omit the superscript k in the arguments below:
thus, we write X (x) instead of Xk(x), and R (x) instead of Rk(x). To prove the theorem it
n n n n
sufficestoprovethat(1)thesequenceofrandomprocesses(R (√kx)/k2 d/2)istightinthespace
kt −
D([0, );C (Rd));and(2)thattheonlypossibleweaklimitisthelocaltimedensityprocessL (x).
b t
These∞condoftheseiseasy,givenLemma1. Thisimpliesthatforanytestfunction ψ C (Rd),
c
∈
1
R (√kx)ψ(x)
k2 kt
x
X
1
= X (√kx)ψ(x)/k
i
k
i kt x
X≤ X
t
X (ψ)ds,
s
→L Z0
10 STEVENP.LALLEYANDXINGHUAZHENG
where(X )isthesuper-Brownian motion started inconfiguration X = µ,runatspeed σ. Hence,
t 0
any weak limit of the sequence (R (√kx)/k2 d/2) must be a density of the occupation measure
kt −
forsuper-Brownian motion. Ontheotherhand,byRemark3andSugitani’s Theorem,
t
X (ψ)ds = L (x)ψ(x)dx.
s t
Z0 Zx
ItfollowsthatL (x)istheonlypossible weaklimit.
t
Thus, to prove Theorem 1 it suffices to prove that the sequence (R (√kx)/k2 d/2) is tight in
kt −
the space D([0, );C (Rd)). In view of hypothesis (6), it is enough to prove the tightness of the
b
∞
re–centered sequence
(28) Y (t,x) := R (√kx) (µkG )(√kx) /k2 d/2.
k kt kt −
−
(cid:16) (cid:17)
This we will accomplish by verifying a form of the Kolmogorov-Centsov criterion. According to
thiscriterion, toprovetightness itsufficestoprovethatforeach compactsubset K of[0, ) Rd
∞ ×
thereexistconstants C < ,α> 0,andδ > d+1suchthatforallpairs(s,a),(t,b) K,
∞ ∈
(29) E Y (t,a) Y (t,b)α C a b δ and
k k
| − | ≤ | − |
(30) E Y (t,a) Y (s,a)α C t s δ.
k k
| − | ≤ | − |
Thetrickistonotworkwithmomentsdirectly,butinstead,followingthestrategyofSugitani(1989),
toworkwithcumulants:
Lemma 6 (Lemma 3.1 in Sugitani (1989)). Let X be a random variable with moment generating
function Eexp(θX) = exp( ∞n=1θnan). Ifforsomeinteger N thereexistsr,b > 0suchthat
P an brn, forn 2N,
| |≤ ≤
thenthereexistsC = C(b,N) > 0suchthat
EX2N Cr2N.
≤
A. Cumulants. In the following discussion we use the notation ν,f to denote the inner product
of a function f and a measure ν on Zd, and we let R be the ohccupiation measure of the nearest
n
neighbor branching random walkwithPoisson(1) offspring distribution. Bytheadditivity andspa-
tial homogeneity of the branching random walk, for any ψ C (Zd) and for each n 1 there
c
∈ ≥
existsafunctionν = νψ C (Zd)suchthatforany(nonrandom) initialconfiguration µ,
n n c
∈
Eµexp( R ,ψ ) = exp( µ,ν ).
n n
h i h i
Note that ν = ψ. The assignment ψ νψ is monotone in ψ, but not in general linear. Setting
1 n
7→
µ = δ andconditioning onthefirstgeneration, weobtain
x
j
1
exp(ν (x)) = Q exp(ψ(x+e)+ν (x+e))
n+1 j n
2d+1
!
j e
X X
where Q istheoffspringdistribution(inthecaseofinterest,thePoissondistributionwithmean1)
j
{ }
andtheinnersumisoverthe2d+1nearestneighbors eoftheorigininZd (recallthattheoriginis