Table Of ContentHarmonic deformation of Delaunay triangulations
Pablo A.Ferraria,∗,Pablo Groismana,∗,Rafael M. Grisib,1,2,∗∗
aDepartamentodeMatema´tica
2 FacultaddeCienciasExactasyNaturales
1
UniversidaddeBuenosAires
0
Pabello´nI,CiudadUniversitaria
2
C1428EGABuenosAires,Argentina.
n
a bInstitutodeMatema´ticaeEstat´ıstica
J UniversidadedeSa˜oPaulo
6 RuadoMata˜o,1010.CidadeUniversita´ria
1
Sa˜oPaulo,SP,Brasil,CEP05508-090.
]
R
P
.
h
Abstract
t
a
m
WeconstructharmonicfunctionsonrandomgraphsgivenbyDelaunaytriangulationsofergodic
[
pointprocessesas thelimitofthezero-temperatureharness process.
2
v Keywords: Harness process, Pointprocesses, Harmonicfunctionsongraphs, Corrector
7
7
6 MSC: 60F17,60G55,60K37
1
.
2
1
1. Introduction
0
1
:
v Let S bean ergodicpointprocess on Rd with intensity1 and S◦ itsPalm version. Call P and
Xi E the probability and expectation associated to S and S◦ (we think that S and S◦ are defined on
r a common probability space). The Voronoi cell of a point s in S◦ is the set of sites in Rd that
a
are closer to s than to any other point in S◦. Two points are neighbors if the intersection of the
closureoftherespectiveVoronoicellshasdimensiond−1. ThegraphwithverticesS◦andedges
given by pairs of neighbors is called the Delaunay triangulation of S◦. The goal is to construct
a function H : S◦ → Rd such that the graph with vertices H(S◦) and edges {(H(s),H(s′)), s
∗Correspondingauthor
∗∗Principalcorrespondingauthor
Emailaddresses:[email protected] (PabloA.Ferrari),[email protected] (PabloGroisman),
[email protected] (RafaelM.Grisi)
URL:http://mate.dm.uba.ar/∼pferrari (PabloA.Ferrari),http://mate.dm.uba.ar/∼pgroisma
(PabloGroisman)
1PresentAdress:CentrodeMatema´tica,Computac¸a˜oeCognic¸a˜o,UniversidadeFederaldoABC,
AvdoEstado,5001,SantoAndre´,SP,Brasil,09210-910
2Phone:(+5511)49968123.
PreprintsubmittedtoElsevier January17,2012
and s′ are neighbors} has the following properties: (a) each vertex H(s) is in the barycenter of
its neighbors and (b) |H(s)−s|/|s| vanishes as |s| grows to infinity along any straight line. If
such an H exists, the resulting graph is the harmonicdeformationof the Delaunay triangulation
of S◦. The search of such H has been proposed by Biskup and Berger [5], who proved its
existencein the graph induced by the supercritical percolation cluster in Zd; their approach was
the motivation of this paper. The harmonic function H was tacitly present in Sidoravicius and
Sznitman[22]and inMatthieuand Piatnitski[21]; thefunctionH(s)−siscalled corrector. See
alsoCaputo, Faggionatoand Prescott[9]for apercolation-typegraphin pointprocesses on Rd.
Figure1: DelaunaytriangulationofaPoissonprocessanditsharmonicdeformation. Thestarindicatestheorigin
(left)andthepointH(0)(right).
ThefunctionsfromS◦ toRarecalledsurfaces. Thecoordinatesh ,...,h ofH areharmonic
1 d
surfaces;thatish (s)istheaverageof{h (s′),s′ neighborofs}. Thesublinearityofthecorrector,
i i
requirement(b)above,amountstoaskthath havetilte,thei-thcanonicalvectorofRd. Roughly
i i
speaking,asurface f hastilt u(aunitvector)if(f(Ku˜)−Ku˜·u)/K convergestozero as K goes
to±infinity foreveryu˜∈Rd (see[6, 8,15]).
Fixing a direction u, we construct a harmonic surface h with tilt u as the limit (and a fixed
point)ofastochasticprocessintroducedbyHammersleycalledtheharnessprocess,[14,18]. The
process is easily described by associatingto each point s of S◦ a one-dimensionalhomogeneous
Poisson process of rate 1. Fix an initial surface h and for each point s at the epochs t of
0
the Poisson process associated to s update h t (s) to the average of the heights {h t −(s′),s′ is a
neighbor of s}. It is clear that if h is harmonic, then h is invariant for this dynamics. We start
theharness processwithh =g ,thehyperplanedefined byg (s)=s ,thei-thcoordinateofsand
0 i
showthath (·)−h (0)convergestohinL (P ×P),whereP isthelawofthepointconfiguration
t t 2
S◦ and P isthelawofthedynamics.
We prove that the tilt is invariant for the harness process for each t and in the limit when
2
t → ¥ . In a finite graph the average of the square of the height differences of neighbors is
decreasing with time for the harness process. Since essentially the same happens in infinite
volume, the gradients of the surface converge under the harness dynamics. It remains to show
that: (1)thelimitofthegradientsisagradientfieldand(2)thelimitisharmonic. Bothstatements
followfromalmostsureconvergencealongsubsequences.
Akeyingredientoftheapproachistheexpressionofthetiltofasurfaceasthescalarproduct
of the gradient of the surface with a specific field (see Section 4). This implies that the limiting
surface hasthesametiltas theinitialone.
2. Preliminariesand mainresult
Point processes and harmonic surfaces Let S bean ergodic point process on Rd with inten-
sity1;callP itslawandE theassociatedexpectation. TheprocessStakesvaluesinN ,thespace
of locally finite point configurations of Rd; we use the notation s for point configurations in N
andSforrandompointprocessesinN . Theelementssofsarecalledpointsandtheelementsxof
Rd arecalledsites. InthesamewayweuseN ◦ forthespaceofconfigurationsinN withapoint
attheoriginands◦forpointconfigurationsinthatspace. LetS◦denotethePalmversionofS. We
can think of S◦ as S conditioned to havea point in the origin. If S is Poisson, then S◦ =S∪{0}.
We abuse the notation and use P and E to denote the law of S◦ and its associated expectation.
For s∈N let theVoronoicell ofs∈s be defined by Vor(s)={x∈Rd : |x−s|≤|x−s′|, forall
s′ ∈s\{s}}. If the intersection of theVoronoi cells of s and s′ is a (d−1)-dimensionalsurface,
we say that s and s′ are Voronoi neighbors. We consider the random graph with vertices s and
edges {(s,s′) : s and s′ are Voronoi neighbors in s}. If S◦ is the Palm version of a Poisson pro-
cess,thegraphisatriangulationa.s. calledtheDelaunaytriangulationofS◦. Toasitex∈Rd we
associatethecenter oftheVoronoicellcontainingx: Cen(x)=Cen(x,s)=s∈sifx∈Vor(s);if
x belongs to the Voronoi cell of more than one point, use lexicographic order of the coordinates
(oranyotherrule)todecidewho isthecenter. Let
X := {(s,s)∈Rd×N :s∈s}
1
X := {(s,s′,s)∈Rd×Rd×N :s,s′ ∈s}.
2
Functions h :X →R are called surfaces and functions z :X →R are called fields. Denote by
1 2
t thetranslationoperator: forxinRd,t s:={s−x:s∈s}. Ifh (s,s)=h (0,t s)foreverys∈s
x x s
we say that h is a translation invariant surface. A field z is covariant if z (s′−s,s′′−s,t s) =
s
z (s′,s′′,s) for all s,s′,s′′ ∈ s. A field z is a flux if z (s,s′,s) = −z (s′,s,s) for all s,s′ ∈ s. The
conductancesinducedby s isthefield adefined by
a(s,s′,s):=1{sand s′ areVoronoineighborsins}. (2.1)
TheLaplacianoperatorisdefined on surfaces h by
Dh (s,s)= (cid:229) a(s,s′,s)[h (s′,s)−h (s,s)] (2.2)
s′∈s
3
Thegradient ofasurfaceh is thefield defined by
h(cid:209) (s,s′,s)=a(s,s′,s)[h (s′,s)−h (s,s)].
Forfields z : X →R thedivergencedivz : X →R isgivenby
2 1
divz (s,s)= (cid:229) a(s,s′,s)z (s,s′,s).
s′∈s
Hence Dh =divh(cid:209) . To simplify notation we may drop the dependence on the point configura-
tion when it is clear from the context. The Laplacian, gradient and divergence depend on the
conductances, but we drop this dependence in the notation, as they are fixed by (2.1) along the
paper.
A surface his called harmonicfors∈N ifD h(s,s)=0for alls∈s.
Pointwise tilt We say that for s∈N a surface h has tilt I(h ,s)=(I (h ,s),...,I (h ,s)) if
e e
1 d
for each u ∈ {e ,...,e } the following limits for K → ±¥ exist, coincide and do not depend
1 d
on x∈Rd
h (Cen(x+Ku),s)−h (Cen(x),s)
I (h ,s):= lim . (2.3)
u
K→±¥ K
Harnessprocess Givenasurfaceh ,letM h bethesurfaceobtainedbysubstitutingtheheight
s
h (s)withtheaverageoftheheightsat theneighborsofs:
1 (cid:229) a(s,s′)h (s′) ifs′ =s,
(M h )(s′)= a(s) (2.4)
s s′∈s
h (s′) ifs′ 6=s,
wherea(s)=(cid:229) a(s,s′). Takeapointconfigurations anddefine thegenerator
s′∈S
L f(h )=(cid:229) [f(M h )− f(h )]. (2.5)
s s
s∈s
That is, at rate 1, the surface height at s is updated to the average of the heightsat the neighbors
ofs. Weconstructthisprocessasafunctionofafamilyofindependentone-dimensionalPoisson
processes T = (T ,n = 1,2...) with law P. Take an arbitrary enumeration of the points, s =
n
(s ,n≥1) (for instance, s may be the n-th closest point to theorigin)and update the surface at
n n
s at the epochs of T . When the pointconfiguration is random, say S◦, ask T to be independent
n n
ofS◦ anddefinetheprocessasabovetoobtainaprocess(h ,t ≥0)asafunctionof(S◦,T),with
t
the product law P ×P, and h . The resulting noiseless harness process is Markov on the space
0
ofsurfaces withgeneratorL . See Section5 forarigurousconstruction.
S◦
Assumptions We assume that S is a stationary point process in Rd with Palm version S◦,
satisfyingthefollowing:
A1 ThelawofS is mixing.
4
A2 ForeveryballB⊂Rd, P(|S∩¶ B|<d+2)=1.
A3 E exp(b a(0,S◦))<¥ forsomepositiveconstantb . Thenumberofneighborsoftheorigin
has afinitepositiveexponentialmoment.
A4 E[(ℓ (¶ Vor(0,S◦)))2]<¥ . Thed−1LebesguemeasureoftheboundaryoftheVoronoi
d−1
cell oftheoriginhas finitesecondmoment.
A5 E[(cid:229) a(0,s)|s|r]<¥ for somer>4.
s∈S◦
A6 P(S isperiodic)=0.
All these assumptions are satisfied if S is a homogeneous Poisson process. Assumption A1
guarantees“onedimensional”ergodicityasin(4.13)later. AssumptionA2issufficienttodefine
the Delaunay triangulation. Notice that A4 implies that the volume of the Voronoi cell of the
originhas finitesecond moment: E[(ℓ (Vor(0,S◦)))2]<¥ .
d
Assumption A6 is used on the one hand in the Appendix to identify the motion of a ran-
dom walk on the Delaunay triangulation with the motion of the enviroment as seen from the
walker. On the otherhand ergodicity and aperiodicity ofthe point process imply that there exist
measurablefunctionss :N →Rd such that
n
B1 s (t S◦)=−s (S◦),
−n sn n
B2 S◦ ={s (S◦);n∈Z}, and
n
B3 t S◦ has thesamedistributionasS◦ foreveryn∈Z.
sn(S◦)
This is used to extend the properties of S◦ to t S◦, for all s∈S◦. The point is that t S◦ has the
s s
same law as S◦ only if s is correctly chosen as was shown in [13, 20] for Poisson processes and
by Timar [23] under the condition that S is ergodic and P-a.s. aperiodic; see Heveling and Last
[19].
Theorem 2.1. Let S◦ be the Palm version of the stationary point process satisfying A1-A6 and
letg beasurfacewithcovariantgradient,tiltI(g )∈Rd andC(|g(cid:209) |r)<¥ forsomer>4. Then:
(a) There exists a harmonic surface h with h(0,S◦) =0 and I(h)=I(g ) P-a.s. (b) if h is the
t
harnessprocesswithinitialconditiong , then,
limEE[h (s )−h (0)−h(s )]2 = 0, (2.6)
t n t n
t→¥
for any n∈Z, with s as in B1-B3. (c) In dimensions d =1 and d =2, h is the only harmonic
n
surfacewith covariantgradientandtiltI(g ).
Letc∈Rd;thehyperplaneg (s,S◦)=c·s,s∈S◦ satisfiesthehypothesesofthetheoremwith
I(g )=c. Items(a)and(b)ofthetheoremsaythatasurfacewithtiltcevolvingalongtheharness
5
process and seen from the height at the origin converges in L (P ×P) to a harmonic surface h
2
withthesametiltand withh(0)=0.
LetH =(h ,...,h ),whereh istheharmonicsurfaceobtainedinTheorem2.1forthetilte.
1 d i i
The graph with vertices H(S◦) = (H(s),s ∈ S◦) and conductances a˜(H(s),H(s′)) := a(s,s′) is
harmonic:
1 (cid:229)
H(s)= a(s,s′)H(s′) (2.7)
a(s)
s′∈S◦
that is, each point is in the barycenter of its neighbors in the neighborhood structure induced by
the Delaunay triangulation of S◦. This graph, called the harmonic deformation of the Delaunay
triangulation,does notcoincidewiththeDelaunay triangulationofH(S◦).
Random walks in random graphs and martingales. Let Y =YS◦, be the random walk on S◦
t t
which jumps from s to s′ at rate a(s,s′). Since H(S◦) is harmonic, the random walk H(Y)
t
on H(S◦) is a martingale and it satisfies the conditions of the martingale central limit theorem
(Durrett, [12, page 417]). So, the invariance principle holds for H(Y). The extension of the
t
invariance principle from the walk H(Y) to the walk Y requires the sublinearity in |s| of the
t t
correctorc (s)=H(s)−s.
Corrector. Mathieu and Piatnitski [21] and Berger and Biskup [5] construct the corrector for
thegraphinducedbythesupercriticalpercolationclusterinZd. Bothpapersprovesharpbounds
on the asymptotic behavior of the corrector and, as a consequence, the quenched invariance
principle for Y for every dimension d ≥ 2. Key ingredients in those proofs are heat kernels
t
estimatesobtainedbyBarlow[1](in[5]theyareusedjustford ≥3). SidoraviciusandSznitman
[22] also used the corrector to obtain the quenched invariant principle for d ≥4. Several papers
obtain generalizations of similar results on subgraphs of Zd [2, 7, 21]. Caputo, Faggionato and
Prescott [9] use the corrector to prove a quenched invariance principle for random walks on
random graphs with vertices in an ergodic point process on Rd and conductances governed by
i.i.d.energy marks.
Uniqueness. The uniqueness of (the gradients of) a harmonic function with a given tilt has
been proved by Biskup and Spohn [8] for the graph with conductances associtated to the bonds
of Zd under “ellipticity conditions” (see (5.1) and Section 5.2 in that paper) and by Biskup and
Prescott [7] in the bond percolation setting in Zd using “heat kernel” estimates, see Section 7
later.
We obtain harmonic surfaces as limits of the zero temperature harness process. The tilt of a
surfaceisobtainedasascalarproductwithaspecificfieldanditisinvariantfortheprocess. This
allowsustoshowthattheharmoniclimitshavethesametiltas theinitialsurface.
Thepaperisorganizedasfollows. InSection3wegivebasicdefinitions,definethespaceH
offieldsasaHilbertspaceandshowausefulintegrationbypartsformula. InSection4weshow
that the coordinates of the tilt of a surface can be seen as the inner product of its gradient with
a specific field in H . In Section 5 we describe the Harris graphical construction of the Harness
process. In Section 6 we prove the main theorem. Section 7 deals with the uniqueness of the
harmonicsurface ind =2.
6
3. Pointprocesses, fields and gradients
Let N =N (Rd) be the set of all locally finite subsets of Rd, that is, for all s ∈N , |s∩B|,
thenumberof pointsin s∩B, is finite for everybounded set B⊂Rd. We considerthe s -algebra
B(N ), the smallest s -algebra containing the sets {s ∈ N : |s∩B| = k}, where B is a bounded
Borel set ofRd and k isapositiveinteger.
Cesa`ro means andthe space H . Let C bethemeasureinX defined on z : X →R by
2 2
C(z )= z dC = 1E (cid:229) a(0,s,S◦)z (0,s,S◦) (3.1)
ZX 2
2 s∈S◦
This measureis absolutely continuouswith respect to the second order Campbell measureasso-
ciated to P with densityZ(u,v,s)=a(u,v,s)d (u). The spaceH :=L (X ,R,C)is Hilbert with
0 2 2
innerproductC(z ·z ′), wherethefield (z ·z ′) isdefined by
(z ·z ′)(s,s′,S◦)=a(s,s′,S◦)z (s,s′,S◦)z ′(s,s′,S◦).
If two fields z and z ′ coincidein the pairs (0,s)for all s neighborof theorigin, then theirdiffer-
encehaszeroC-measureandhenceafieldinH ischaracterizedbyitsvaluesat((0,s),sneighbor
oftheorigin). Definetheequivalencerelationz ∼z ′ ifandonlyifz (0,s,s◦)=z ′(0,s,s◦),forall
neighbor s of the origin. Each class of equivalence in H has a canonical covariant representant
obtained by z (s,s′,s):=z (0,s′−s,t s) for s,s′ ∈s. So hereafter, when we refer to a field in H ,
s
weassumethat itisthecovariantrepresentant.
ThespaceH waspreviouslyconsideredbyMathieuandPiatnitski[21]when(S◦,a)aregiven
by the infinite cluster for supercritical percolation in Zd. The Hilbert structure of this space is
usefulto obtainweak convergenceforthedynamics.
Define theCesa`ro limitofafield z : X →R by
2
C(z ):= lim 1 (cid:229) a(s,s′,S)z (s,s′,S), (3.2)
L րRd 2|L |{s,s′}∩L 6=0/
whereL =L (K):=[−K,K]d ⊂Rd. SinceSisergodic,thePointwiseErgodicTheorem[10,pp.
318] impliesC(z )=C(z ), P-a.s. Analogously,for translationinvariant surfaces h we define its
Cesa`ro meanC(h )(witha slightabuseofnotation)and wehaveC(h )=C(h )=E(h (0,So)).
Lemma 3.1 (Mass Transport Principle [3, 4, 17, 20]). Let z :X →R be a covariant field such
2
thateitherz is nonnegativeorE (cid:229) |z (0,s,S◦)|<¥ . Then
s∈S◦
E (cid:229) z (0,s,S◦)=E (cid:229) z (s,0,S◦). (3.3)
s∈S◦ s∈S◦
Proof. Let s be the maps introduced in B1-B3. Use B2 and Fubini in the first identity and
n
covarianceofz in thesecond onetoobtain
E (cid:229) z (0,s,S◦) = (cid:229) Ez (0,s (S◦),S◦) = (cid:229) Ez (−s (S◦),0,t S◦)
n n sn(S◦)
s∈S◦ n∈Z n∈Z
= (cid:229) Ez (s (t S◦),0,t S◦) = (cid:229) Ez (s (S◦),0,S◦) = E (cid:229) z (s,0,S◦),
−n sn(S◦) sn(S◦) −n
n∈Z n∈Z s∈S◦
7
where weused B1 in thethird identity,B3 in thefourth oneand Fubiniand B2 again in thefifth
one.
Lemma 3.2 (Integrationby parts formula). Let z ∈H bea fluxand f bea translationinvariant
surfacesatisfyingE[a(0)f 2(0)]<¥ . Then
C(f(cid:209) ·z )=−C(f ·divz ). (3.4)
Proof. Notethat
C(f(cid:209) ·z )= 1E (cid:229) a(0,s,S◦)f(cid:209) (0,s,S◦)z (0,s,S◦)
2
s∈S◦
= 1E (cid:229) a(0,s,S◦)f (s,So)z (0,s,S◦)−1E (cid:229) a(0,s,S◦)f (0,S◦)z (0,s,S◦)
2 2
s∈S◦ s∈S◦
= 1E (cid:229) a(0,s,S◦)f (s,S◦)z (0,s,S◦)−1E[f (0,S◦)divz (0,S◦)].
2 2
s∈S◦
Sincez andaarecovariantandf istranslationinvariant,a(s,s′,S◦)f (s′,S◦)z (s,s′,S◦)iscovariant
and Lemma3.1implies
E (cid:229) a(0,s,S◦)f (s,S◦)z (0,s,S◦) = E (cid:229) a(s,0,S◦)f (0,S◦)z (s,0,S◦)
s∈S◦ s∈S◦
= −E (cid:229) a(0,s,S◦)f (0,S◦)z (0,s,S◦) = −E[f (0,S◦)divz (0,S◦)].
s∈S◦
Weused thatz is aflux anda issymmetric.
4. Tilt
Wedefineherethe“integratedtilt”J(h )forsurfacesh withcovariantgradienth(cid:209) ∈H . The
coordinatesofJ(h )aredefined astheinnerproductofthegradientfieldh(cid:209) withaconveniently
chosen field. We thenprovethatthepointwisetiltI(h ,S◦)coincides withJ(h ),P-a.s.
Take a unit vector u and a point configuration s◦. For neighbors s of the origin, let b(0,s,s◦)
be the (d−1)-dimensional sidein common of theVoronoi cells of 0 and s and let b (0,s,s◦) be
u
theprojectionofb(0,s,s◦)overthehyperplaneperpendicularto u,seeFigure2. Definethefield
w by
u
w (0,s,s◦):=sg(s·u)a(0,s,s◦)ℓ (b (0,s,s◦)). (4.1)
u d−1 u
whereℓ isthe(d−1)-dimensionalLebesguemeasure. ByassumptionA4,w ∈H andsince
d−1 u
h(cid:209) isalso inH , wecan define
J (h ):=C(h(cid:209) ·w ) and J(h ):=(J (h ),...,J (h )). (4.2)
u u e e
1 d
8
s
b(0,s,s◦)
b (0,s,s◦)
u
0
Figure2: Definitionofthefieldw foru=e .
u 1
Proposition4.1. Let h bea surfacewithcovarianth(cid:209) ∈H . Then
I(h ,S◦)=J(h ), P-almostsurely. (4.3)
Beforeprovingthepropositionweshowatechnicallemma. LetO bethed−1dimensional
u
hyperplaneorthogonaltou: O ={y∈Rd : y·u=0}.
u
For y ∈ O let l (y) = {y+a u;a ∈ R}, the line containing y with direction u. Fix s ∈ N ,
u u
define L (y,s):={s∈s: Vor(s)∩l (y)6=0/}, the set of centers of the Voronoi cells intersecting
u u
l (y). Definew:Rd×X →{0,1}by
u 2
1 ifb(s,s′,s)∩l (y)6=0/;
w(y;s,s′,s)= u ,
0 otherwise
(
the indicator that s and s′ are neighbors and its boundary intersects the line l (y). Define also
u
q :Rd×X →R by
1
q (y;s,s)= (cid:229) s′a+(s,s′,s)w(y;s,s′,s),
s′∈s
wherea+(s,s′,S)=a(s,s′,S)1{(s′·u)>(s·u)}. In words,fors∈L (y,s),q (y;s,s)istheneigh-
u
borofsin thedirectionu such thattheirboundaryintersectsl (y).
u
Forx∈Rd,letx∗∈O betheprojectionofxoverthehyperplaneO . Observethatwsatisfies
u u
w(y;s,s′,s)=w(y−x∗;s−x,s′−x,t s), (4.4)
x
and
q (y;s,s)−x=q (y−x∗;s−x,t s), (4.5)
x
forall x∈Rd.
9
q (y;s,S) s
K
v K
Figure3:PointsofL(0,s)(redandbig).Thehorizontallineisl .
y
Lemma 4.2. Let z ∈H bea flux, u aunitvector andy∈Rd. Then
E (cid:229) z (s,q (y;s,S),S)1 (s)1 (s·u)=ℓ (A)C(z ·w ) (4.6)
Lu(y,S) A 1 u
s∈S
forallA∈B(R)with 1-dimensionalLebesguemeasureℓ (A)<¥ .
1
The random set {(s·u) : s ∈ L (y,S)} is the one-dimensional stationary point process ob-
u
tained by projecting the points of L (y,S) to l (y). One can think that each point s has a weight
u u
z (s,q (y;s,S),S). Theexpressionontheleftof(4.6)istheaverageoftheseweightsforthepoints
projected over A. The expression on the right of (4.6) says that this average contributes to the
expressionas muchastheLebesguemeasureoftheprojectionoverO oftheboundarybetween
u
s anditsneighborinL toitsright.
Proof. By translation invariance we can take y=0 and, for simplicitywe take u=e , the other
1
directions are treated analogously. In this case O = {x ∈ Rd : x = 0}, s·u = s , the first
u 1 1
coordinateofs and x∗ =(0,x ,...,x ). Define
2 d
g(s,s):=z (s,q (0;s,s),s)1 (s)1 (s ).
Lu(0,s) A 1
From theGeneralized Campbellformula,(4.5)and Fubini,
E (cid:229) |g(s,S)| = E|g(x,t S◦)|dx
−x
ZRd
s∈S
= E|z (0,q (−x∗;0,S◦),S◦)|1 (0)1 (x )dx
ZRd Lu(−x∗,S◦) A 1
= ℓ1(A)ZRd−1E (cid:229) |z (0,s,S◦)|1{q (x∗;0,S◦)=s}1Lu(x∗,S◦)(0)dx2...dxd
s∈S◦
= ℓ1(A)E (cid:229) |z (0,s,S◦)|ZRd−11{q (x∗;0,S◦)=s}1Lu(x∗,S◦)(0)dx2...dxd (4.7)
s∈S◦
Fors∈S◦ such thata+(0,s,S◦)=1,
{s=q (x∗;0,S◦), 0∈L (x∗,S◦)} = {l (x∗)∩b(0,s,S◦)6=0/} = {x∗ ∈b (0,s,S◦)}.
u u u
10
