The Stationary Distribution of Reflected Brownian Motion in a Planar Region
Author(s): J. M. Harrison, H. J. Landau, L. A. Shepp
Source: The Annals of Probability, Vol. 13, No. 3 (Aug., 1985), pp. 744-757
Published by: Institute of Mathematical Statistics
Stable URL: http://www.jstor.org/stable/2243711
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The Annals ofProbability
1985,Vol. 13, No. 3, 744-757
THE STATIONARY DISTRIBUTION OF REFLECTED
BROWNIAN MOTION IN A PLANAR REGION
BY J. M. HARRISON,1H. J. LANDAU,AND L. A. SHEPP
and AT&T BellLaboratories
StanfordUniversity
Supposegivena smooth,compactplanarregionS and a smoothinward
processZ
pointingvectorfieldon MS.It is knownthatthereis a diffusion
instanwhichbehaveslikestandardBrownianmotioninsideS and reflects
in thedirection
taneously
at theboundary
specified
bythevectorfield.It is
We finda simple,generalexplicit
alsoknownZ hasa stationary
distributionp.
formula
forp in termsoftheconformal
mapofS ontotheupperhalfplane.
We also showthatthisformula
remainsvalidwhenS is a boundedpolygon
casearisesas the
andthevectorfieldis constanton eachside.Thispolygonal
forcertaintwo-dimensional
diffusion
queueing
heavytraffic
approximation
andstorageprocesses.
1. Introductionand summary. In thispaperwe calculatethestationary
diffusion
distribution
fora particulartype of two-dimensional
process.The
processis denotedbyZ = IZ(t), t 2 0}. Its statespaceis a compactplanarregion
ofS likestandardBrownianmotion(uncorrelated
S, anditbehavesintheinterior
and
unit variance).At the boundaryZ reflects
zero
drift
componentswith
instantaneously,
and the directionof reflection
mayvarywithlocation.This
featureoftheprocessunderstudy,and
behavioris thedistinguishing
boundary
it willbe explainedfurther
shortly.
We firsttreatthecase ofa smoothstatespaceand smoothly
direction
varying
on thework
ofreflection,
picturedin Figure1. In thiscase one can builddirectly
ofStroock-Varadhan
(1971).We takeas givena boundedand simplyconnected
regionG oftheform
G = tz E R2: 0(z) 2 0}
(1.1)
where
0: R2
--
R is twice continuously differentiable,is bounded with
(1.2) boundedfirstand second-order
partials,and satisfiesIV0q(z)I 2 1 on
Iz: k(z) = O}.
Hereafterlet us summarize(1.1) and (1.2) withthe statementthat G is a C2
domain.OurstatespaceS is theclosureofG,so
aS = OG = 1zE R2: 0(z) = 0}.
(1.3)
fieldon theboundary,
let 0 be a continuously
To specifythereflection
differenReceivedJanuary
1984;revisedJuly1984.
1 Stanford
Thisworkwasdonepartially
at Bell
University.
whileJ.M. Harrisonwasa consultant
andwaspartially
Laboratories,
supported
byNSF grantECS-8017867.
AMS 1980subjectclassifications.
Primary
60J65;secondary
60K30.
Key wordsand phrases.Diffusionprocess,reflecting
barrier,invariantmeasures,conformal
valueproblem.
mapping,
boundary
744
REFLECTED BROWNIAN MOTION
745
8(01
at a boundary
FIG. 1. Angleofreflection
point
tiablefunction
on 9G,withI0(a) I < 4r2 forall a E 9G. Interpret
0(a) as an
angle of reflectionat boundary point a, measured clockwise fromthe inward
pointingnormalas shownin Figure1. To understand
theboundary
behaviorof
consider
a
process
that
behaves
like
standard
Brownian
motion
in
G
andjumps
Z,
a distancee whenever
specified
by
aG is struck,thisjumpbeingin thedirection
0(.). One may thinkof Z as the limitof this processas e 4 0; a precise
mathematical
willbe givenin Section2. Therewe also showthatthe
description
MarkovprocessZ has a unique stationarydistribution,
that this stationary
distribution
concentrates
all its mass on the interiorG of S, thatit also has a
densityfunction
p, and thatp is givenbytheformula
p(z) = c RefexpL(z)}, z E G.
(1.4)
Herec is a normalization
constantchosenso thatp integrates
to one,and L is a
certaincomplex-valued
function
ofthecomplexvariablez. To be specific,let F
be anyconformal
and letg0be the(unique)
mappingofS to theupperhalf-plane,
pointofaG suchthatF(uo) = ??.Then L is theanalyticfunction
(1.5)
L Xz)=
1 ('0(a)
-
[F()-
-
0(go)1
dF(a)
-
i0(co), z E G.
Note thatF is real-valuedon OG (it maps OG ontothe boundaryof the upper
half-plane),
so the realand imaginary
partsofthe integralin (1.5) can each be
definedin theordinary
Riemann-Stieltjes
sense;each integralis to be computed
movingcounterclockwise
aroundtheboundary.
In Section3 we turnto the typeof case picturedin Figure2. Here S is a
,
polygonwithverticesu1, *.., IfK (in counterclockwise
order).For k = 1,
K - 1 we definesidek as the openline segmentbetweenukand Uk+1. Similarly,
sideK is thelinebetweenuK and u1,excluding
theendpoints.Let tk denotethe
interior
anglemadebythetwosidesmeetingat vertexk,as shownin Figure2.
Also givenare angles01, ***,OK satisfying
IOkI < ir/2.For futurepurposes,set
MarkovprocessZ withthefollowing
00 = OK.The objectofourstudyis a strong
fourproperties.
(a) It behaveslikestandardBrownianmotionin G. (b) It reflects
instantaneously
at angleOkon sidek,as shownin Figure2. (c) It spendsno time
at theverticesofS. (d) It has continuous
that
samplepaths.(Note in particular
746
HARRISON, LANDAU, SHEPP
O3
FIG. 2. An example
ofthepolygonal
case
have been associatedwiththe vertices.)It turnsout
no directions
ofreflection
that these fourproperties,translatedinto precise mathematicallanguage,
ofZ if
thedistribution
uniquelydetermine
(1.6)
Ok-1 <
Ok+
forall k = 1,
2bk
,K.
On the otherhand,thereis no diffusion
processwiththe statedpropertiesif
(1.6) fails.This followsfromthe workofVaradhanand Williams(1983),as we
shallexplainin Section3.
As statedearlier,formula(1.4)-(1.5) givesthe stationarydistribution
of Z
whenS is a smoothregionand 0( *) variessmoothly
overtheboundary.
But this
formulacontinuesto makesense,at leastformally,
in manysituationswhereS
and 0(.) are notsmooth;one naturallysuspectsthatthe formularemainsvalid
in such situations.We provein Section3 thatthis suspicionis correctforthe
also simplifies
in that
polygonalcase,and thatthegeneralformula
considerably
case. As theconformal
mappingF, we taketheinverseofthestandardSchwarzChristoffel
map fromthe upperhalf-planeto S, chosenso thatgOis on side K
and hence0(co) = OK. (See pages 189-196of Nehari(1952) fora discussionof
Schwarz-Christoffel
mappings.)On each side of S, the integrandin (1.5) is a
oflog[F(z) - F(cr)]. Thus the integralcan
constanttimestheexactdifferential
be evaluatedexplicitly,
and (1.4)-(1.5) eventually
reducesto
(1.7)
P(z) = c Retexp(-iOK) f
k=l
[F(z) -
F(uk)
z E GI
aI
where
(1.8)
ak
=
(1/7r)(Ok
-
for k = 1, *
Ok-)
,
K.
Notethat(1.7) can be re-expressed,
usingrealvariablesonly,as
(1.9)
p(Z)
Here yj(z),
=
**,
CcS[oK[X
YK(Z)
akYk(Z)
-
oK]
llK=l
IF(z)
-
F(9k)
Ik.
are the phase angles picturedin Figure 3, and
REFLECTED
747
BROWNIAN MOTION
F(z)
an
F(N1)
;
Ha
F(o-2)
Ad X~~~~Y(Z)
*o*K
F(o)
REAL AXIS
FIG. 3. Phase anglesfiguringin finalsolutionforthepolygonalcase
| F(z) - F(uk) I is thelengthofthelineconnecting
F(z) and F(uk). The density
withsingularities
at verticesofS, and from(1.9) we see
function
p is harmonic,
thatitsvalueon oG is givenby
(1.10)
(j = 1,
p(a)
***,
=
c cos Oj k=rl
IF(a) -
F(90
I k
on side j
K). A notable featureof (1.10), and more generallyof (1.4)-(1.5), is
thatboundaryvalues ofp dependonlyon the restriction
of F to 'G, whichis
real-valued.
We conjecturethat (1.4)-(1.5) remainsvalid fora muchbroaderclass of
processesthan consideredhere.For example,to unifythe twocases discussed
above,one mightconsidera boundedand piecewisesmoothstate space, with
piecewisecontinuousangleofreflection.
Also,extensionsto unboundedregions
are doubtlesspossible.These potentialdirectionsforfutureresearchwill be
in Section4.
discussedbriefly
The originalmotivation
forour studyof reflecting
Brownianmotioncomes
fromqueueingandstoragetheory.
Harrison(1978)showedthatreflecting
Brownian motionon the quadrant,withnormalreflection
at one axis and oblique
reflection
at theother,is thenaturaldiffusion
fora tandemqueue.
approximation
Moreprecisely,
thetwo-dimensional
queuelengthprocess,properly
normalized,
underconditionsof heavytraffic
convergesweaklyto the indicateddiffusion
(approximate
equalityof averagearrivalrate and averageservicerates).This
resultwas greatly
generalized
byReiman(1983),whoconsidered
themultidimensionalqueuelengthprocessassociatedwitha generalopennetwork
ofn service
facilities.Reimanshowedthatthe normalizedqueue lengthprocessconverges
weakly,underheavytrafficconditions,to reflecting
Brownianmotionon the
n-dimensional
positiveorthant,the directionof reflection
beingconstantover
each boundaryhyperplane.A similarsort of limittheoremwas provedby
Wenocur(1982) forthe multidimensional
inventory
processassociatedwitha
production
network.
By considering
systemswithfinitestoragecapacityat each
Wenocurobtaineda diffusion
productionfacility,
limitwhosestate space is a
boundedpolygonalregion,againwithconstantdirectionofreflection
overeach
This earlierworkshowswhatdiffusion
boundaryhyperplane.
processshouldbe
used to approximate
the heavytraffic
behaviorof a multidimensional
queueing
or storageprocess,but it does not showhowto calculateinteresting
quantities
748
HARRISON, LANDAU, SHEPP
diffusion.
associatedwiththe approximating
Using our formula(1.9) and the
programdescribedby Trefethen(1980), one can numerically
evaluatethe staof the approximating
diffusion
in a varietyof interesting
tionarydistribution
two-dimensional
cases. Therearealso interesting
specialcases wheretheconforwhichallows directnumerical
mal mappingF can be writtenout explicitly,
evaluationof (1.9); one ofthesewas studiedearlierby Harrison-Shepp
(1983).
For moreon the theoryof reflecting
Brownianmotion,and the associated
analyticalproblems,see Harrison-Reiman(1981a, 1981b), Foschini (1982),
Harrison-Shepp(1983), Foddy (1983), Williams (1983), and VaradhanWilliams(1983).
Independently
oftheworkdescribed
inthepreceding
paragraph,
Newell(1979)
studieddiffusion
approximations
fortandemqueueingsystems,
focusing
on the
andcertaincloselyrelatedquantities.
Newell
stationary
queuelengthdistribution
didnotdiscussapproximation
ofthequeuelengthprocessbya diffusion
process.
Instead,usinga generalapproachfamiliarin mathematical
physics,he directly
formulated
a partialdifferential
thetimeequationwhosesolutionapproximates
ofthequeuelengthprocessunderheavytraffic
conditions.
dependent
distribution
This is a diffusion
equation(heat equation),withauxiliaryconditionsof an
versionof
unusualand difficult
type.Newellwas able to solvethe steady-state
hisequationforcertainspecialcases,thusobtaining
an approximate
equilibrium
distribution
forthe originalqueueingsystem.We beganourstudyby verifying
that,exceptforwhatseemsto be a typographical
error,Newell'ssteady-state
forthe reflecting
solutionis in facta stationary
distribution
Brownianmotion
his queue lengthprocess.Generalizing
thisresultby stages,
thatapproximates
weeventually
arrivedat formula
(1.4)-(1.5).Thus Newell'sanalysisprovidedthe
essentialenergyfor our investigation,
althoughmost readerswould see no
connection
betweenthetwoworksin finalform.
2. Bounded smoothregion with smoothlyvarying angle. Let G be a
on aG meeting
theconditions
boundedC2domain,S itsclosure,and 0 a function
thispaper,we denoteby C2(S) the set of
spelledout in Section1. Throughout
all real-valuedfunctions
differentiable
on a domain
thatare twicecontinuously
S. For eachf E C2(S) and a E aG, let
containing
(2.1)
Df(a) = (a/an)f() + tan0( )(a/a)) f(a),
wherea/ansignifiesthe inward-pointing
normalderivative,
and a/auis the
in
direction.
the
the
counterclockwise
derivative
tangential
along boundary
Thus,
is
directional
derivative
off in
for
on
the
a
constant
except
depending a, Df(a)
thedirection
ofreflection
picturedin Figure1.
of
To defineZ precisely,
we adoptthelanguageand mathematical
machinery
Stroock-Varadhan
(1971). Let Q consistof all continuousfunctionsmapping
ofuniform
on finitetime
convergence
[0,oo)-- S. EndowingQ withthetopology
let, be theBorela-algebraon R. Finally,letZ be theidentity
intervals,
map
(2.2)
Z(t, w) = w(t) forall t : 0 and w E Q.
REFLECTED BROWNIAN MOTION
Let F be theclass ofall functions
fE
C2(S)
749
suchthat
Df(a) 2 0 forall a E 0G.
(2.3)
We wishto associatewitheach startingstatez E S a probability
measureP, on
(Q2,S)such that
PztZ(O) = zI = 1, z E S, and
(2.4)
(2.5)
f(Z(t)) --
2 o
ff(Z(s)) ds,
to
is a submartingale
on (Q, YX P,) forall z E S and f E A,
whereAfis theLaplacian.AnysuchfamilyIPz,z E SI willbe calleda solutionof
thesubmartingale
problem(2.4)-(2.5).
THEOREM2.6. The submartingale
problem(2.4)-(2.5) has a uniquesolution.
thefamilyofprobability
measuresJP, z E SI has thestrongMarkov
Moreover,
property.
PROOF. This is a variationof the centralresultprovedby Stroockand
we state
Varadhan(1971). Because our diffusion
processis timehomogeneous,
of stateonly,whereasStroockand
the submartingale
problemusingfunctions
driftand diffusion
Varadhanstudiedprocesseswithtime-dependent
coefficients,
on bothstateandtime.
so theywereobligedto considertestfunctions
depending
Theirargument
can be modified
to provethe currentproposition
exactlyas in
Varadhan-Williams
(1983).
A probability
measureiron S is saidtobe a stationary
distribution
(orinvariant
probability
measure)forJP, z E SI if
(2.7)
1
ir(dz)E-[f(Z(t))]
f
=
ir(dz) f(z)
forall t> 0 and all bounded,measurable
f: S -- R. Weiss (1981) developedthe
of stationary
following
analyticalcharacterization
distributions.
We shall commentbriefly
on theproofafterhis theoremhas beenstatedprecisely.
THEOREM2.8. Thereexistsa uniquestationarydistribution.
Moreover,a
distribution
measureiris thestationary
probability
ifand onlyifir(oG) = 0 and
(2.9)
f
ir(dz)zAf(z) c 0 forall f E W.
The proofof existencedependscritically
on the assumedboundednessof S.
750
HARRISON, LANDAU, SHEPP
measureit on S, let
Startingwithanyprobability
f tf,(dz)Pz{Z(t)E AI
wrt(A)=-
t o
oftimespentin set
irt(A)as theexpectedfraction
fort - 0. One mayinterpret
to pi.The set ofall
according
A up to timet whentheinitialstateis randomized
probability
measureson a compactset is itselfweaklycompact,so [{rn, n = 1,
measure-r. A
2, - ** convergesweaklyalong a subsequenceto a probability
thenshowsthatirsatisfies(2.7). To proveuniquenessofthe
standardargument
ofLaplace'soperator
oneusestheuniform
plus
ellipticity
distribution,
stationary
standardresultsfromergodictheory.FromtheresultsofStroockand Varadhan
(1971),it followsthatPzIZ(t) E OGI = 0 forall z E S and t - 0, and hence
distribution
satisfies(2.9), let f E F be
ir(aG) = 0. To see thatthe stationary
arbitrary,
and notethat(2.5) implies
(2.10)
Ez f(Z(t)) -
f
Af(Z(s)) ds]
f(z)
bothsidesof (2.10) withrespectto r(dz), and use
forall z E S. Now integrate
(2.7) and Fubini'stheoremto obtain
(2.11) 0
fr(dz)Ez[
J
Af(Z)(s)) ds] =
A
I
ir(dz)E[Af(Z(s)) dsj.
The innerintegralon thelast lineequalsfG r (dz)Af(z) by (2.7),so we conclude
the proofof
that (2.9) is necessaryfor ir to be the stationarydistribution;
is muchmoredifficult,
and we willnotdiscussit.
sufficiency
in Section1
thattheprobability
measureidentified
It remainsonlyto verify
deduced
satisfiesWeiss'condition(2.9). As theproofwillsuggest,we originally
from(2.9) thata well-behaved
density
p wouldhaveto satisfy
stationary
(2.12)
(2.13)
Ap(z) = 0 in G,
(a/an)p(a) = (a/Oa)[p(o)tan 0(o)]
on 0G.
Formula(1.4)-(1.5) was thenobtainedbysolvingthisboundary
valueproblem.
LEMMA 2.14. The function
L definedby (1.5) is boundedand analyticin G
withIm L = -0 on oG.
PROOF. SupposefirstthatG is theunitdiskandF is thestandardconformal
mapping
REFLECTED BROWNIAN MOTION
751
Thus OGis theunitcircleC, and o0= 1. Specializing(1.5),we have
L _z)
(2.16)
[F%-F(u)1
r
I
=-ir Jc [ (a)
-0
(1) ] (z
dF(o)-i0(o)
(1 - ) (z) duo
()
- - i0(l).
-
on OG,we see fromthelastlineof(2.16)
Since0 is assumedLipschitz-continuous
forIz I < 1. Hereafterset
in
L(z) is factconvergent
thatthe integraldefining
g(a) = 0(a) - 0(1) and define
(2.17)
Tz
=
(a)
J
Izi < 1,
that the integralis singularwhen Iz I = 1. Equation
withthe understanding
so T is a boundedanalyticfunction
formula,
Poissonintegral
(2.17)is thefamiliar
withRe T = g on C, cf.Chapter4 ofAhlfors(1979). Now the kernelappearing
as
on thelastlineof(2.16) can be written
(1 - z)du
(Z a)(1 - a)
As+ 1\ du
a- 1 2a
as + z\ du
a - z 2c'
so (2.16) can be rewrittenL(z) = iT(l) - iT(z) - i0(). Of course T(1) = g(1) =
0, so this reducesto L(z) = -iT(z) - i0(1). Fromour previouscharacterization of T it followsthat L is a boundedanalyticfunctionwithIm L(U)
-
-g (a)
- 0(1) = -0(a)
on C, as desired.
F(z) as i[1 + 4)(z)]/[1-(z)],
In the generalcase, we can alwaysrepresent
mappingfromS to the unitdisksuchthat 4)(o) = 1.
where4) is a conformal
mapping
oftheconformal
as thecomposition
Thatis,F can alwaysbe represented
(2.15) withsucha 4).Let us makethechangeofvariablew = 4)(z), let s = (a)
denotea genericpointon theunitcircle,and setL*(w) = L(z) and 0*(s) = 0(a).
(1.3) ofOGit followsthat
of0 and thedescription
FromtheLipschitz-continuity
aboveshowsthatL* is bounded
on C, so theargument
0*is Lipschitz-continuous
and analyticwithIm L* =-0* on C. The desiredconclusionforL thenfollows
directly.
is ir(dz) = p(z)dz, where
distribution
THEOREM 2.18. The uniquestationary
by(1.4)-(1.5).
p is defined
we take
positiveconstantforthemoment,
PROOF. Viewingc as an arbitrary
p(z) = c ReIexp L(z)} and q(z) = c ImIexp L(z)}, meaningthat
(2.19)
p(z) = c expIRe L(z)jcos{Im L(z)},
(2.20)
q(z) = c expIRe L(z)lsin{Im L(z)}.
FromLemma(2.14) we knowthatexp L(z) is itselfboundedand analytic,so p
HARRISON, LANDAU, SHEPP
752
and thatIm L(a-) = -0 (a) on aG. Now I0(a) I < r/2
and q are bothharmonic,
principle.
Thus
so I Im L(z) I < r/2forall z E G bythemaximum
byassumption,
thatc canbe settomakep a probability
positiveandbounded,insuring
p is strictly
density.From(2.19),(2.20) and (2.14),
(2.21)
q(u)/p(a) = tan{Im L(o)l = -tan 0(u)
on aG.
that0/On
we have (remember
Finally,sincep and q are conjugatefunctions,
normalderivative)
denotestheinward-pointing
(2.22)
(a/an)p(o) = -(alan)q(o).
ofboth
derivative
Nowmultiply
bothsidesof(2.21) byp (a), takethetangential
thedevelopment
(2.22) to obtain(2.13). Summarizing
sides,and thensubstitute
density
up to here,we haveseenthat(1.4)-(1.5) does in factdefinea probability
p, and thatp satisfies(2.12)-(2.13).
Usingt2.12)and Green'ssecond
Let r(dz) = p (z)dz,and letf E Wbe arbitrary.
normalderivathata/an is an inward-pointing
and againremembering
identity,
tive,we have
f
ir(dz)Af(z) =
(2.23)
f
p(z) Af (z) dz
t'r
1
a
JOGL[f(a) -an p()
-23
-p(a)
-
an f(a)j
do.
of W Integrating
by
But (a/an)f(a) - -tan 0(a)(a/aa)f(() by the definition
partsaroundaG, we thushave
OG
(2.24)
p(a)
-
On
f(a) dou?
=
Combining
(2.23) and (2.24) gives
rG(z)
Af' )-
G
{a
p (a)
p(a)tan 0(a)
dJOG
f (a) -[p(a)tan
dG
a
- f(a) du
Ocr
0(u)] du.
aa [p(o)tan ?(a)]ff(a)
du.
zero,so ir satisfies(2.9), and
on the rightis identically
By (2.13),theintegrand
theproofis complete.
3. The polygonal case. All notationand assumptionsare as described
recallthatG is theinterior
ofa polygonS with
earlierin Section1. In particular,
vertices1, **, UK. In thecurrent
context,we define
(3.1)
Df(a) = (a/an)f(a) + tanOk(a/aof)f(W) on side k,
withDf undefined
at vertices.Also,letWbe theset ofall f E C2(S) suchthat
(3.2)
f is a constantovera neighborhood
of each vertex
(3.3)
Df 2 0 on each side of S.
REFLECTED BROWNIAN MOTION
753
processZ, we nowseeka familyofmeasuresIP_,z E8 S
To definethediffusion
S8
on (Q, Y) suchthat,foreach z E
(3.4)
f(Z(t)) - 2
(3.5)
(3.6)
l t
2
P2'1Z(O) = z = 1,
Af(Z(s)) ds, t - 0,
on (Q, 9 P,) foreachf E A,and
is a submartingale
EZ[f
lZ(s)=,kl
dsl=
0 for k = 1, -.., K.
Condition(3.6) requiresthatthe residencetimeofZ at each vertex0k be zero
state.This conditionis an essentialcompleofstarting
almostsurely,regardless
thatare flataround
mentto (3.5),becausethelatterinvolvesonlytestfunctions
vertices,and thus(3.5) does not specifythe behaviorof Z at the vertices.The
is provedby Varadhanand Williams(1983) forthe case whereS is a
following
is accomplishedby an easy localization
wedge;the extensionto a polyhedron
whichwe omit.
argument,
problem(3.4)-(3.6) has a solutionifand
THEOREM 3.7. The submartingale
data satisfy(1.6). In thiscase,thesolutionis unique,and the
onlyiftheproblem
familytPz,z E SI has thestrongMarkovproperty.
Hereafter
we assume(1.6) holds.It followsfromtheresultsofVaradhanand
statez E S, vertexSk is hitwithprobability
Williams(1983)thatforanystarting
We shallhaveno need
zerootherwise.
1 ifOk < Ok-i, and Sk is hitwithprobability
forthisresultin ourstudy.
are definedexactlyas in Section2. The following
Stationarydistributions
is identicalto (2.8), exceptthat now 9' is defined
analyticalcharacterization
a
is
This
theorem
compositeofresultsbyWeiss(1981) and Williams
differently.
(1983),as we shallexplain.
Moreover,a
distribution.
THEOREM 3.8. Thereexistsa uniquestationary
ifand onlyifir( G) = 0 and
distribution
measureir is thestationary
probability
(3.9)
f
ir(dz)Af(z)
c 0 forall f E W.
is provedexactlyas in thecase ofsmooth
distribution
Existenceofa stationary
data, usingthe boundednessof S. By imitatingthe analysisin Chapter7 of
in thefinetopology,
and thus
Williams(1983),one can showthatZ is recurrent
of
its invariantmeasureis unique.Because Z spendsno timeon OG,regardless
of
musthaveir(aG) = 0. The necessity
distribution
state,thestationary
starting
(3.9) is provedexactlyas in the case of smoothdata, and Theorem3 of Weiss
(Weisstreatsa moregeneralproblemwithpiece(1981) establishessufficiency.
problemis
submartingale
wise smoothdata, assumingthat the corresponding
wellposed.)
HARRISON, LANDAU, SHEPP
754
p definedby (1.7). Let us firstcheckthat
Considernowthedensityfunction
This comes
density.
set
to
makep a probability
can
be
so
c
over
G.
p is integrable
from
Recall
vertex
around
an
arbitrary
of
integrability
downto thequestion
Sk.
A
at
0k.
sides
meeting
made
the
two
interior
is
by
2
the
angle
Figure thatok
is
that
mapping
of
the
Schwarz-Christoffel
property
fundamental
-F
I F(z)
I
Z - ak /(k)
mk)I = O(I
as
Iz
Ik I -O.
thiswith(1.9) gives
Combining
= O(|
p(Z)
Z -
JkI k)
as I Z - Sk
,
I
(1.6) says thatflk >-2, so p is
wherefk = (Ok - Ok-l)/Ak- Our keyrestriction
ofSk, as required.
overanyneighborhood
integrable
In provingthatWeiss' condition(3.9) is satisfiedby r(dz) = p (z)dz, we shall
showthat
Ap(z) = 0 in G,
(3.10)
(3.11)
(3.12)
(a/an)p(a)
Jr -p(a)
an
= tan
on side k, and
0k(a/ao)p(W)
du = p(b)tan 0(b)
-
p(a)tan 0(a),
wherea and b are boundarypointslocatedon side i and sidej, respectively,
curvethatleads froma to b
0(a) = 0iand 0(b) = Oj,and r is anydifferentiable
analogousto (2.12)-(2.13)
that
being
think
(3.10)-(3.11),
G. One might
through
determine
would
p, butthisis notthecase.
case
smooth
uniquely
data,
the
of
for
the
uniform
satisfied
density.)The
are
conditions
by
that
these
always
(Note
extrarequirement
(3.12) is essentialhere.
is lr(z) = p(z)dz, wherep
distribution
THEOREM3.13. The uniquestationary
is defined
by(1.7).
function
PROOF. Let us definethecomplex-valued
= logjexp(-i0K)
(3.14) L(z)
[F(z)
Hfl -=F(ok)I
-
Ek=i
aklog[F(z)
- F(k)]
-iK
forz E G. Let H be the upperhalf-planeand R the real line,and let L* (w)
L(z), wherew = F(z). Also set 0*(u) = O(o) on R, whereu = F(o), withthe
that0(u) = Okon sidek. From(3.14) we have
obviousconvention
-
(3.15)
L*(w) =
,k~l
(1/lr)(Ok - Ok-1)lg(W
-uk)
-OK,
whereuk= F(rk). Recall that log(w) is analyticin the upperhalf-planewith
thiswith(3.15),
Im log(u) = 0 foru < 0, and Im log(u) = irforu < 0. Combining
withIm L*(u) = 0*(u) on R.
we see thatL* is analyticin theupperhalf-plane
Thus L is analyticin G withIm L(u) = -0k on side k.
Noting that p(z) = c ReIexp L(z)} by (3.14) and (1.7), let us defineq(z) = c
exactlyas in theproofofTheorem(2.18),we findthat
ImIexpL(z)}. Proceeding
REFLECTED
755
BROWNIAN MOTION
p and q are conjugateharmonicfunctions
satisfying
-q(a) = tan OkP(o)
(3.16)
on side k.
ofeach sideof(3.16),and substitute
Nowtakethetangential
derivative
-(a/a)q(o)=
(3.17)
(a/an)p(u)
curve
as in theproofof(2.18).This gives(3.11).'Next,let F be anydifferentiable
thatleads froma to b through
G, wherea, b, E 9Gare notvertices.From(3.16)
we have
(3.18)
p(b)tan 0(b) - p(a)tan 0(a)
=
-[q(b) - q(a)].
But [q(b) - q(a)] can be written
as theintegralofthetangentialderivative
ofq
forthetangentialderivative,
overF. Using(3.17) to substitute
we finallyarrive
at (3.12).Thusp has beenshownto satisfy(3.10)-(3.12).
Finally,toprovethat(3.9) is satisfied
by r (dz) = p(z)dz, letf E F be arbitrary.
Becausef is flataroundeach vertex,we can cutthecornersofG, as picturedin
containing
thesmall
Figure4, so thatfhas constantvalueCk overa neighborhood
triangleeliminatednear Sk. Let a,, * , ax and bi, ***, bKbe as shown in Figure
4, and letFk be thelinesegmentleadingfromak to bk. Finally,let W be theopen
polygonalregionboundedby F,, ** , FK and thesidesofS, as shownin Figure
4. NotethatAf= 0 in each ofthesmalltriangles
thatmakeup G - W,sincef is
constantovereach suchtriangle.Thus,using(3.10) and Green'ssecondidentity
exactlyas in theproofof(2.8),we have
T
r(dz)Af(z) =
(3.19)
f
p(z)Af(z) dz
=1
=
f
p(z)Af(z) dz
(a) do.
1 p(u) - p(u)
O9n
O9n
For the last stage of the argument,
let 0(.) be definedon OW so that it is
continuously
differentiable,
I0(*) I < r/2,and 0(a) = Ok at each pointa E8 AW
thatlies on sidek. Recallthat
X
(3.20)
(a/an)f()
f()
- -tan 0( )(a/a))f (a)
on each sideofS bydefinition
ofW.This weakinequalityextendsto all a E8aW,
03
a3
b3
W
a1
04/
FIG.
b2
N,0-
4PoyoareinWfrebyctigteca2
FIG. 4. Polygonal
regionWformed
bycutting
thecorners
ofG
756
HARRISON, LANDAU, SHEPP
becauseboththenormaland tangentialderivatives
off are zeroon F1, *K*, FK.
Justas in the proofof (2.8), we substitute(3.20) into (3.19) and integrate
by
partsaroundaW to arriveat
(3.21)
-p(a)
f ir(dz)f (z) ' I { tan
-
a Oo]f~)
o
a[p()tan
0(a)] f(a) du.
a poa
By (3.11),thequantityin bracesvanisheswhereaW coincideswithaG, so (3.21)
reducesto
(3.22)
(d)
f(z) <_ Ek=1 Ck
k
tan P(a)
-
-
[p(o)tan 0(u)] do
But each of the integralson the rightside of (3.22) is zeroby (3.12),so (3.22)
reducesto (3.8),and theproofis complete.
4. Concludingremarks. To unifythetwocases treatedin thispaper,one
mightconsidera piecewisesmoothregionof the typestudiedby Weiss (1981).
Let G be the unionand/orintersection
of finitely
manyboundedC2 domains,
withthe restriction
that thereare no cusps on aG, and let 0(.) be Lipschitz
continuous
on each sectionofaG, withI 0(- ) I < r/2.Let S be theclosureofG,
and let F be a conformal
mappingfromS to the upperhalf-planesuch that
F(uo) = 0o forsome o E aG thatis nota singularity.
The submartingale
problem
forthiscase is statedverymuchas in Section3, usingtestfunctions
f thatare
flataroundsingularities
ofaG. We conjecture
thatifthesubmartingale
problem
is well posed,then thereis a unique stationarydistribution,
and its density
is givenby formula(1.4)-(1.5). This submartingale
function
problem(existence
and uniquenessof a diffusion
withthe specifieddata) has notbeen studiedas
is correct,
thenit can onlybe wellposedwhenthefunction
yet.Ifourconjecture
overG; it maywellbe thatintegrability
of
p definedby (1.4)-(1.5) is integrable
thisfunction
is necessary
and sufficient
forthesubmartingale
problemto havea
(unique)solution.
In Sections2 and 3 we have onlyused theboundedness
ofS to provethata
distribution
stationary
exists;thendirectcalculationsshowthatthe stationary
function
we conjecture
densityis givenbya particular
p. Forunbounded
regions,
p is theuniquestationary
thatthissamefunction
ifit is integrable,
distribution
and thatno stationary
If theanglesof
distribution
existsifit is notintegrable.
reflection
providesufficient
restorative
force,it is certainly
possibleto have a
withan unboundedregion:see Harrison-Shepp
stationary
distribution
(1983)for
a concreteexampleofthis(S is thesemi-infinite
stripand theangleofreflection
is constanton each side).
Readers interestedin the behaviorof reflectedBrownianmotionaround
singularboundarypoints may consultDynkin (1964, 1967) and VaradhanWilliams(1983) formoreon thisfascinating
and intricate
subject.We also refer
to Gakhov(1966) and Smirnov(1964) forextensivediscussionofobliquederivativeboundary
valueproblemsofthetypeencountered
in Sections2 and 3.
REFLECTED BROWNIAN MOTION
757
Acknowledgement. We are indebtedto S. R. S. Varadhanand R. J.
Williamsformanyhelpfulconversations.
REFERENCES
NewYork.
AHLFORS,L. V. (1979).Complex
Analysis(ThirdEdition).McGraw-Hill,
valueproblem
solutionsofa boundary
DYNKIN,E. B. (1964).Martinboundaries
and non-negative
in MarkovProcessesand RelatedProblemsof
witha directionalderivative,
reprinted
University
Analysis,
LondonMathematical
SocietyLectureNote Series#54.Cambridge
Press,1982.
forsomediffusion
processes.Proc.FifthBerkeley
DYNKIN, E. B. (1967).Generallateralconditions
ofCalifornia
II Part2, 17-49.
Press,Berkeley
Symp.Math.Stat.andProbab.,University
to
forBrownianmotionwithdriftconfined
ofsomefunctionals
FODDY, M. E. (1983).Computation
Stanford
Dept. of Mathematics,
a quadrantby obliquereflection.
Ph.D. dissertation,
University.
modelsof pairs of communicating
computersFoSCHINI, G. J. (1982). Equilibriafordiffusion
Theory28 273-284.
symmetric
case.IEEE Trans.Inform.
F. D. (1966).BoundaryValueProblems.
Addison-Wesley,
Reading,Mass.
GAKHOV,
Adv.in
fortandemqueuesin heavytraffic.
HARRISON,J. M. (1978).The diffusion
approximation
Appl.Probab.10 886-905.
Ann.Probab.
Brownian
motiononan orthant.
HARRISON,J. M. and REIMAN,M. I. (1981a). Reflected
9 302-308.
of multidimensional
reflected
HARRISON,J. M. and REIMAN,M. I. (1981b).On the distribution
Brownian
motion.SIAM J.Appl.Math.41 345-361.
anditsdiffusion
system
limit.Stochastic
HARRISON,J.M. andSHEPP, L. A. (1984).A tandemstorage
Process.Appl.16 257-274.
NEHARI,Z. (1952).Conformal
Mapping,
paperback edition (1975), Dover Publications,New York.
NEWELL, G. F. (1979).Approximate
BehaviorofTandemQueues.Lecture Notes in Economics and
Berlin.
Mathematical Systems 171, Springer-Verlag,
REIMAN,M. I. (1983). Open queueing networksin heavy traffic.Math.Oper.Res.To appear.
Volume III, Part Two (Complex Variables
SMIRNOV,V. I. (1964). A CourseofHigherMathematics,
and Special Functions),Addison-Wesley,Reading, Mass.
STROOCK,D. W. and VARADHAN,
S. R. S. (1971). Diffusionprocesses with boundary conditions.
Comm.PureAppl.Math.24 147-225.
transformation.
SIAM
TREFETHEN, L. N. (1980). Numericalcomputationof the Schwarz-Christoffel
J. Sci. Statist.Comput.1 82-102.
VARADHAN,
S. R. S. and WILLIAMS,R. J. (1985). Brownianmotionin a wedgewithoblique reflection.
Comm.PureAppl.Math.,to appear.
WEISS, A. (1981). Invariant measures of diffusionprocesses on domains with boundaries. Ph.D.
dissertation.Dept. of Mathematics,New York University.
WENOCUR,M. (1982). A productionnetworkmodel and its diffusionapproximation.Ph.D. dissertation. Dept. of Statistics,StanfordUniversity.
WILLIAMS, R. J. (1983). Reflected Brownian motion in a wedge with oblique reflectionat the
boundary.Ph.D. dissertation,Dept. of Mathematics,StanfordUniversity.
J.M. HARRISON
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