On bounded continuous solutions of the archetypal equation with rescaling

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Article:
Bogachev, LV, Derfel, G and Molchanov, SA (2015) On bounded continuous solutions of
the archetypal equation with rescaling. Proceedings of the Royal Society A: Mathematical,
Physical and Engineering Sciences, 471 (2180). 20150351. ISSN 1364-5021
https://doi.org/10.1098/rspa.2015.0351
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On bounded continuous
solutions of the archetypal
rspa.royalsocietypublishing.org
equation with rescaling
1 2
Leonid V. Bogachev , Gregory Derfel and
Research
3
Stanislav A. Molchanov
1
Articlesubmittedtojournal DepartmentofStatistics,SchoolofMathematics,
UniversityofLeeds,LeedsLS29JT,UK
2
DepartmentofMathematics,BenGurionUniversityof
SubjectAreas:
theNegev,Be’erSheva84105,Israel
mathematicalanalysis,functional
3
DepartmentofMathematics,UniversityofNorth
equations,probabilitytheory
CarolinaatCharlotte,CharlotteNC28223,USA
Keywords:
functional&functional-differential The ‘archetypal’ equation with rescaling is given
equations,pantographequation, by y(x)= R2y(a(x−b))µ(da,db) (x∈R), where
µ is a prRoRbability measure; equivalently, y(x)=
Markovchain,harmonicfunction,
E{y(α(x−β))}, with random α,β and E denoting
martingale,stoppingtime
expectation.Examplesinclude:(i)functionalequation
y(x)= ipiy(ai(x−bi)); (ii) functional-differential
Authorforcorrespondence: (‘pantoPgraph’)equationy′(x)+y(x)= ipiy(ai(x−
LeonidV.Bogachev ci)) (pi>0, ipi=1). Interpreting sPolutions y(x)
as harmonicPfunctions of the associated Markov
e-mail:[email protected]
chain(Xn),weobtainLiouville-typeresultsasserting
that any bounded continuous solution is constant.
In particular, in the ‘critical’ case E{ln|α|}=0 such
a theorem holds subject to uniform continuity of
y(x); the latter is guaranteed under mild regularity
assumptions on β, satisfied e.g. for the pantograph
equation (ii). For equation (i) with ai=qmi (mi∈
Z, ipimi=0), the result can be proved without
thePuniformcontinuityassumption.Theproofsutilize
the iterated equation y(x)=E{y(Xτ)|X0=x} (with
a suitable stopping time τ) due to Doob’s optional
stoppingtheoremappliedtothemartingaley(Xn).
1. Introduction
(a) The archetypal equation
Thispaperconcernstheequationwithrescaling(referred
toas‘archetypal’)oftheform
y(x)= y(a(x−b))µ(da,db), x∈R, (1.1)
ZZ
R2
where µ(da,db) is a probability measure on R2. The
integral in (1.1) has the meaning of expectation with
(cid:13)c TheAuthor(s)PublishedbytheRoyalSociety.Allrightsreserved.
respecttoarandompair(α,β)withdistributionP{(α,β)∈da×db}=µ(da,db);thus,equation
2
(1.1)canbewrittenincompactformas
y(x)=E{y(α(x−β))}, x∈R. (1.2) ....rspa
AshasbeenobservedbyDerfel[1](andwillbeillustratedbelowin§1(b)),thisequationisa .....roy
richsourceofvariousfunctionalandfunctional-differentialequationswithrescaling,specifiedby .a
..ls
asuitablechoiceoftheprobabilitymeasureµ(i.e.thedistributionof(α,β)).Itisforthatreason .o
.c
thatweproposetocall(1.2)(aswellasitsintegralcounterpart(1.1))thearchetypalequation(AE). ..ie
..ty
Thestudyofthisequationallowsonetoenhanceandunifyearlierresultsforparticularsubclasses .p
ofequationswithrescaling,whilemakingtheanalysismoretransparentandefficient(cf.[2]). ...ub
..lis
Notingthatanyfunctiony(x)≡constsatisfiestheAE(1.2),itisnaturaltoinvestigatewhether .h
..in
tnhaetrueraalrleyaanriysensonin-trthiveiaclo(ni.tee.xntoonf-cfuonncsttiaonnta)lbaonudndfuedncctoionntainl-udoifufser(ebn.cti.)alsoeqluutaiotinosn.sSwucihthareqsuceasltiinogn, ......g.org
where the possible existence of bounded solutions (e.g. periodic, almost periodic, compactly ..P
supported,etc.)isofmajorinterestinphysicalandotherapplications(seee.g.[3–6]).Solutions ....roc
understudymayalsobeboundedbynature,e.g.representingaruinprobabilityasafunctionof ..R
.
ainsitaiafilrcsatpstiteaplt[o7,w8]a.rOdnsathfeuloltdheesrchriapntdio,ncoonffithneinagsyomnepsteolftitcobbeohuanvdioeudrsooflusotilountisomnsa.ybeconsidered ......SocA
Thus,thegoalofthepresentpaperistogiveconditionsonthedistributionµoftherandom ..0
coefficients(α,β),underwhichanyb.c.-solutionofequation(1.2)isconstantonR.Forshorthand, ....000
cwoemrpelfeexrtaonasltyasteismaenndtshoafrmthoisnikcinfudnactsioLniotuhveiloler-yt,ybpeeatrhineogrienmmsibnydatnhaaltoyg(yx)wiinth(1s.i2m)iislaarwreesuiglhtsteidn .....000
averageofothervalues,thusresemblingtheusualharmonicfunction.Moredetailshighlighting
thepertinenceof‘harmonicity’inthecontextofequation(1.2)areprovidedin§1(d).
Remark 1.1. Continuity of y(x) (or some other regularity assumption) is needed to avoid
pathologicalsolutions,asiswellknowninthetheoryoffunctionalequations(cf.[9,Ch.2]).For
example,allb.c.-solutionsoftheequationy(x)=py 1(x+1) +(1−p)y 1(x−1) (0≤p≤1)
2 2
areconstantbyTheorem1.1(a)statedbelow,butift(cid:0)hecontin(cid:1)uityrequirem(cid:0) entisdr(cid:1)oppedthen
one can easily construct other bounded solutions, e.g. the Dirichlet function y(x)=1Q(x) (i.e.
y(x)=1ifxisrationalandy(x)=0otherwise),whichiseverywherediscontinuous.
(b) Some subclasses of the archetypal equation; historical remarks
Beforeoutliningourresults,weillustratetheremarkablecapacityofequation(1.2)justifying
thename‘archetypal’.Generalsurveysoffunctionalandfunctional-differentialequationswith
rescalingarefoundinDerfel[10]andBaron&Jarczyk[11],bothwithextensivebibliographies.
(i)Functionalequationsandself-similarmeasures
Tostartwith,inthesimplestcaseα≡1equations(1.1),(1.2)arereducedto
y(x)=Z y(x−t)µβ(dt) ⇐⇒ y(x)=E{y(x−β)}, (1.3)
R
where µβ(dt):=P(β∈dt). This equation (sometimes called the integrated Cauchy functional
equation [12]) plays a central role in potential theory and harmonic analysis on groups [13,14],
andisalsoprominentinprobabilitytheoryinrelationtorenewaltheorems[15,§XI.9],Markov
chains[16,Ch.5],queues[17,§III.6],characterizationofprobabilitydistributions[12,Ch.2],etc.
ALiouville-typeresultinthiscaseisrenderedbythecelebratedChoquet–Denytheorem[18](see
also[12]andreferencestherein).
Notethatequation(1.3)canbewrittenintheconvolution1formy=y⋆µβ.Moregenerally,if
αhasadiscretedistribution(withatomsaiandmassespi)then,denotingbyµβi theconditional
1Theconvolutionbetweenfunctiony(x)andmeasureσinRisdefinedasy⋆σ(x):=RRy(x−t)σ(dt).
distributionofβgivenα=ai,theAE(1.2)isconvenientlyexpressedinconvolutions, 3
y(x)= p y(a x)⋆µi.
i i β
i
Forapurelydiscretemeasureµ,withatomsX(ai,bi)andmassespi=P(α=ai,β=bi),theAE(1.2) ....rspa
specializesto .....roy
y(x)=Xipiy(ai(x−bi)). (1.4) ....also
Ifallai>1then(1.4)isanexampleofHutchinson’sequation[19]forthedistributionfunctionof ...cie
aself-similarprobabilitymeasurewhichisinvariantunderafamilyofcontractions(here,affine ..ty
.p
transformationsx7→bi+x/ai).Animportantsubclassof(1.4),withai≡a>1,isexemplifiedby ...ub
y(x)=12y(a(x+1))+ 12y(a(x−1)). .....lishin
Twhhiesreeqtuhaetsioignndseasrcercibheossethnein(sdeelfp-esinmdielnartl)ydwisittrhibpurtoiobnabfiulintycti21o.nCohfatrhaectrearnizdaotmionseorfiethsiPsd∞nis=t0ri±buat−ionn, .......g.org
forRdeiftfuerrneinntgat>oe1qiusatthioento(p1i.4ca)lwBietrhnaouil≡liaco>nv1o,luthtieondsenprsoitbylezm(x[2)0:=]. y′(x)(ifitexists)satisfies .....Proc
..R
.
z(x)=aXipiz(a(x−bi)), (1.5) ....Soc
oftencalledthetwo-scaledifferenceequationorrefinementequation[21].Constructionofcompactly ..A
supportedcontinuoussolutionsof(1.5)playsacrucialroleinwavelettheory[22,23]andalsoin ...00
subdivisionschemesandcurvedesign[4,24],whichisarapidlygrowingbranchofapproximation ...00
.0
theory.Aspecialversionof(1.5)knownasSchilling’sequation ...00
.
z(x)=α 1z(αx+1)+ 1z(αx)+ 1z(αx−1)
4 2 4
(cid:2) (cid:3)
arises in solid state physics in relation to spatially chaotic structures in amorphous materials
[5, p.230], where the existence of compactly supported continuous solutions is again of major
interest;see[25]forafullcharacterizationofthisproblemintermsofarithmeticalpropertiesofα.
(ii)Functional-differentialequations
Let us now turn to the situation where the distribution of β conditioned on α is absolutely
continuous(i.e.hasadensity).ItappearsthatforcertainsimpledensitiestheAE(1.2)produces
some well-known functional-differential equations. An important example is the celebrated
pantograph equation, introduced by Ockendon & Tayler [26] as a mathematical model of the
overheadcurrentcollectionsystemonanelectriclocomotive.2 Initsclassical(one-dimensional)
formthepantographequationreads
y′(x)=c0y(x)+c1y(αx). (1.6)
Thisequationanditsramificationshaveemergedinastrikingrangeofapplications,including
number theory [7], astrophysics [28], queues & risk theory [29], stochastic games [8], quantum
theory [6], and population dynamics [30]. The common feature of all such examples is some
self-similarity of the system under study. Thorough asymptotic analysis of equation (1.6) was
given by Kato & McLeod [31]. A more general first-order pantograph equation (with matrix
coefficients, and also allowing for a term with a rescaled derivative) was studied by Iserles
[27], where a fine geometric structure of almost-periodic solutions was also described. Further
developmentsincludeanalysisinthecomplexdomain[32],higher-orderequations[27,33],and
stochasticversions[34].AmongrecentimportantanalyticresultsisaproofbydaCostaetal.[35]
oftheunimodalityofsolutionswhichplaysasignificantroleinmedicalimagingoftumours[36].
Abalancedversionofthepantographequationisgivenby(see[1,2])
y′(x)+y(x)= piy(ai(x−ci)), pi>0, pi=1. (1.7)
i i
X X
Asexplainedin§3(c),equation(1.7)isessentiallyequivalenttotheAE(1.2)wherebyαisdiscrete,
withP(α=ai)=pi,andβconditionedonα=aihastheunitexponentialdistributionon(ci,∞),
2Theterm‘pantographequation’wascoinedbyIserles[27].
with the density function eci−t1 (t). The discreteness of α is not significant here, and a
(ci,∞) 4
similarconnectionwiththeAEholdsformoregeneralintegro-differentialequations(cf.[37])
where γ is a randyo′(mx)v+aryia(bxl)e=anEd{yµ(αα,(γx(d−a,γd)}c)≡=ZZPR(2αy∈(ad(ax,−γ∈c)d)cµ)αi,sγ(thdae,ddics)t,ribution of (α(,1γ.8)). .........rspa.roya
Higher-orderpantographequationscanalsobededucedfromtheAE,e.g. ..ls
.o
.c
−y′′(x)+y(x)= p y(a (x−c )), p >0, p =1, ..ie
i i i i i i i ..ty
X X .p
andmoregenerally(cf.[2]) ...ub
C2y′′(x)+C1y′(x)+y(x)=E{y(α(x−γ)} (C1,C2∈R, C12−4C2≥0). .....lishin
onF[−or1a,n1]e,xthamenpelequoaftaiodnif(f1e.r2e)nitskreinddu,cteadketoα≡2andassumethatβhastheunformdistribution ......g.org
2 2 ..P
y(x)=Zx+1/2y(2u)du. (1.9) ......rocR
x−1/2 .
.S
Differentiating(1.9),forz(x):=y′(x)weobtainRvachev’sequation[3] ...oc
..A
z′(x)=2 z(2x+1)−z(2x−1) . (1.10) ...00
(cid:2) (cid:3) ...00
Acompactlysupportedsolutionof(1.10)(calledthe‘up-function’)anditsgeneralizations(unified .0
underthenameatomicfunctions)haveextensiveapplicationsinapproximationtheory(see[3,24] ...00
.
andreferencestherein);allsuchfunctionscanbeobtainedassolutionsofsuitableversionsofthe
AE(1.2)(see[1]).
(c) Main results
Letussummarizeourresults.First,certaindegeneratecaseswarrantaseparateanalysisbut
needtobeexcludedingeneraltheory,namely:(i)α=0withpositiveprobability;(ii)|α|≡1;and
(iii) α(c−β)≡c for some c∈R (resonance). Note that (ii) includes the case α≡1 settled in the
Choquet–Denytheoremmentionedin§1(b)-i;in§2(b)wegeneralizethisresult(Theorem2.3).As
forcases(i)and(iii),aLiouvilletheoremholdshereunconditionally,whichiseasytoprovefor
(i),analyticallyandprobabilisticallyalike(seeTheorem2.1).Intheresonancecase(iii),theproof
ismoreinvolvedrelyingheavilyontheChoquet–Denytheorem(see§2(c)),buttheresultitselfis
quitelucidandappealing.
Inthenon-degeneratesituation,existenceofnon-trivialb.c.-solutionsisessentiallygoverned
bythesignofK:= ln|a|µ(da×db)=E{ln|α|}.Moreprecisely,onecanprove(see[38])the
R2
followingdichotomyRRbetweenthesubcritical(K<0)andsupercritical(K>0)regimes.
Theorem1.1. SupposethatK=E{ln|α|}isfiniteandE{lnmax(|β|,1)}<∞.
(a) IfK<0thenanyb.c.-solutionoftheAE(1.2)isconstant.
(b) If K>0 and α>0, then there is a b.c.-solution of (1.2) given by the distribution function
FΥ(x):=P(Υ ≤x), where Υ := ∞n=1βn jn=−11αj−1 and {(αn,βn)} is a sequence of
independentidenticallydistributed(Pi.i.d.)randQompairswiththesamedistributionµas(α,β).
Remark1.2. Almostsure(a.s.)convergenceoftherandomseriesΥ (forα6=0)andcontinuityof
FΥ(x)onRwereprovedbyGrintsevichyus[39].
Remark 1.3. The result of Theorem 1.1 was obtained by Derfel [1] under a stronger moment
condition E{|β|}<∞ and only for α>0 (which is essential in (b) but not in (a)); however, his
argumentsholdinthegeneralcasewithminorchanges.
Remark1.4. IncontrastwiththesubcriticalcaseK<0,whichisinsensitivetothesignofα(see
Theorem1.1(a)),thesupercriticalcaseK>0ismoredelicate:ifP(α<0)>0theny=FΥ(x)isno
longerasolutionoftheAE(1.2);e.g.ifα<0(a.s.)thenthisfunctionsatisfiestheequationy(x)=
5
1−E{y(α(x−β))}(cf.[39,Eq.(5)]).Moreover,onecanprove[38]thatanyboundedsolutionof
(1.2) with limits at ±∞ is constant; thus, any non-trivial solution must be oscillating, which is
drasticallydifferentfromthecaseα>0(a.s.). ....rspa
The critical case K=0 is much more challenging, and it has remained open since [1]. More .....roy
.a
recently, for a pantograph equation (1.8) without shift (i.e. γ≡0) and some second-order ..ls
.o
extensions, aLiouville theoremin thecaseK=0 wasestablished byBogachev et al.[2]. In the ...cie
presentpaper,weprovethefollowinggeneralresult(cf.Theorem3.1below). ..ty
.p
...ub
Theorem1.2. AssumethatP(|α|=6 1}>0andK=E{ln|α|}=0.If y(x)isaboundedsolutionofthe ...lish
AE(1.2)whichisuniformlycontinuousonR,thenitisconstant. ........ing.org
.
Although an unwanted restriction, the uniform continuity assumption can be shown to be .P
satisfiedprovidedthereexiststheprobabilitydensityofβ conditionedonα(Theorem3.3).An ....roc
alternativecriteriontailoredtothemodelβ=γ+ξ withξ independentof(α,γ)(Theorem3.4) ..R
.
.S
is applicable to a large class of examples including the pantograph equation (1.8) and its ...oc
generalizations (§3(c)). As a consequence, we obtain a Liouville theorem for the general ..A
(balanced)pantographequationinthecriticalcase(cf.Theorem3.8),significantlyextendingthe ..0
.0
resultofBogachevetal.[2].Inparticular,forthefirst-orderpantographequation(1.8)wehave ...00
.0
...00
Theorem1.3. If P(|α|=6 1}>0andK=E{ln|α|}=0,thenanyboundedsolutionof (1.8)isconstant. .
Remark1.5. Asexplainedin§3(d)below,Theorem1.3extendstothecase|α|=1(a.s.)byvirtue
ofthegeneralizedChoquet–Denytheoremprovedin§2(b).
On the other hand, for a subclass of functional equations (1.4) with multiplicatively
commensurable coefficients {ai} (i.e. ai=qmi, q>1, mi∈Z), where the random shift β is
discrete and therefore the criteria of Theorems 3.3, 3.4 do not apply, the Liouville theorem in
thecriticalcaseK=0(i.e. ipimi=0)canbeprovedbyadifferentmethodthatcircumvents
thehypothesisofuniformcPontinuity(Theorem3.9).
(d) The method: associated Markov chain and iterations
Inthissubsection,wedescribetheprobabilisticapproachtotheAEbasedonMarkovchains
andmartingales,andintroducesomebasicnotationanddefinitions.
ConsideraMarkovchain(Xn)onRdefinedrecursivelyby
Xn=αn(Xn−1−βn) (n∈N), X0=x, (1.11)
where {(αn,βn)}n∈N is a sequence of i.i.d. random pairs with the same distribution as (α,β)
(see(1.2)).NotethattheAE(1.2)canthenbeexpressedas
y(x)=Ex{y(X1)}, x∈R, (1.12)
whereindexxintheexpectationreferstotheinitialconditionin(1.11).Thatistosay,anysolution
oftheAE(1.2)isaharmonicfunctionoftheMarkovchain(Xn)(cf.[16,p.40]).
Remark1.6. BoundedharmonicfunctionsplayaparamountroleinthegeneraltheoryofMarkov
chains(seemoredetailsandsomereferencesinappendixB.
Remark 1.7. Stochastic recursion (1.11) is well known in the literature as the random difference
equation(seee.g.[40–43]andfurtherreferencestherein).
Notethatequation(1.12)propagatesalongtheMarkovchain(Xn),i.e.foranyn∈N 6
y(x)=Ex{y(Xn)}, x∈R. (1.13)
Equivalently,anintegralformofequation(1.13)canbeobtainedbyiteratingforwardtheAE(1.1). ....rspa
Therecursion(1.11)canalsobeiteratedtogiveexplicitly .....roy
.a
Xn=Anx−Dn (n∈N), (1.14) ....lsoc
..ie
where ..ty
.p
n n n ...ub
An:= αk, Dn:= βk αj. (1.15) ..lis
kY=1 kX=1 jY=k ...hin
Recall that K=E{ln|α|}. In the subcritical case (K<0), for the proof of Theorem 1.1(a) it ......g.org
sufficestoconsideriterations(1.13)asn→∞.Indeed,Kolmogorov’sstronglawoflargenumbers .
.P
impliesthatSn:= nk=1ln|αk|→−∞andhence|An|=exp(Sn)→0a.s.;inviewof(1.14),this ....roc
indicatesthattheriPght-handsideof(1.13)eventuallybecomesx-free(seemoredetailsin[38]).In ..R
.
(tha.es.c)r;ihtiecnaclec,aasets(Kom=e0r)a,ntdheomrantidmoemswτǫawlke(hSanv)eis|Areτcǫu|r=reenxtpb(uStτnǫ)on<eǫth(feolreassnlyimǫ>inf0n),→w∞hiSchn=can−b∞e ......SocA
usedtoinferthaty(x)≡constinasimilarfashionasbefore. ..0
.0
Expanding this idea, our approach to the analysis of equation (1.2), first probed in [2], is ...00
basedonreplacingafixednintheiteratedequation(1.13)byasuitablestoppingtimeτ,defined .0
as a random (integer-valued) variable such that for any n∈N the event {τ≤n} is determined ....00
by(α1,β1),...,(αn,βn).Itsufficesforourpurposestoworkwith‘hittingtimes’τB:=inf{n≥
1: An∈B}≤∞,3whereAn=α1···αn(see(1.15))andB⊂Risanintervalorasinglepoint.
Weshallroutinelyusethefollowingcentrallemma(wherecontinuityofy(x)isnotrequired).
Lemma1.4. Let(Xn)betheassociatedMarkovchain(1.11),andτ astoppingtimesuchthatτ<∞a.s.
If y(x)isaboundedsolutionoftheAE(1.2)thenitsatisfiesthe‘stopped’equation
y(x)=Ex{y(Xτ)}, x∈R. (1.16)
Thecrucialfactisthaty(Xn)isamartingale(cf.[16,p.43,Proposition1.8]);indeed,by(1.11)
Ex{y(Xn)|(αk,βk), k<n}=Ex{y(αn(Xn−1−βn))|Xn−1}=y(Xn−1) (a.s.), (1.17)
which verifies the martingale property [44, §10.3, p.94]. The lemma then readily follows by
Doob’s optional stopping theorem (e.g. [44, p.100, Theorem 10.10(b)]). For the sake of a more
self-containedexposition,adirectproofofLemma1.4isincludedasappendixA.
Layout
The rest of the paper is organized as follows. In §2 we work out the degenerate cases
mentionedatthebeginningof§1(c),namely:P(α=0)>0(§2(a)),|α|≡1(§2(b)),andα(c−β)≡c
(§2(c)). In §3(a) we prove our main result for the critical case (Theorem 3.1; cf. Theorem 1.2),
backed up in §3(b) by simple sufficient conditions for the uniform continuity of solutions
(Theorems3.3and3.4).In§3(c)weexplainindetailtheremarkablelinkbetweenthepantograph
equations and the AE, which enables us to prove a Liouville theorem for the general (integro-
differential)pantographequationofanyorder(Theorem3.8).Thisiscomplementedin§3(d)by
a Liouville theorem for the functional equation (1.4) with ai=qmi (Theorem 3.9). Appendix A
providesanelementaryproofofLemma1.4,andappendixBgivesabriefcompendiumofbasic
factsilluminatingthefundamentalroleofboundedharmonicfunctionsinthegeneraltheoryof
Markovchains.
3Hereandbelow,weadopttheconventionthatinf∅:=∞.
2. Three degenerate cases
7
BeforeembarkingonageneraldiscussionoftheAE,weneedtostudytheproblemofbounded
csoolnustiidoenrsatiinoncethrtearienafstpeer.cRiaelccaallstehseonfoptaotsisoinblAenv:a=luQesnko=f1ααkan(sdeeβ(,1.w15h)i)c.h will be excluded from ........rspa.roy
.a
(a) Vanishing of the scaling coefficient ...lso
.c
..ie
Let us consider the case where the scaling coefficient α may take the value zero. Note that ..ty
.p
continuityofsolutionsy(x)isnotassumedapriori. ...ub
..lis
.h
TheWoerefimrs2t.g1.ivSeuapnpoesleemp0e:n=taPr(yα‘a=na0l)y>tic0’.pTrhoeonfaonfytbhoiusnsdimedpsloeluthtieoonroefmthaenAdEt(h1e.2n)pisrecsoennsttaanntootnhRer. .........ing.org
.P
prooftoillustratethemethodbasedonLemma1.4. ....roc
..R
Proofof Theorem2.1. Withy0(x):=y(x)−y(0),equation(1.1)maybewrittenintheform ..S
...oc
y0(x)=(1−p0)ZZR2\{a=0}y0(a(x−b))µ˜(da,db), x∈R, (2.1) .....A00
wnohremreoµ˜n:R=,(fr1o−mp(02).1−)1wµ,esoobttahiant µ˜(R2\{a=0})=1. Denoting by kfk:=supx∈R|f(x)| the sup- .......00000
.
|y0(x)|≤(1−p0)ky0k (x∈R) =⇒ ky0k≤(1−p0)ky0k. (2.2)
Since1−p0<1,thesecondinequalityin(2.2)immediatelyimpliesthatky0k=0,andthenthe
firstinequalitygivesy0(x)≡0,i.e.y(x)≡y(0),asclaimed.
Alternativeproofof Theorem2.1. Considerthestoppingtimeτ0:=inf{n≥1: An=0}.Notethat
P(τ0>n)=P(An6=0)=P(α16=0,...,αn6=0)=(1−p0)n→0 (n→∞),
henceτ0<∞a.s.Now,usingtheiterationformulas(1.14),(1.15),andnotingthatAτ0=0a.s.,by
Lemma1.4weobtain
y(x)=Ex{y(Xτ0)}=E{y(xAτ0 −Dτ0)}=E{y(−Dτ0)}, x∈R. (2.3)
Sincetheright-handsideof(2.3)doesnotdependonx,itfollowsthaty(x)=const.
(b) An extension of the Choquet–Deny theorem
(i)Theclassicalcaseα≡1
Asmentionedin§1(b),theAE(1.2)withα≡1isreducedto
y(x)=E{y(x−β)}, x∈R. (2.4)
ThefamousChoquet–Denytheorem[18](cf.[15,p.382,Corollary]or[16,p.161,Theorem1.3])
assertsthatanyb.c.-solutionof(2.4)isconstantprovidedthat(thedistributionof)theshiftβ is
non-arithmetic,i.e.notsupportedonanysetλZ={λk, k∈Z}(withspanλ∈R).
Remark 2.1. In connection with the uniform continuity condition in Theorem 1.2, it may be of
interest to note that some proofs of the Choquet–Deny theorem (e.g. [15, p.382]) deploy the
convolutiony˜(x)=y⋆ϕ0,σ2(x)ofab.c.-solutiony(x)withthedensityfunctionϕ0,σ2(x)ofthe
normal distribution with zero mean and variance σ2, whereby the function y˜(x) is uniformly
continuousandstillsatisfiesequation(2.4).Onceithasbeenprovedthaty˜(x)isconstant,thisis
extendedtotheoriginalsolutiony(x)bytakingthelimitasσ→0.
A discrete version of the Choquet–Deny theorem [18] (see also [45, §XIII.11] or [46, p.276,
8
Theorem T1]) refers to the case where equation (2.4) is considered on Z and β is integer-
valued.Namely,assumethatthesmallestadditivegroupcontainingtheset{x∈Z: P(β=x)>0}
coincideswithZ;thenthetheoremassertsthaty(x)≡constforallx∈Z. ....rspa
of bIn.c.t-hsoelcuotinotnesxtinoftheequaartiitohnm(e2t.i4c)coanseth(eexwcluhdoilneglinthee, tdheisgeennearbalteescoanseetβo≡gi0v)e. Tahfeulnledxetsrcersiputlitoins ......roya
essentiallywellknowninfolklore;wegiveitsproofforthesakeofcompleteness. ...lso
.c
..ie
Theorem2.2. Assumethatthedistributionofβ isarithmetic,i.e.itssupportΛiscontainedintheset ..ty
.p
λZwithmaximalspanλ>0.Thenthegeneralb.c.-solutionofequation(2.4)isgivenbyy(x)=g(x/λ), ...ub
whereg(·)isanycontinuousperiodicfunctionofperiod1. ..lis
.h
..in
wPriotohf.λWZ.eInstdaeretdb,yfosrhnow∈iNngletthdantt∈heλNsmbaelltehsetgardedaitteisvtecsoumbmgroonupdiGvi⊂soλrZofgtehnee(rfianteitde)bsyetΛΛcno:i=nc{idse∈s ......g.org
Λin:te|gse|r≤sλmni},⊂anGd.BityfoBlélozwoust’tshiadtednntit∈yG(s(ene∈e.Ng.).[4S7i,n§c1e.2th])ewseeqhuaevnecedndn=/Pλ∈siN∈Λinsmnoisni-iwncitrheassoinmge, ......Proc
thereexiststhelimitk∗:=limn→∞dn/λ=dn∗/λ∈NandsoΛ⊂k∗λZ.Butλ>0isthemaximal ...R
spaNn,ohwe,nitceiske∗a=sy1toansdeeththuastλeq=udatni∗o∈nG(2,.4w)hsipclhitisminptloiesseGpa=raλtZe,daissccrleatiemeeqdu.ationsz˜(k)=E{z˜(k− ......SocA
β/λ)} on every coset x0+λZ (x0∈[0,λ)), where z˜(k):=z(x0+kλ) (k∈Z). The discrete ...00
Choquet–Denytheoremshowsthatz˜(k)isconstantonZ;inotherwords,anyboundedsolution ...00
of(2.13)onRisλ-periodic,andtheclaimofthetheoremeasilyfollows. ....000
.
Remark2.2. Itisevidentthatanyfunctiony(x)=g(x/λ)satisfiesequation(2.4);themainpointof
Theorem2.2isthattherearenootherb.c.-solutions.
Remark2.3. Laczkovich[48]givesafullcharacterizationofnon-negativemeasurablesolutionsof
theequationy(x)= ℓi=1Ciy(x−bi)(witharbitrarycoefficientsC1,...,Cℓ>0)intermsofthe
realrootsofthecharPacteristicequation ℓi=1Cie−bis=1.
P
Remark 2.4. The original proof by Choquet & Deny [18] (as well as many subsequent proofs
and extensions) is based on a reduction to a uniformly continuous solution (cf. Remark 2.1)
andestablishingthatthelattermustreachitsmaximumatafinitepointx0∈R.Inviewofthe
martingaletechniquesusedinthepresentpaperforageneralAE,itisofinteresttopointoutan
elegantmartingaleprooffoundbySzékely&Zeng[49](cf.Rao&Shanbhag[12,Ch.3]).
(ii)Case|α|≡1
WeprovehereanextensionoftheChoquet–Denytheoremforαtakingthevalues±1;tothebest
ofourknowledge,sucharesulthasnotyetbeenmentionedintheliterature.
Theorem2.3. Supposethat|α|≡1andP(α=1)<1.Letβ+,β−havethedistributionof βconditioned
onα=1andα=−1,respectively(incaseα≡−1,setβ+≡0).
(a) If β+isnon-arithmetictheneveryb.c.-solutionofequation(1.2)isconstant.
(b) Letβ+bearithmeticwithspanλ6=0.
(b-i) If the distribution of β−is not supported on any set λ0+λZ (λ0∈R), then every
b.c.-solutionofequation(1.2)isconstant.
(b-ii) Otherwise,thegeneralb.c.-solutionofequation(1.2)isoftheformy(x)=g(x/λ),where
g(·)isacontinuous1-periodicfunctionsymmetricaboutpointx0=21λ0/λ,i.e.g(x0−
x)=g(x0+x) (x∈R).
Remark 2.5. In part (b-ii), all functions with the required symmetry property may be
represented (though not uniquely) as g(x)=g0(x−x0)+g0(−x+x0), where g0(·) is an
arbitrary (continuous) 1-periodic function. It is straightforward to check that so constructed
functions y(x)=g(x/λ) satisfy equation (1.2) (with |α|≡1), but part (b-ii) asserts that all
b.c.-solutionsarecontainedinthisclass.
Proof of Theorem2.3. Considerthestoppingtimeτ1:=inf{n≥1: An=1}.Withp1:=P(α=1)< 9
1andq1:=P(α=−1)=1−p1>0,thedistributionofτ1isgivenby4
P(τ1=1)=p1, P(τ1=n)=q12p1n−2 (n≥2), (2.5) ....rspa
henceτ1<∞a.s.SinceAτ1=1(a.s.),from(1.14)wehaveXτ1=x−Dτ1,andbyLemma1.4 .....roy
.a
y(x)=E{y(x−Dτ1)}, x∈R. (2.6) ...lso
.c
Now, in view of the Choquet–Deny theorem, we need to investigate whether the random shift ..ie
aDsτβ1L+eista(nnβodn+n-)βaa−rni,tdrhem(sβpen−etic)ct,ibvie.eetl.wyP.o(CDsoeτnq1du∈ietλinoZcne)isn<ogf1oif.nio.dτr1.arnaanynddλou∈msiRnv.gar(i1a.1b5le)saenadch(2,.w5)i,thwtehoebstaaminedistribution ...........typublishin
P(Dτ1∈λZ)=p1P(β1+∈λZ)+ ∞ q12p1n−2P(β1−−β2+−···−βn+−1−βn−∈λZ) ......g.org
nX=3 ..P
≤p1+q12 ∞ p1n−2=1. (2.7) ......rocR
nX=2 ..S
Ifp1>0then(2.7)impliesthatP(Dτ1∈λZ)<1unlessβ1+∈λZ(a.s.)andforalln≥2 .....ocA
β1−−β2+−···−βn+−1−βn−∈λZ (a.s.). (2.8) ......0000
Snin≥c2e;ahllenβci+e,acroeni.di.idti.o,nthse(2fi.8rs)taorfetrheedsueciendclutosioβn−s−imβp−l∈iesλZtha(at.sβ.2)+. +In·t·u·r+n,βtn+h−e1la∈stλZcoan.ds.itfioornailsl .....000
1 2
equivalenttoβ−∈λ0+λZforsomeλ0∈R.Indeed,applyingLebesgue’sdecompositiontheorem
[15,p.142]toeachofthei.i.d.randomvariablesβ−andβ−,itisevidentthatthecontinuouspart
1 2
oftheircommondistributionmustvanish,sothatthisdistributionispurelydiscrete;furthermore,
its(countable)support{bi}satisfiestheconditionbi−bj∈λZforalli,j,andtheclaimfollows.
Asimilarargumentisalsovalidforp1=0,wherebyβ+=0, β−=β,and(2.7)simplifiesto
P(Dτ1∈λZ)=P(β1−β2∈λZ)≤1.
Thiscompletestheproofofparts(a)and(b-i).
Finally,weprovepart(b-ii),wherebyβ+∈λZ,β−∈λ0+λZandDτ1∈λZ(a.s.).ByTheorem
2.2,anyb.c.-solutionofequation(2.6)mustbeoftheformy(x)=g(x/λ),withsome1-periodic
functiong(·).Substitutingthisintotheoriginalequation(2.6)(with|α|≡1)weget
g(x/λ)=p1E{g((x−β+)/λ)}+q1E{g((−x+β−)/λ)}
=p1g(x/λ)+q1g((−x+λ0)/λ), (2.9)
andsinceq16=0itfollowsthatg(·)satisfiesthefunctionalequationg(x)=g(−x+λ0/λ) (x∈R),
whichisequivalenttothesymmetryconditionstatedinthetheorem.
Remark2.6. Ifα≡1thenβ+≡β, β−≡0,andparts(a)and(b-ii)ofTheorem2.3formallyrecover
theChoquet–Denytheorem.Noextrarequirementong(·)arisesfrom(2.9),sinceq1=0.
Remark 2.7. Note that the values α=1 and α=−1 (and the corresponding conditional
distributions of β represented by β+ and β−, respectively) feature in Theorem 2.3 in a non-
symmetricway:e.g.ifβ+hasanon-arithmeticdistributionthen,accordingtopart(a),thereareno
b.c.-solutionsexceptconstants,irrespectivelyofβ−;however,ifβ+isarithmeticwithspanλ6=0
whilst β−is non-arithmetic but supported on a set λ0+λZ (i.e. with λ06=0 incommensurable
withλ)thenthereexistnon-trivialb.c.-solutions,accordingtopart(b-ii).
Example2.1. Theorem2.3withα≡−1isexemplifiedbytheequations
(a) y(x)= ∞y(t−x)e−tdt (equivalenttothepantographequationy′(x)+y(x)=y(−x),
0
cf.(1.7)R),whichbypart(b-i)hasonlyconstantb.c.-solutions;
4Theseformulasincludethecasep1=0(undertheconvention00:=1),wherebyτ1=2a.s.