A comparison principle based on couplings of partial integro-differential operators

Della Co e, Se ena; Fuchs, Fabian; K aaij, Richa d; Nendel, Max
Wo king Pape
A compa ison p inciple based on couplings o pa ial
in eg o-di e en ial ope a o s
Cen e o Ma hema ical Economics Wo king Pape s, No. 696
P o ided in Coope a ion wi h:
Cen e o Ma hema ical Economics (IMW), Biele eld Uni e si y
Sugges ed Ci a ion: Della Co e, Se ena; Fuchs, Fabian; K aaij, Richa d; Nendel, Max (2024) : A
compa ison p inciple based on couplings o pa ial in eg o-di e en ial ope a o s, Cen e o
Ma hema ical Economics Wo king Pape s, No. 696, Biele eld Uni e si y, Cen e o Ma hema ical
Economics (IMW), Biele eld,
h ps://nbn- esol ing.de/u n:nbn:de:0070-pub-29941902
This Ve sion is a ailable a :
h ps://hdl.handle.ne /10419/306536
S anda d-Nu zungsbedingungen:
Die Dokumen e au EconS o dü en zu eigenen wissenscha lichen
Zwecken und zum P i a geb auch gespeiche und kopie we den.
Sie dü en die Dokumen e nich ü ö en liche ode komme zielle
Zwecke e iel äl igen, ö en lich auss ellen, ö en lich zugänglich
machen, e eiben ode ande wei ig nu zen.
So e n die Ve asse die Dokumen e un e Open-Con en -Lizenzen
(insbesonde e CC-Lizenzen) zu Ve ügung ges ell haben soll en,
gel en abweichend on diesen Nu zungsbedingungen die in de do
genann en Lizenz gewäh en Nu zungs ech e.
Te ms o use:
Documen s in EconS o may be sa ed and copied o you pe sonal
and schola ly pu poses.
You a e no o copy documen s o public o comme cial pu poses, o
exhibi he documen s publicly, o make hem publicly a ailable on he
in e ne , o o dis ibu e o o he wise use he documen s in public.
I he documen s ha e been made a ailable unde an Open Con en
Licence (especially C ea i e Commons Licences), you may exe cise
u he usage igh s as speci ied in he indica ed licence.
h ps://c ea i ecommons.o g/licenses/by/4.0/
696
Oc obe 2024
A COMPARISON PRINCIPLE
BASED ON COUPLINGS OF PARTIAL
INTEGRO-DIFFERENTIAL OPERATORS
Se ena Della Co e, Fabian Fuchs, Richa d C. K aaij, and Max Nendel
Cen e o Ma hema ical Economics (IMW)
Biele eld Uni e si y
Uni e si ¨a ss aße 25
D-33615 Biele eld ·Ge many
e-mail: [email p o ec ed]
uni-biele eld.de/zwe/imw/ esea ch/wo king-pape s
ISSN: 0931-6558
Unless o he wise no ed, his wo k is licensed unde a C ea i e Commons
A ibu ion 4.0 In e na ional (CC BY) license. Fu he in o ma ion:
h ps://c ea i ecommons.o g/licenses/by/4.0/deed.en
h ps://c ea i ecommons.o g/licenses/by/4.0/legalcode.en
A COMPARISON PRINCIPLE BASED ON COUPLINGS OF
PARTIAL INTEGRO-DIFFERENTIAL OPERATORS
SERENA DELLA CORTE, FABIAN FUCHS, RICHARD C. KRAAIJ, AND MAX NENDEL
Abs ac . This pape is conce ned wi h a compa ison p inciple o iscosi y solu-
ions o Hamil on–Jacobi (HJ), –Bellman (HJB), and –Isaacs (HJI) equa ions o gen-
e al classes o pa ial in eg o-di e en ial ope a o s. Ou app oach inno a es in h ee
ways: (1) We ein e p e he classical doubling-o - a iables me hod in he con ex
o second-o de equa ions by cas ing he Ishii–C andall Lemma in o a es unc ion
amewo k. This adap a ion allows us o e ec i ely handle non-local in eg al ope a-
o s, such as hose associa ed wi h Lé y p ocesses. (2) We ansla e he key es ima e
on he di e ence o Hamil onians in e ms o an adap a ion o he p obabilis ic no-
ion o couplings, p o iding a uni ied app oach ha applies o di e en ial, di e ence,
and in eg al ope a o s. (3) We s eng hen he sup-no m con ac i i y esul ing om
he compa ison p inciple o one ha encodes con inui y in he s ic opology. We
apply ou heo y o a a ie y o examples, in pa icula , o second-o de di e en ial
ope a o s and, mo e gene ally, gene a o s o spa ially inhomogeneous Lé y p ocesses.
Keywo ds: Compa ison p inciple, iscosi y solu ion, Hamil on–Jacobi-Bellman–Isaacs
equa ion, coupling o ope a o s, Lyapuno unc ion, Jensen pe u ba ion, mixed
opology.
MSC 2020 classi ica ion: P ima y 35J60; 35D40; 45K05; Seconda y 49L25; 49Q22.
1. In oduc ion
In his wo k, we p o ide a new pe spec i e on compa ison p inciples o iscosi y
solu ions o he Hamil on–Jacobi equa ion
−λH =h, λ > 0, h ∈Cb(Rq),(1.1)
o Hamil onians Ho he ype
H (x) = ⟨b(x),∇ (x)⟩+1
2T ΣΣT(x)D2 (x)
+Z (x+z)− (x)−χB1(0)(z)⟨z,∇ (x)⟩µx(dz) + H(∇ (x)) (1.2)
and, mo e gene ally, o hose in Bellman and Isaacs o m
H (x) = sup
θ∈Θ{Hθ (x)−I(x, θ)}and
H (x) = sup
θ1∈Θ1
in
θ2∈Θ2{Hθ1,θ2 (x)−I(x, θ1, θ2)},
wi h Hθand Hθ1,θ2as in (1.2) bu wi h θand (θ1, θ2)dependen coe icien s, espec i ely,
and an app op ia e cos unc ional I.
Mo i a ed by con ex Hamil onians, o which no unique classical o weak solu ions
exis in gene al, [17] in oduced he no ion o iscosi y solu ions. The seminal wo ks
[13], [15], [26], [27], [31] explo e his amewo k o i s -o de equa ions.
Da e: Oc obe 28, 2024.
This wo k was unded by he Deu sche Fo schungsgemeinscha (DFG, Ge man Resea ch Founda-
ion) – SFB 1283/2 2021 – 317210226 and by The Ne he lands O ganisa ion o Scien i ic Resea ch
(NWO), g an numbe 613.009.148.
1
a Xi :2410.19566 1 [ma h.AP] 25 Oc 2024
2 SERENA DELLA CORTE, FABIAN FUCHS, RICHARD C. KRAAIJ, AND MAX NENDEL
Mos mode n compa ison p oo s o ope a o s con aining second-o de e ms a e
based on esul s o [28], [29]. Using hen ecen ad ances o gene alized di e en ials,
[14] p o ided wha is nowadays known as he C andall–Ishii Lemma. An o e iew o e
uniqueness esul s o iscosi y solu ions o degene a e ellip ic equa ions is gi en in he
Use ’s Guide [16].
The ea men o non-local ope a o s was ini ially mo i a ed by p oblems in op imal
con ol heo y; see [2], [3], [34] o ea ly examples wi h non-local ope a o s. The wo k [5]
gi es a non-local e sion o he C andall–Ishii Lemma by adap ing he o iginal p ocedu e
in [16], and [23] ex ends hese esul s o unbounded solu ions. We also e e o [20] o
an o e iew o he Hilbe ian se ing, [8] o compa ison p inciples o con ex mono one
semig oups on spaces o con inuous unc ions, o [18] o he classical well-posedness o
con ex Cauchy p oblems on Lp, o [25] o a compa ison p inciple in he amewo k o
G-Lé y p ocesses, and o [7] o a compa ison p inciple o HJB equa ions on he se o
p obabili y measu es.
Ou app oach and ou main esul s, Theo em 3.1 and Co olla y 3.2, inno a e upon
classical compa ison p inciples in he ollowing h ee ways:
(1) We ein e p e he classical doubling-o - a iables me hod in he con ex o second-
o de equa ions by cas ing he C andall–Ishii Lemma in o a es unc ion ame-
wo k. This adap a ion allows us o e ec i ely handle non-local in eg al ope -
a o s, such as gene a o s o Lé y p ocesses, in he same amewo k as second-
o de ope a o s, pa ing he way o s abili y esul s.
(2) We ansla e he key es ima e on he di e ence o Hamil onians in e ms o an
adap a ion o he p obabilis ic no ion o couplings, p o iding a uni ied app oach
ha applies o bo h con inuous and disc e e ope a o s. We poin ou ha [15]
also discusses a coupling poin o iew, bu only o i s o de ope a o s.
(3) We s eng hen he ypical compa ison p inciple using Lyapuno unc ionals om
a sup-no m con ac i i y esul o wha we call he s ic compa ison p inciple,
c . De ini ion 2.4, which encodes con inui y in he s ic o some imes also called
mixed opology, c . [9], [33].
The esul s a e illus a ed in a ious examples in Sec ion 4. To in oduce he i s wo
inno a ions, we heu is ically ace back he classical doubling-o - a iables p ocedu e
used o ob ain compa ison p inciples o i s and second-o de equa ions. Fo he sake
o exposi ion, we ocus on he θ-independen case.
Gi en a subsolu ion uand supe solu ion o an equa ion o ype (1.1) and, o α > 1,
op imize s (xα, yα) o
u(xα)− (yα)−α
2d2(xα, yα) = sup
x,y∈Rqnu(x)− (y)−α
2d2(x, y)o,(1.3)
one es ima es
sup
x∈Rq
u(x)− (x)≤h(xα)−h(yα)+λhHα
2d2(·, yα)(xα)−H−α
2d2(xα,·)(yα)i.
Consequen ly, compa ison hen holds, i
lim in
α→∞ Hα
2d2(·, yα)(xα)−H−α
2d2(xα,·)(yα)≤0.(1.4)
The es ima e (1.4), hen ansla es in o explici condi ions on H.
When His, o example, o he o m
H (x) = ⟨b(x),∇ (x)⟩+1
2|∇ (x)|2,
A COMPARISON PRINCIPLE BASED ON COUPLINGS 3
he es ima e (1.4) ansla es in o
Hα
2d2(·, yα)(xα)−H−α
2d2(xα,·)(yα)
=⟨b(xα), α(xα−yα)⟩+α2
2d2(xα, yα)−⟨b(yα), α(xα−yα)⟩+α2
2d2(xα, yα)
≤ ⟨b(xα)−b(yα), α(xα−yα)⟩,
which goes o 0 o α→ ∞, i bis one-sided Lipschi z.
Fo second o de ope a o s, howe e , he same s a egy ails since, conside ing, o
example, he Laplacian H (x) = 1
2∆ (x) = 1
2T D2 (x), we ge
Hα
2d2(·, yα)(xα)−H−α
2d2(xα,·)(yα)=2α,
which di e ges as α→ ∞.
The wo ks [28], [29] use he key insigh ha , while he i s o de - iscosi y solu ion
me hod explo es he sequences o op imize s o (1.3) sepa a ely ( ix yαand a y x o
he subsolu ion pa and ice e sa), o second o de equa ions, one needs o ea he
wo sequences join ly. This insigh was la e o malized in [14] and as Theo em 3.2 in
he Use ’s Guide [16], now known as he C andall–Ishii Lemma. The lemma s a es o
equa ions o ype H (x) = 1
2T D2 (x) ha , gi en Xα=D2u(xα)and Yα=D2 (yα)
o hei app op ia e gene aliza ions, we ha e he es ima e
Xα0
0−Yα≤3α
1
−
1
−
1 1
.
Conjuga ing he ma ices wi h
C:=1
√2
1 1
1 1
,(1.5)
i.e. essen ially using C o couple he subsolu ion and supe solu ion p oblems, we a i e
a he desi ed es ima e
1
2T (Xα)−1
2T (Yα) = 1
4T Xα−YαXα−Yα
Xα−YαXα−Yα≤0.(1.6)
We now b ie ly desc ibe he h ee inno a ions (1)–(3).
Inno a ion 1: A es unc ion amewo k. Examining he p oo o he C andall–
Ishii Lemma, we can in e p e he p ocedu e as he cons uc ion o wo es unc ions
ϕα, ψα∈C2(Rq) ha a e squeezed be ween uand on one-hand and α
2d2on he o he .
To be mo e p ecise, we ind ϕα, ψα∈C2(Rq)such ha
u(xα)−ϕα(xα) = sup
x∈Rq{u(x)−ϕα(x)}and (yα)−ψα(yα) = in
y∈Rq{ (y)−ψα(y)},
and
ϕα(xα)−ψα(yα)−α
2d2(xα, yα) = sup
x,y∈Rqnu(x)− (y)−α
2d2(x, y)o.(1.7)
As be o e, compa ison now ollows om he es ima e
lim in
α→∞ Hϕα(xα)−Hψα(yα)≤0.
Fo he Laplacian H (x) = 1
2T D2 (x), his ansla es o
Hϕα(xα)−Hψα(yα) = 1
2T (D2ϕα(xα)) −1
2T (D2ψα(yα)).(1.8)
A his poin in p oo s using he C andall–Ishii Lemma, he es ima e (1.6) is pe o med
by conjuga ion wi h he ma ix Cin (1.5). We o malize his s ep by adap ing he

4 SERENA DELLA CORTE, FABIAN FUCHS, RICHARD C. KRAAIJ, AND MAX NENDEL
p obabilis ic no ion o couplings, c . [10], [30], [37], and iden i y he choice o he ma ix
Cin (1.5) wi h he synch onous coupling (also called co-mono one coupling).
Inno a ion 2: The coupling app oach. Indeed, gi en wo B ownian mo ions
s a ing in xand y, one can cons uc a coupling o he wo by conside ing
(X( ), Y ( )) = (x+B( ), y +B( )),(1.9)
whe e B( )is a s anda d B ownian mo ion. The gene a o o he coupled p ocess (1.9)
is gi en by
b
Hg(x, y) := 1
2(∂x+∂y)2g(x, y) = 1
2T 
1 1
1 1
D2g(x, y)=1
2T CD2g(x, y)CT,
whe e we eco e he ma ix Co (1.5). No e ha b
His indeed a coupling: Fo 1, 2∈
Cb(Rq)and ( 1⊕ 2)(x, y) := 1(x) + 2(y), we ha e
b
H( 1⊕ 2)(x, y) = H 1(x) + H 2(y).
Using he coupling b
H, we can now ew i e (1.8) as
Hϕα(xα)−Hψα(yα) = b
H(ϕα⊕−ψα) (xα, yα)
≤b
Hα
2d2(xα, yα)=0,(1.10)
whe e he i s equali y ollows by he de ini ion o a coupling, he inequali y is based
on he posi i e maximum p inciple wi h he op imize s om equa ion (1.7), and he
inal equali y is due o he ac ha he synch onous coupling con ols dis ance g ow h.
A simila s a egy can be used o ea a disc e ized e sion o he B ownian Mo ion
by conside ing he gene a o H (x) = 1
2[ (x+ 1) − (x)] + 1
2[ (x−1) − (x)] o a
andom walk: We synch onously couple he andom walk wi h i sel using he ope a o
b
H (x, y) = 1
2[ (x+ 1, y + 1) − (x, y)] + 1
2[ (x−1, y −1) − (x, y)] .
The a gumen in (1.10) hen wo ks o he andom walk exac ly as i did o he B ow-
nian mo ion.
This coupling app oach is one o he main con ibu ions o his pape , allowing o a
uni ying amewo k o show compa ison o Hamil on–Jacobi equa ions wi h Hamil oni-
ans o ype (1.2) and hei Bellman and Isaacs e sions, c . Theo em 3.1 and Co olla y
3.2.
Inno a ion 3: The s ic compa ison p inciple. Ou hi d inno a ion is on he
inal es ima e ha is ob ained as he compa ison p inciple. Fo a subsolu ion u o
−λH =h1
and a supe solu ion o
−λH =h2
he compa ison p inciple amoun s o es ablishing ha
sup
x∈Rq
u(x)− (x)≤sup
x∈Rq
h1(x)−h2(x).
The compa ison p inciple, once es ablished, hus implies sup-no m con ac i i y o
he solu ion map R(λ) : Cb(Rq)→Cb(Rq), whe e R(λ)his he unique iscosi y solu ion
o he Hamil on–Jacobi equa ion (1.1).
I is well-known om examples, c . [4], [11], [22], [39], ha he map R(λ)h akes
he o m o an exponen ially discoun ed Ma ko ian con ol p oblem. I he dynamics
admi s a Lyapuno unc ion V, ha ing compac suble el se s and sa is ying HV ≤c,
hen he con olled Ma ko p ocesses sa is y igh ness p ope ies. Mo e p ecisely, i he
con olled p ocess s a s in a compac se K, one can ind, o any ime ho izon T > 0
A COMPARISON PRINCIPLE BASED ON COUPLINGS 5
and ε > 0, a compac se b
K⊇K, gi en in e ms o he suble el se s o Vsuch ha ,
wi h p obabili y 1−ε, he p ocess emains in b
Kup o ime T. Rew i ing his in e ms
o an es ima e on he solu ion map R(λ), we hen ind
sup
x∈K
R(λ)h1(x)−R(λ)h2(x)≤ε||h1−h2||+ sup
x∈
b
K
h1(x)−h2(x).(1.11)
Es ima es o his ype a e indeed cha ac e ized by he s ic opology, as was i s
es ablished o linea unc ionals in [33, Theo em 5.1] and o con ex, mono one unc-
ionals in [32, Co olla y 2.10]. No e ha in his pape , we do no es ablish con exi y o
h7→ R(λ)h, bu wan o poin ou ha gi en a con ex H, con exi y o R(λ)his o be
expec ed by pe o ming a compa ison p inciple in e ms o h ee a iables using a ian s
o he, e.g., h ee dimensional Theo em 3.2 o [16], see also he domina ion p inciple o
Theo em 2.22 and Co olla y 2.26 o [23]. We lea e his o u u e wo k.
Building upon he no ion o Lyapuno unc ions, we will show ha we can di ec ly
es ablish a a ian o (1.11) o a subsolu ion uand a supe solu ion . Gi en i s mo i-
a ion, we will call his es ima e he s ic compa ison p inciple, see De ini ion 2.4 and
he main esul , Theo em 3.1, below.
O ganiza ion o he pape . The es o he pape is o ganized as ollows: Sec ion 2
in oduces he no a ion and de ini ions. Sec ion 3in oduces he amewo k by s a ing
he necessa y assump ions and o malizing he main esul s. In Sec ion 4, we show
how o apply ou amewo k o ope a o s o he o m (1.2). Sec ion 5con ains he
cons uc ion o he equi ed op imizing poin s and es unc ions. Finally, Sec ion 6
con ains he p oo o he main heo ems.
2. P elimina ies and gene al se ing
2.1. No a ion and P elimina ies. Th oughou he pape , le q∈Nand E=Rq. We
w i e C(E) o he se o all eal- alued con inuous unc ions on E, whe e Eis endowed
wi h he opology induced by he Euclidean dis ance don Rq.
Le C(E)and Cb(E)be he se o con inuous and bounded con inuous unc ions.
Fo k∈N, le Ck(E)deno e he space o all eal- alued unc ions on E ha a e k-
imes con inuously di e en iable. Le Ck
b(E) he se o all unc ions in Ck(E)wi h
bounded de i a i es up o o de k. We deno e he space o all smoo h unc ions ha
a e cons an ou side o a compac se by C∞
c(E). We w i e Cu(E)and Cl(E) o he
se o con inuous unc ions on E ha a e uni o mly bounded om abo e and below,
espec i ely. Mo eo e , we w i e
C+(E) := { ∈C(E)| has compac sub-le el se s},
C−(E) := { ∈C(E)| has compac supe -le el se s},
Cc(E) := { ∈C(E)| is cons an ou side o a compac se }.
We u he mo e de ine he ollowing in e sec ions: C2
c(E) = Cc(E)∩C2(E),
C2
+(E):=C+(E)∩C2(E), C2
−(E):=C−(E)∩C2(E).
Fo a, b ∈R, we w i e a∨b:= max{a, b}and a∧b:= min{a, b}. We deno e he
sup emum no m by ||·||, ha is
|| || = sup
x∈E| (x)|,
o ∈Cb(E), while, o u∈C(E), we use he no a ion
⌈u⌉:= sup
x∈E
u(x),⌊u⌋:= in
x∈Eu(x)
6 SERENA DELLA CORTE, FABIAN FUCHS, RICHARD C. KRAAIJ, AND MAX NENDEL
o a sup emum o in imum o e he en i e space and
⌈u⌉C:= sup
x∈C
u(x),⌊u⌋C:= in
x∈Cu(x)
o a sup emum o in imum o e a subse C⊆E.
We say ha a unc ion ω: [0,∞)→[0,∞)is a modulus o con inui y, i ωis uppe
semi-con inuous wi h ω(0) = 0. We say ha a unc ion ∈C(E)admi s a modu-
lus o con inui y, i , o e e y compac K⊆E, he e exis s a modulus o con inui y
ωK: [0,∞)→[0,∞)such ha , o all x, y ∈K, we ha e
| (x)− (y)| ≤ ωK(d(x, y)).
A unc ion ϕ:E→Ris called semi-con ex wi h cons an κ∈Ri o any x0∈E
he map
x7→ ϕ(x) + κ
2d2(x, x0)
is con ex. Mo eo e , ϕis called semi-conca e wi h cons an κ∈Ri −ϕis semi-con ex
wi h cons an −κ.
We say ha a unc ion ∈C(E, Rq)is one-sided Lipschi z i , o all x, y ∈Eand
some cons an C∈R, we ha e
⟨x−y, (x)− (y)⟩ ≤ Cd2(x, y).
Fo any z∈E, le sz:E→Rqbe he shi map
sz(x) = x−z.
Fo any z1, z2∈E, le
dz1,z2(x, y):=d(sz1(x), sz2(y)) .
Le 1, 2∈C(E). Then, we de ine he di ec sum 1⊕ 2, 1⊖ 2∈C(E×E)as
( 1⊕ 2)(x1, x2):= 1(x1) + 2(x2)and ( 1⊖ 2)(x1, x2):= 1(x1)− 2(x2)
o all x1, x2∈E. Fo wo se s o unc ions F1, F2⊆C(E), we de ine
F1⊕F2:={ 1⊕ 2| 1∈F1, 2∈F2}and F1⊖F2:={ 1⊖ 2| 1∈F1, 2∈F2}.
2.2. Ope a o no ions. We conside ope a o s H⊆C(E)×C(E), whe e we iden i y
Hby i s g aph. As usual, he domain o His gi en by
D(H):={ ∈C(E)|∃g∈C(E): ( , g)∈H}.
Le H1, H2⊆C(E)×C(E). We de ine
H1+H2:={( , g1+g2)|( , g1)∈H1,( , g2)∈H2},
which is an ope a o wi h domain
D(H1+H2):=D(H1)∩D(H2).
We say ha His linea on i s domain i , o any , g ∈ D(H)and a∈Rsuch ha
a +g∈ D(H), we ha e
H(a +g) = aH +Hg.
We will p o e he compa ison p inciple o he equa ion in e ms o Hby ela ing i
o wo equa ions in e ms o wo es ic ions o H. To do so, we will need o be able
o cons uc es unc ions in he domain o H om unc ions in he domain o he
es ic ions. In pa icula , we will need he ollowing no ion.
De ini ion 2.1 (Sequen ial Denseness).Le D ⊆ Cb(E),D+⊆C+(E), and D−⊆
C−(E).
A COMPARISON PRINCIPLE BASED ON COUPLINGS 7
•We say ha Dis upwa d sequen ially dense in D+i , o any †∈ D+and
cons an a∈R, he e exis s a unc ion †,a ∈ D such ha
( †,a(x) = †(x)i †(x)≤a,
a < †,a(x)≤ †(x)i †(x)> a.
•We say ha Dis downwa d sequen ially dense in D−i , o any ‡∈ D−and
cons an a∈R, he e exis s a unc ion ‡,a ∈ D such ha
( ‡,a(x) = †(x)i ‡(x)≥a,
a > ‡,a(x)≥ ‡(x)i ‡(x)< a.
2.3. Viscosi y solu ions. Fo λ > 0, conside h1∈Cl(E)and h2∈Cu(E)and wo
ope a o s H1⊆Cl(E)×C(E)and H2⊆Cu(E)×C(E). We s udy he pai o equa ions
−λH1 ≤h1,(2.1)
−λH2 ≥h2.(2.2)
The no ion o iscosi y solu ion is buil upon he maximum p inciple.
De ini ion 2.2 (Maximum p inciple).We say ha an ope a o H⊆C(E)×C(E)
sa is ies he maximum p inciple i , o all 1, 2∈ D(H)and x0∈Ewi h
1(x0)− 2(x0) = sup
x∈E{ 1(x)− 2(x)},
we ha e
H 1(x0)≤H 2(x0)
and, analogously, o all 1, 2∈ D(H)and x0∈Ewi h
1(x0)− 2(x0) = in
x∈E{ 1(x)− 2(x)},
we ha e
H 1(x0)≥H 2(x0).
Obse e ha e e y ope a o H⊆C(E)×C(E) ha sa is ies he maximum p inciple
is single- alued, i.e., o all ∈ D(H),
#{g∈C(E)|( , g)∈H}= 1.
De ini ion 2.3 (Viscosi y sub- and supe solu ions).Le H1⊆Cl(E)×C(E)and
H2⊆Cu(E)×C(E)be wo ope a o s wi h domains D(H1)and D(H2), espec i ely.
Mo eo e , le λ > 0,h1∈Cl(E), and h2∈Cu(E).
(a) A bounded, uppe semicon inuous unc ion u:E→Ris called a ( iscosi y)
subsolu ion o (2.1) i , o all ( , g)∈H1, he e exis s a sequence (xn)n∈N⊆E
such ha
lim
n→∞ u(xn)− (xn) = sup
x∈E
u(x)− (x),
lim sup
n→∞ u(xn)−λg(xn)−h1(xn)≤0.
(b) A bounded, lowe semicon inuous unc ion :E→Ris called a ( iscosi y)
supe solu ion o (2.2) i , o all ( , g)∈H2, he e exis s a sequence (xn)n∈N⊆E
such ha
lim
n→∞ (xn)− (xn) = in
x∈E (x)− (x),
lim in
n→∞ (xn)−λg(xn)−h2(xn)≥0.
14 SERENA DELLA CORTE, FABIAN FUCHS, RICHARD C. KRAAIJ, AND MAX NENDEL
i s domain and compa ible wi h V,{ζz,p}z∈E,p∈Rq, and {ζz}z∈Eand wi h associa ed
con olled g ow h coupling b
A:= b
A1+b
A2.
Rema k 4.3. Le B1,B2⊆C(E)×C(E)be compa ible wi h V,{ζz,p}z∈E,p∈Rq, and
{ζz}z∈Eand con ex semi-mono one ope a o s. Then B:= B1+B2is compa ible wi h
V,{ζz,p}z∈E,p∈Rq, and {ζz}z∈Eand con ex semi-mono one ope a o .
The es o his sec ion is o ganized as ollows:
•In Sec ion 4.1, we conside d i e ms and con ex i s -o de Hamil onians;
•In Sec ion 4.2, we conside di usion ope a o s;
•In Sec ion 4.3, we conside in eg al ope a o s.
4.1. De e minis ic Example: D i e ms and con ex i s -o de Hamil oni-
ans. In his sec ion, we conside he de e minis ic pa o he ope a o (4.1).
P oposi ion 4.4. Suppose ha Bis gi en by
B (x) = ⟨b(x),∇ (x)⟩+H(∇ (x))
wi h he d i e m x7→ b(x)locally, one-sided Lipschi z wi h cons an Lb,K and ||b(x)|| ≤
cb
2(1 + ||x||) o some cons an cb>0, and p7→ H(p)con inuous and con ex.
Then, Bis compa ible wi h bo h collec ions o De ini ion 4.1, c . Assump ion 3.5 (b),
and con ex semi-mono one. Fu he mo e, V= log(1 + x2
2)is a Lyapuno unc ion:
sup
x∈E
BV(x)<∞.
P oo . Con ex semi-mono onici y: Clea ly, Bis locally i s -o de wi h B (x) =
⟨b(x),∇ (x)⟩+H(∇ (x)) = B(x, ∇ (x)). Addi ionally, o any compac se K⊆E
α > 0, and x, x′∈K, we ha e
B(x, α(x−x′)) −B(y, α(x−x′)) = b(x), α(x−x′)+H(α(x−x′))
−b(x′), α(x−x′)−H(α(x−x′))
=b(x)−b(x′), α(x−x′)
+H(α(x−x′)) −H(α(x−x′))
≤αLb,Kd2(x, x′),
es ablishing semi-mono onici y. As con exi y o p7→ B(x, p)is immedia e, we conclude
ha Bis con ex semi-mono one.
Lyapuno con ol: Using ha V(x) = log 1 + x2
2,∇V(x) = 2x
2+|x|2is bounded
as a unc ion o x,bhas linea g ow h, and ha His con inuous, we ind ha
sup
x∈E
BV(x) = sup
x∈Eb(x),2x
2 + |x|2+H2x
2 + |x|2<∞.
Compa ibili y: We show he compa ibili y o B, c . Assump ion 3.5 (b), by e alu-
a ion o he pe u ba ion and con ainmen unc ion in he ope a o .
Using ξz(x) = 1
2d2(x, z)and ζz,p(x) = ⟨p, x −z⟩, we ind o z0, z1, z ∈Eand p∈B1(0)
B(Ξz0,p,z1◦sz)(x) = ⟨b(x),(x−z−z0) + p+ (x−z−z1)⟩
+H((x−z−z0) + p+ (x−z−z1)) ,
which is con inuous in (x, z0, p, z1, z)as band Ha e con inuous. Fo V(x) = log 1 + 1
2x2
and z∈E, we ind
B(V◦sz)(x) = b(x),2(x−z)
2 + |x−z|2+H2(x−z)
2 + |x−z|2,
which is con inuous in (x, z)as band Ha e con inuous. Thus, Bis compa ible. □

A COMPARISON PRINCIPLE BASED ON COUPLINGS 15
4.2. S ochas ic Example: Di usion ope a o s. In his sec ion, we ocus on di u-
sion ope a o s o he o m
A (x) = 1
2T Σ(x)ΣT(x)D2 (x),
whe e Σ(x)is a posi i e semi-de ini e ma ix o each ixed x∈E.
Ou main goal is o cons uc a con olled g ow h coupling o he ope a o A. To
illus a e he idea behind ou app oach, conside he simple case o he Laplacian
ope a o
A0 (x) = 1
2T (D2 (x)),
which is he in ini esimal gene a o o B ownian mo ion. The well-known synch onous
coupling o wo B ownian mo ions s a ed om xand x′, espec i ely, is gi en by
(X( ), X′( )) = (x+B( ), x′+B( )) wi h B( )a s anda d B ownian mo ion, ha ing
gene a o
b
A0g(x, x′) = 1
2(∂x+∂x′)2g(x, x′),
which sa is ies b
A0d2= 0. Aiming o gene alize his, we ew i e
b
A0g(x, x′) = T CCTD2g(x, x′)wi h C=1
√2
1 1
1 1
.
In gene al we ob ain he ollowing esul .
P oposi ion 4.5. Suppose ha Ais gi en by
A (x) = 1
2T Σ(x)ΣT(x)D2 (x)
wi h Σ(x)posi i e semi-de ini e o all x∈E,x7→ Σ(x)locally Lipschi z wi h cons an
LΣ,K and ||b(x)|| ≤ cΣ
2(1 + ||x||) o some cons an cΣ>0. Conside
b
A (x, y) := T b
Σ2(x, x′)D2 (x, x′),
whe e
b
Σ2(x, y) := Σ(x)ΣT(x) Σ(x′)ΣT(x)
Σ(x)ΣT(x′) Σ(x′)ΣT(x′).
Then, Ais compa ible, c . Assump ion 3.5 (a), linea on i s domain, and admi ing he
con olled g ow h coupling b
A. Fu he mo e, V= log(1 + x2
2)is a Lyapuno unc ion:
sup
x
AV(x)<∞.
Fo he p oo we make use o he ollowing auxilia y lemma.
Lemma 4.6. Fo each x∈E, le B(x)be a posi i e semi-de ini e ma ix and conside
A (x) = 1
2T B(x)D2 (x).
Fo any x, x′∈E, le b
B(x, x′)be a posi i e semi-de ini e ma ix ha ing block-s uc u e
b
B(x, x′) = B(x)B(x, x′)
B(x, x′)TB(x′).
De ine
b
A (x, x′) := 1
2T b
B(x, x′)D2 (x, x′).
Then, b
Ais a coupling o A.
16 SERENA DELLA CORTE, FABIAN FUCHS, RICHARD C. KRAAIJ, AND MAX NENDEL
P oo .
b
A( 1⊕ 2)(x, y) = 1
2T b
B(x, y)D2( 1⊕ 2)(x, y)
=1
2T B(x)D2 (x)+1
2T B(y)D2 (y)
=A 1+A 2
and i sa is ies he maximum p inciple. □
P oo o P oposi ion 4.5.Con olled g ow h coupling: By Lemma 4.6,b
Ais a cou-
pling o A. We hus e i y ha b
Ahas con olled g ow h. Conside α > 1,K⊆Ea
compac se , and x, x′, y, y′∈K. Then,
b
Aα
2d2
x−y,x′−y′(x, x′) = 1
2T b
Σ2(x, x′)D2α
2d2
x−y,x′−y′(x, x′)
=1
2T b
Σ2(x, x′)α
1
−
1
−
1 1
(x, x′)
=α
2T ((ΣT(x)−ΣT(y))(Σ(x)−Σ(y)))
≤αL2
Σ,Kd2(x, x′),
es ablishing con olled g ow h.
Lyapuno con ol: Using V(x) = log(1+ x2
2)and he ac ha Σhas linea g ow h,
we ind ha
sup
x∈E
AV(x) = sup
x∈E
1
2T Σ(x)ΣT(x)D2V(x)<∞.(4.2)
Compa ibili y: Using ξz(x) = 1
2d2(x, z),ζz,p(x) = ⟨p, x −z⟩and V(x) = log(1 +
x2
2), we ind o z0, z1, z ∈Eand p∈B1(0)
A(Ξ ◦sz)(x) = 2 T (Σ(x)ΣT(x)),
A(V◦sz)(x) = 1
2T Σ(x)ΣT(x)D2(V◦sz)(x),
which, by an analogous calcula ion as in equa ion (4.2), is con inuous in (x, z0, p, z1, z)
and (x, z). Consequen ly, Ais compa ible. □
4.3. S ochas ic Example: In eg al ope a o s. In his sec ion, we co e examples
o spa ially inhomogeneous Lé y p ocesses ha ha e gene a o s o he ype
A (x) = Z (x+z)− (x)−χB1(0)(z)⟨z,∇ (x)⟩µx(dz),(4.3)
whe e χB1(0)(z) = l(|z|) o some smoo h non-dec easing unc ion lsa is ying l= 1 on
a neighbo hood o 0and l( ) = 0 o ≥1.
We nex speci y he space om which we can ake ou jump measu es µx. Fo his,
we need o con ol he mass close o 0as o la ge alues o z. The ollowing unc ion
con ols bo h:
W(z) := χB1(0)(z)|z|2+ (1 −χB1(0)(z)) log 1 + |z|2.
We ake he amily o jump measu es {µx}x∈E om he se o equi alence classes
MW(Rq) := M(Rq)/∼wi h
M(Rq):=µ∈M(Rq)ZW(z)µ(dz)<∞,
whe e M(Rq)is he se o all Bo el measu es on Rqand whe e
µ∼νi and only i µ|Rq {0}=ν|Rq {0}.
A COMPARISON PRINCIPLE BASED ON COUPLINGS 17
We opologize he se MW(Rq)by he weak opology σWinduced by he pai ings
µ7→ Zg(z)µ(dz)∀g∈CW,(4.4)
whe e
CW:=(g∈C(Rq)g(0) = 0,and sup
z=0
|g(z)|
W(z)<∞.).
Below, we cons uc con olled g ow h couplings o ope a o s o he ype (4.3). To
cla i y he concep s, we conside he example o an uncompensa ed p ocess, i.e., ha ing
an ope a o o he ype
A (x) = Z (x+z)− (x)µx(dz).
Couplings o his ype o ope a o a e o he o m
b
A (x, x′) = Z (x+z1, y +z2)− (x, x′)πx,x′(dz1,z2),(4.5)
whe e πx,x′couples µxand µx′. In he ollowing example, we illus a e he need o being
able o couple jumps synch onously.
Example 4.7 (Random Walk).Conside he simple andom walk on Rmaking jumps
o size 1, i.e µx=µ=δ−1+δ1leading o he ope a o
A (x) = [ (x−1) + (x+ 1) −2 (x)] .
Well known couplings include walks wi h simul aneous jumps bu independen di ec-
ions, ully independen jumps, and synch onous jumps. The co esponding gene a o s
a e gi en as in (4.5) wi h jump measu es
π1:= µ⊗µ,
π2:= δ(−1,0) +δ(1,0) +δ(0,−1) +δ(0,1),
π3:= δ(−1,−1) +δ(1,1),
espec i ely. This leads o he ope a o s
b
A1 (x, x′) = (x−1, x′−1) + (x−1, x′+ 1)
+ (x+ 1, x′−1) + (x+ 1, x′+ 1) −4 (x, x′),
b
A2 (x, x′) = (x−1, x′) + (x+ 1, x′)−2 (x, x′)
+ (x, x′−1) + (x, x′+ 1) −2 (x, x′),
b
A3 (x, x′) = (x−1, x′−1) + (x+ 1, x′+ 1) −2 (x, x′).
Only o he inal example, we see ha b
A3d2≤0, poin ing a he necessi y o he
alignmen o jumps.
No e ha he hi d coupling abo e has di e en o al mass, and we hus wo k ou side
he ealm o he ypical no ion o couplings o p obabili y measu es. A second ea u e
o coupling jump measu es, no p esen in he example abo e, is ha we can make one
p ocess jump, whe eas he o he does no .
We o malize his in he ollowing de ini ion.
De ini ion 4.8. Le µ, ν ∈ MW(Rq). We say ha π∈M(Rq×Rq)is an ex ended
coupling o µand ν, i
π((A {0})×Rq) = µ(A {0})∀A∈ B(Rq),
π(Rq×(B {0})) = ν(B {0})∀B∈ B(Rq).
18 SERENA DELLA CORTE, FABIAN FUCHS, RICHARD C. KRAAIJ, AND MAX NENDEL
Rema k 4.9. A a ian o his coupling was in oduced in [21]. The e mass can be
mo ed o he bounda y o a domain. In ou con ex , his bounda y is he poin 0.
De ini ion 4.10. Le x7→ µxbe a map om Ein o M(Rq). Le (x, x′)7→ πx,x′be a
map om E2in o M(Rq×Rq).
(a) We say ha (x, x′)7→ πx,x′is an ex ended coupling o x7→ µx, i o all x, x′∈E,
we ha e ha πx,x′is an ex ended coupling o µxand µ′
x.
(b) We say ha (x, x′)7→ πx,x′is locally Lipschi z, i , o any compac se K⊆E,
he e exi s a cons an Lπ,K such ha , o x, x′∈K, we ha e
Zd2(z1,z2)πx,x′(dz1,dz2)≤Lπ,Kd2(x, x′).
Rema k 4.11. No e ha condi ions (12),(34), and (35) in [5] o µand jco espond
o ou choice o MW(Rq)and locally Lipschi z ex ended coupling πx,x′.
Rema k 4.12. Le η:R→Rbe any locally Lipschi z map wi h local Lipschi z
cons an s Lη,K. Se µx:= δη(x)
1
η(x)=0(x)and πx,x′=δ(η(x),η(x′)). Then, (x, x′)7→ πx,x′
is a locally Lipschi z coupling o x7→ µxwi h Lπ,K =Lη,K.
The main p oposi ion o his subsec ion below aims o show ha in eg al ope a o s
o he o m (4.3) can be ea ed analogous o he o he examples abo e. We wo k wi h
he second collec ion o penaliza ion unc ions, c . De ini ion 4.1, o a oid in eg abili y
issues.
P oposi ion 4.13. Conside
A (x) = Z (x+z)− (x)−χB1(0) ⟨z,∇ (x)⟩µx(dz).
Suppose he e exis s a σW-con inuous map x7→ µxin MW(Rq), c . (4.4), and ha he e
exis s a locally Lipschi z ex ended coupling (x, x′)7→ πx,x′o x7→ µxwi h Lipschi z
cons an Lπ,K and, o bχ(z1,z2) := χB1(0)(z1)χB1(0)(z2), se
b
Ag(x, x′) := Zhg(x+z1, x′+z2)−g(x, x′)
−bχ(z1,z2)(z1,z2)T,∇g(x, x′)iπx,x′(dz1,dz2).
Assume u he mo e ha
sup
x∈EZlog 1 +
1
2|z|2+⟨x, z⟩
1 + 1
2|x|2!µx(dz)<∞.
Then, Ais compa ible, c . De ini ion 3.5 (a), and linea on i s domain admi ing he
con olled g ow h coupling b
A. Fu he mo e, V= log(1 + x2
2)is a Lyapuno unc ion:
sup
x∈E
AV(x)<∞.
Rema k 4.14. Co esponding o Rema k 3.3, we e e o [6, Co olla y 2.3] o a unique-
ness esul o a Lé y p ocess ma ingale p oblem.
The p oo o P oposi ion 4.13 is based on he ollowing wo auxilia y lemmas. In he
i s , we ob ain bounds on he in eg and o ou ope a o ac ing on he Lyapuno unc ion
V. In he second, we compu e he in eg and o ou Lé y ype ope a o ac ing on he
shi ed squa ed me ic. We p o e hese wo lemmas ollowing he p oo o P oposi ion
4.13.
A COMPARISON PRINCIPLE BASED ON COUPLINGS 19
Lemma 4.15. Fix x, z ∈E.
(a) Fo z∈Rq, we ha e
−log 1 + 1
2|x−z|2≤V◦sz(x+z)−V◦sz(x)
≤log 1 +
1
2|z|2+⟨x−z, z⟩
1 + 1
2(x−z)2!
≤log 1 + |z|2.
(b) Fo z∈B1(0), we ha e
|V◦sz(x+z)−V◦sz(x)−⟨z,∇(V◦sz)(x)⟩| ≤ 1
2|z|2.
Lemma 4.16. We ha e
1
2d2
x−y,x′−y′(x+z1, x′+z2)−1
2d2
x−y,x′−y′(x, x′)
−bχ(z1,z2)z1
z2,∇1
2d2
x−y,x′−y′(x, x′)
≤1−1
2bχ(z1,z2)d2(z1,z2) + (1 −bχ(z1,z2))1
2d2(y, y′).
P oo o P oposi ion 4.13.Con olled g ow h coupling: As (x, x′)7→ πx,x′is a lo-
cally Lipschi z ex ended coupling o x7→ µx, c . De ini ion 4.10, we ha e ha b
Ais a
coupling. Thus, we need o e i y he con olled g ow h p ope y o b
A.
Le x, x′, y, y′∈K o K⊆Ea compac se . Using Lemma 4.16, we hen ha e
b
Aα
2d2
x−y,x′−y′(x, x′)≤α
2Z1−1
2bχ(z1,z2)d2(z1,z2)πx,x′(dz1,dz2)
+α
2Z(1 −bχ(z1,z2))1
2d2(y, y′)πx,x′(dz1,dz2)
≤α
2Lπ,Kd2(x, x′) + α
4c′
πd2(y, y′),
whe e he second inequali y is due o he local Lipschi z p ope y o he map (x, x′)7→
πx,x′and c′
π>0exis s since, o e e y x, x′∈E,πx,x′∈M(Rq×Rq). As such, A
admi s he con olled g ow h coupling b
A.
Lyapuno con ol: Using Lemma 4.15, we ind
sup
x∈E
AV(x)≤sup
x∈EZ(1 −χB1(0)) log(1 + |z|2) + χB1(0)|z|2µx(dz)<∞.
Compa ibili y: We s a by es ablishing he con inui y o (x, z)7→ A(V◦sz)(x). Le
(xn, zn)con e ge o (x, z). We aim o apply Lemma C.1 wi h X=Rq {0},νn=µxn,
and
ϕn(z) := V◦szn(xn+z)−V◦szn(xn)−χB1(0) ⟨z,∇(V◦szn)(xn)⟩,
ϕ∞(z) := V◦sz(x+z)−V◦sz(x)−χB1(0) ⟨z,∇(V◦sz)(x)⟩.
As ϕnis con inuous, i emains o show ha supn∈Nsupz=0
|ϕn(z)|
W(z)<∞. By Lemma
4.15, we can es ima e
|ϕn(z)| ≤ χB1(0)
1
2|z|2+ (1 −χB1(0)) max −log 1 + 1
2|xn−zn|2,log 1 + |z|2.
Since (xn, zn)is con e gen , hence bounded, we ob ain he desi ed es ima e. Con inui y
o (x, z)7→ A(V◦sz)(x)now ollows by Lemma C.1.

20 SERENA DELLA CORTE, FABIAN FUCHS, RICHARD C. KRAAIJ, AND MAX NENDEL
Using he pa icula o m o Ξz0,p,z1, c . De ini ion 4.1, one eadily e i ies ha he
map (x, z0, p, z1, z)7→ A(Ξz0,p,z1◦sz) (x)is con inuous wi h an analogous a gumen a-
ion. □
P oo o Lemma 4.15.Le y=x−z, hen we can w i e
V◦sz(x+z)−V◦sz(x)
= log 1 + 1
2(y+z)2−log 1 + 1
2|y|2= log 1 +
1
2|z|2+⟨y, z⟩
1 + 1
2|y|2!.
Applying Young’s inequali y o ⟨y, z⟩leads o he uppe bound
V◦sz(x+z)−V◦sz(x)≤log 1 + |z|2+1
2|y|2
1 + 1
2|y|2!= log 2 + |z|2−1
1 + 1
2|y|2!≤log 1 + |z|2.
Using ha he i s e m is posi i e, we ob ain he lowe bound
V◦sz(x+z)−V◦sz(x)≥log 1−
1
2|y|2
1 + 1
2|y|2!=−log 1 + 1
2|y|2.
This es ablishes (a). Fo he p oo o (b), we apply Taylo ’s Theo em o ob ain
|V◦sz(x+z)−V◦sz(x)−⟨∇(V◦sz)(x),z⟩| ≤ 1
2|z|2sup
z∈B1(0)
sup
i,j ∇2
i,jV(y+z)
≤1
2|z|2,
which ollows by a di ec inspec ion o
∇2
i,jV(x) = 2δi,j 1 + 1
2|x|2−2xixj
(1 + 1
2|x|2)2.
□
P oo o Lemma 4.16.E alua ing he shi maps, calcula ing he g adien o he squa ed
Euclidean dis ance, and expanding he squa es leads o
1
2d2
x−y,x′−y′(x+z1, x′+z2)−1
2d2
x−y,x′−y′(x, x′)
−bχ(z1,z2)z1
z2,∇1
2d2
x−y,x′−y′(x, x′)
=1
2d2(y+z1, y′+z2)−1
2d2(y, y′)−bχ(z1,z2)y−y′,z1−z2
=1
2d2(z1,z2) + y−y′,z1−z2−bχ(z1,z2)y−y′,z1−z2
≤1−1
2bχ(z1,z2)d2(z1,z2) + (1 −bχ(z1,z2))1
2d2(y, y′),
whe e in he second equali y we use p ope ies o he Euclidean dis ance dand he inal
line is due o Young’s inequali y. □
5. Cons uc ion o es unc ions
In classical p oo s o compa ison p inciples, he app oach o es ima e sup u− o
a subsolu ion uand supe solu ion is a iable doubling o quad uplica ion, c . [4,
Theo em 3.1] o [16, in oduc ion o Sec ion 3]: Fo α > 1
sup
x∈E
u(x)− (x)≤sup
x,x′∈E
u(x)− (x′)−α
2d2(x, x′).(5.1)
A COMPARISON PRINCIPLE BASED ON COUPLINGS 21
Le ing α→ ∞, o ces op imizing poin s, i hey exis , o he igh -hand side oge he .
In addi ion, by a ying ei he o he wo componen s, one ob ains basic es unc ions
in e ms o α
2d2 o he use in he de ini ion o he sub- and supe solu ion p ope ies o
uand .
To ensu e ha op imize s in (5.1) exis , we will conside ins ead, o small ε > 0, he
ollowing p oblem ha includes he con ainmen unc ion Vand uppe bounds sup u−
up o a e m o o de ε:
sup
x∈E
1
1−εu(x)−1
1 + ε (x)
≤sup
x,x′∈E
1
1−εu(x)−1
1 + ε (x′)−α
2d2(x, x′)−ε
1−εV(x)−ε
1 + εV(x′).(5.2)
The pa icula o m o he ac o s 1−εand 1 + εis mo i a ed by con exi y based
a gumen s, which will show up in he p oo s o P oposi ion 6.3 and Theo em 3.1 below.
The p ocedu e in (5.2) would be su icien o a s anda d, i s -o de Hamil on–Jacobi
equa ion. The es unc ions p oduced by his p ocedu e, howe e , will no be su icien
o ea second-o de o in eg al ope a o s. This p oblem was conside ed in [16] and
[5]. We will ollow hei app oach by conside ing a quad uplica ion o a iables, which
we also ph ase in e ms o sup- and in -con olu ions. We hen pe o m a Jensen- ype
pe u ba ion.
As we aim o uni y p oo s o bo h in eg al and di e en ial ope a o s, we e isi he
ull p oo and s a e ou esul in e ms o es unc ions.
In P oposi ions 5.1 and 5.3 below, which can be conside ed o be an ex ended wo-
a iable a ian o he C andall–Ishii cons uc ion [16, Theo em 3.2], we s a ou by
conside ing he op imiza ion (5.2) in e ms o he sup- and in -con olu ion o uand ,
espec i ely, e ec i ely leading o a quad uplica ion p oblem, see (5.3) below.
We hen pe o m he Jensen pe u ba ion, see (5.4). The es o he p oposi ion deals
wi h a ious p ope ies o he op imize s in ela ion o uand .
In P oposi ion 5.3, we ca y ou an addi ional laye o smoo hing ope a ions o ob ain
C∞- es unc ions. Consequen ly, we can mo e away om he no ion o solu ions in
e ms o sub- and supe je s, which is o pa amoun impo ance o e ec i ely ea
di usi e and jump- ype p ocesses in a common amewo k.
Fo eadabili y, we exp ess sup ema and in ima using ⌈·⌉ and ⌊·⌋, espec i ely, as
de ined in Sec ion 2.1.
P oposi ion 5.1 (Cons uc ion o op imize s).Le ube bounded and uppe semi-
con inuous, be bounded and lowe semi-con inuous, Vbe a con ainmen unc ion as
in De ini ion 2.12, and {ζz,p}z∈E,p∈Rq⊂C(E)and {ξz}z∈E⊂C1(E)be collec ions o
unc ions as in De ini ion 2.13. Fix ε∈(0,1) and φ∈(0,1].
Then, he e exis compac se s Kε,0⊆Kε⊆Eand, o any α > 1, h ee pai s
o a iables (yα,0, y′
α,0),(yα, y′
α),(xα, x′
α)in E2and pα, p′
α∈B1/α(0) such ha he
ollowing ou se s o p ope ies hold.
P ope ies o yα,0, y′
α,0:
The a iables yα,0, y′
α,0op imize ⌈Λα⌉, whe e
Λα(y, y′) := 1
1−εPα[u](y)−1
1 + εPα[ ](y′)−α
2d2(y, y′)
−ε
1−ε(1 −φ)V(y)−ε
1 + ε(1 −φ)V(y′)(5.3)
and sa is y he ollowing p ope y
(a) yα,0, y′
α,0∈Kε,0.
22 SERENA DELLA CORTE, FABIAN FUCHS, RICHARD C. KRAAIJ, AND MAX NENDEL
P ope ies o yα, y′
αand pα, p′
α:
The pai yα, y′
αop imizes Λα−ε
1−εφΞ0
1−ε
1 + εφΞ0
2(5.4)
and uniquely op imizes Λα−ε
1−εφΞ1−ε
1 + εφΞ2(5.5)
whe e Λαis as in (5.3)and
Ξ0
1(y):= Ξ0
yα,0,pα(y),Ξ0
2(y′):= Ξ0
y′
α,0,p′
α(y′),
Ξ1(y):= Ξyα,0,pα,yα(y),Ξ2(y′):= Ξy′
α,0,p′
α,y′
α(y′)
as in De ini ion 2.13. Mo eo e , he op imize s yα, y′
αo (5.4)and (5.5)sa is y
(b) We ha e
d(yα, yα,0)≤1
α, d(y′
α, y′
α,0)≤1
α.
(c) Pα[u]and Pα[ ]a e wice di e en iable in yαand y′
α, espec i ely.
P ope ies o xα, x′
α:
The a iables xα, x′
αop imize
Pα[u](yα) = u(xα)−α
2d2(xα, yα),
Pα[ ](y′
α) = (x′
α) + α
2d2(x′
α, y′
α),
and sa is y
(d) xαand x′
αa e he unique op imize s in he de ini ion o Pα[u](yα)and Pα[ ](y′
α),
espec i ely.
(e) We ha e ha
u(xα)−Pα[u]◦sxα−yα(xα) = ⌈u−Pα[u]◦sxα−yα⌉,
(x′
α)−Pα[ ]◦sx′
α−y′
α(x′
α) =  −Pα[ ]◦sx′
α−y′
α.
Beha iou as α→ ∞:
( ) We ha e limα→∞ αd2(yα,0, y′
α,0)=0.
(g) We ha e
lim
α→∞ αd(xα, yα) + dyα, y′
α+dy′
α, x′
α2= 0.
(h) xα, yα, y′
α, x′
α∈Kε.
In addi ion, he ollowing es ima e on u− holds: Fo any compac se K⊆E, he e
is a compac se b
K=b
K(K, ε, u, )gi en by
b
K:= z∈EV(z)≤||u||+|| ||
ε+⌈V⌉K,
such ha
(i) Fo any compac se K⊆E,
⌈u− ⌉K≤1
1−εu(xα)−1
1 + ε (xα) + ε(cε,φ +o(1)) ,
whe e
cε,φ := 2
1−ε2(1 −φ)⌈V⌉K−1
1−εu−1
1 + ε K
,
and o(1) is in e ms o α→ ∞ o ixed εand φ.
A COMPARISON PRINCIPLE BASED ON COUPLINGS 23
(j) Any limi poin o he sequence (xα, yα, yα,0, y′
α,0, y′
α, x′
α)as α→ ∞ is o he
o m (z, z, z, z, z, z)wi h z∈b
K.
Figu e 1 isualizes he ela ion be ween he di e en op imizing poin s.
xαyαy′
αx′
α
yα,0y′
α,0
5.2.(b) 5.1.(g) 5.2.(b)
5.1.(b)
5.1.( )
5.1.(b)
Figu e 1. Rela ion be ween he op imizing poin s wi h a no e which pa s o
he p oposi ions gi e us dis ance con ol.
The p oo o P oposi ion 5.1 uses a ious p ope ies o sup- and in -con olu ions,
which we ga he in he nex lemma. I s p oo is elega ed o Appendix D.2.
Lemma 5.2. Le u:E→Rbe bounded and uppe semi-con inuous and :E→Rbe
bounded and lowe semi-con inuous. Fo α > 1, se
Pα[u](y):= sup
x∈Enu(x)−α
2d2(x, y)o=lu−α
2d2(·, y)m,(5.6)
Pα[ ](y):= in
x∈En (x) + α
2d2(x, y)o=ju+α
2d2(·, y)k.(5.7)
Then,
(a) we ha e ||Pα[u]|| ≤ ||u|| and ||Pα[ ]|| ≤ || ||.
(b) o any x, y ∈Esuch ha
Pα[u](y) = u(x)−α
2d2(x, y),
we ha e α
2d2(x, y)≤u(x)−u(y). Simila ly, o any x, y ∈Ewi h
Pα[ ](y) = (x) + α
2d2(x, y),
we ha e α
2d2(x, y)≤ (y)− (x).
(c) Pα[u]and −Pα[ ]a e dec easing in α.
(d) Pα[u]and −Pα[ ]a e semi-con ex wi h semi-con exi y cons an α. As a conse-
quence, bo h a e locally Lipschi z con inuous.
(e) i Pα[u]is di e en iable a y0, hen he e exis s a unique op imize x0in (5.6)
such ha
Pα[u](y0) = u(x0)−α
2d2(x0, y0)
and DPα[u](y0) = α(x0−y0). Simila ly, i Pα[ ]is di e en iable a y0, hen
he e is a unique op imize x0in (5.7)such ha
Pα[ ](y0) = (x0) + α
2d2(x0, y0)
and DPα[ ](y0) = −α(x0−y0).
P oo o P oposi ion 5.1.P oo o (a):As uand a e bounded, by Lemma 5.2 (a),
he same holds o ||Pα[u]|| and ||Pα[ ]||. Using ha Vhas compac suble else s, c .
De ini ion 2.12, he exis ence o op imize s (yα,0, y′
α,0) o ⌈Λα⌉ ollows.
The de ini ion o Λαand he con olu ions Pα[u]and Pα[ ]imply ha
ε
1−ε(1 −φ)V(yα,0) + ε
1 + ε(1 −φ)V(y′
α,0)≤1
1−ε⌈u⌉− 1
1 + ε⌊ ⌋−⌈Λα⌉.(5.8)
30 SERENA DELLA CORTE, FABIAN FUCHS, RICHARD C. KRAAIJ, AND MAX NENDEL
Case y∈Ac
1and y′∈A2:We ha e
b
1(y)−b
2(y′)−α
2d2(y, y′)≤b
1(y)−b
2(y′)
< 1(yα)− 2(y′
α)−⌊ 2⌋−α
2d2(yα, y′
α)− 2(y′)
= 1(yα)− 2(y′
α)−α
2d2(yα, y′
α)− 2(y′)−⌊ 2⌋
≤ 1(yα)− 2(y′
α)−α
2d2(yα, y′
α).
Case y∈A1and y′∈Ac
2:Follows analogously o he case y∈Ac
1and y′∈A2.
Case y∈Ac
1and y′∈Ac
2:We ha e
b
1(y)−b
2(y′)−α
2d2(y, y′)
≤b
1(y)−b
2(y′)
< 1(yα)− 2(y′
α)−⌊ 2⌋−α
2d2(yα, y′
α)
− 2(y′
α)+(⌈ 1⌉− 1(yα)) + α
2d2(yα, y′
α)
≤ 1(yα)− 2(y′
α)−2α
2d2(yα, y′
α)− 2(y′
α)−⌊ 2⌋−(⌈ 1⌉− 1(yα))
≤ 1(yα)− 2(y′
α)−α
2d2(yα, y′
α).
We conclude ha he pai (yα, y′
α)is also he unique op imize o lb
1−b
2−α
2d2m.
Applying he shi maps sxα−yαand sx′
α−y′
α, espec i ely, we ind ha (xα, x′
α)uniquely
op imize l 1◦sxα−yα− 2◦sx′
α−y′
α−α
2d2
xα−yα,x′
α−y′
αm. Addi ionally, as M1≥m1and
M2≤m2, we ha e b
1,b
2∈C∞
c(E), es ablishing (a).
We nex p o e (b). As ≤Ω−
M1( ),
1
1−εPα[u](y)−ε
1−ε(1 −φ)V(y)−ε
1−εφΞ1(y)=Π1(y)≤Ω−
M1◦Π1(y)≤b
1(y),
which, a e ea angemen o e ms, implies (b).
We p oceed wi h he p oo o (c). By (b) and P oposi ion 5.1 (e),
†(x)− †(xα) = b
†◦sxα−yα(x)−b
†◦sxα−yα(xα)
≥(Pα[u]◦sxα−yα) (x)−(Pα[u]◦sxα−yα) (xα)
≥u(x)−α
2d2(x, sxα−yα(x)−u(xα)−α
2d2(xα, sxα−yα(xα)
=u(x)−u(xα)
wi h equali y uniquely ealized a xα, es ablishing (c).
We conclude wi h he p oo o (d). Fi s o all, no e ha he equali y o i s and
second o de de i a i es o †and b
†as well as o ‡and b
‡ ollows by he chain ule.
The exp essions o Db
†(yα)and Db
‡(y′
α) ollow om (b) and P oposi ion 5.1 (c) and
(d).□

A COMPARISON PRINCIPLE BASED ON COUPLINGS 31
6. P oo o he s ic compa ison p inciple
In his sec ion, we p o e Theo em 3.1. The p oo is based on a a ian o he a iable
quad uplica ion p ocedu e on he basis o
sup
x∈E
1
1−εu(x)−1
1 + ε (x)
≤sup
x,,y,y′x′∈E
1
1−εu(x)−1
1 + ε (x′)−α
2(1 −ε)d2(x, y)−α
2d2(y, y′)
−α
2(1 + ε)d2(y′, x′)−ε
1 + εV(x)−ε
1 + εV(x′),
which we ha e o malized in e ms o es unc ions †, ‡in P oposi ions 5.1 and 5.3.
In a i s s ep, we ela e sub- and supe solu ions o he Hamil on–Jacobi equa ion
o H o hose o H+and H−: This will be ca ied ou in Lemma 6.1. A second s ep
is o show ha †∈ D(H+)and ‡∈ D(H−): This will be ca ied ou in Lemma 6.2.
A e es ablishing hese echnical poin s, we p oceed o ame he compa ison p in-
ciple in e ms o an es ima e on
H+ †
1−ε−H− ‡
1 + ε.(6.1)
This educ ion will be ca ied ou in P oposi ion 6.3, he s a emen o which is mo e
in ol ed han ypically in he li e a u e, bu leads o he imp o ed s ic compa ison
p inciple. I s o mula ion and p oo hinges on he use o Vas a Lyapuno unc ion.
The s a emen s o Lemmas 6.1,6.2, and P oposi ion 6.3 can be ound in Sec ion 6.1,
hei p oo s in Sec ion 6.2.
We inish in Sec ion 6.3 by es ima ing (6.1) in wo s eps leading o ou inal esul . We
i s es ablish in Lemma 6.4 ha he p e- ac o s (1 −ε)−1and (1 + ε)−1wo k well wi h
he combina ions o unc ions ha de ine †, ‡in P oposi ion 5.3. We conclude his
sec ion wi h he p oo o Theo em 3.1, whe e we use his spli , he coupling assump ion
on A, he semi-mono onici y o B, modulus o con inui y con ol on Iand, again, ha
Vis a Lyapuno unc ion o a i e a ou inal esul .
6.1. Compa ison in e ms o es ima ing he di e ence o Hamil onians. We
s a wi h connec ing he no ion o sub- and supe solu ions o H o hose o H+and
H−, espec i ely.
Lemma 6.1. Le Hand Hsa is y Assump ion 3.4. Then, o any h∈Cb(E)and
λ > 0, we ha e he ollowing:
(a) Any iscosi y subsolu ion o −λH =his also a iscosi y subsolu ion o
−λH+ =h.
(b) Any iscosi y supe solu ion o −λH =his also a iscosi y supe solu ion o
−λH− =h.
The p oo ollows in Sec ion 6.2 below. In he nex lemma we show ha he es
unc ions ha we cons uc ed in he p e ious sec ion a e in he domain o H+and H−.
Lemma 6.2. Le Hbe an ope a o sa is ying Assump ions 3.4 and 3.5. Le b
†, †and
b
‡, ‡be as in P oposi ion 5.3. Then, b
†, †∈ D(H+)and b
‡, ‡∈ D(H−).
The p oo o he lemma is ou lined in Sec ion 6.2 below. We nex s a e ou key
p oposi ion, which ela es he s ic compa ison p inciple o an es ima e on he di e ence
o Hamil onians.
32 SERENA DELLA CORTE, FABIAN FUCHS, RICHARD C. KRAAIJ, AND MAX NENDEL
P oposi ion 6.3. Le H⊆C(E)×C(E)sa is y Assump ions 3.4 and 3.5. Le h1, h2∈
Cb(E), and λ > 0. Conside he equa ions
−λH+ ≤h1,(6.2)
−λH− ≥h2.(6.3)
Le uand by iscosi y sub- and supe solu ions o (6.2)and (6.3), espec i ely. Fo
each ε∈(0,1),φ∈(0,1] and α > 1, conside he cons uc ion o op imize s xα, x′
αand
es unc ions †, ‡as in P oposi ions 5.1 and 5.3.
Suppose he e exis s a map ε7→ C0
ε, and o any ε∈(0,1) a non-nega i e map
φ7→ Cε,φ sa is ying lim supε↓0C0
ε<∞and limφ↓0Cε,φ = 0 such ha
lim in
α→∞
H+ †(xα)
1−ε−H− ‡(x′
α)
1 + ε≤εC0
ε+Cε,φ.(6.4)
Then, o any compac se K⊆Eand ε∈(0,1),
sup
x∈K
u(x)− (x)≤εCε+ sup
x∈
b
K
h1(x)−h2(x),
whe e b
Kε:= b
Kε(K, u, )and Cε:= Cε(K, u, , h1, h2)a e gi en by
b
Kε:= z∈EV(z)≤||u||+|| ||
ε+⌈V⌉K,
Cε:= λC0
ε+2
1−ε2⌈V⌉K+1
1−ε||h1||+1
1−ε||h2||−1
1−εu−1
1 + ε K
.
In pa icula , he s ic compa ison p inciple holds o (6.2)and (6.3).
6.2. P oo o Lemmas 6.1,6.2, and P oposi ion 6.3.
P oo o Lemma 6.1.We only p o e he i s s a emen , he second one ollows analo-
gously. Le ube a subsolu ion o −λH =hand le ( , g)∈H+. Ou claim hus
ollows i he e exis s x0sa is ying
u(x0)− (x0) = ⌈u− ⌉,(6.5)
u(x0)−λg(x0)≤h(x0).(6.6)
As uis uppe semi-con inous and bounded, and has compac suble el se s, he exis-
ence o x0sa is ying (6.5) is immedia e. We hus p oceed wi h (6.6) using he sequen ial
upwa d denseness o D(H)in D(H+), c . Assump ion 3.4 (c). Se
a:= (x0) + ⌈u⌉−u(x0), A := {x| (x)≤a}.
We can hus ind ( a, ga)∈Hwi h asa is ying
( a(x) = (x)i x∈A,
a< a(x)≤ (x)i x /∈A.
We i s es ablish ha
u(x0)− a(x0) = ⌈u− a⌉.(6.7)
Using (6.5) and ha = aon A, (6.7) ollows by e i ying ha
u(x)− a(x)< u(x0)− (x0), x ∈Ac,
which ollows om he de ini ion o a:
u(x)− (x)< u(x)−a
=u(x)−( (x0) + ⌈u⌉−u(x0))
=u(x0)− (x0)−(⌈u⌉−u(x))
≤u(x0)− (x0).
A COMPARISON PRINCIPLE BASED ON COUPLINGS 33
Thus, by (6.7), we can use he subsolu ion inequali y o ( a, ga)in he poin x0. We
ob ain:
u(x0)−λga(x0)≤h(x0).(6.8)
Recalling ha a(x0) = (x0)and a≤ , we ha e
a(x0)− (x0) = ⌈ a− ⌉.
Using he posi i e maximum p inciple o H, c . Assump ion 3.4 (a), hus yields
ga(x0)≤g(x0).(6.9)
Combining (6.8) and (6.9), leads o
u(x0)−λg(x0)≤u(x0)−λga(x0)≤h(x0),
es ablishing (6.6) and consquen ly ha uis a subsolu ion o −λH+ =h.□
P oo o Lemma 6.2.As 1, 2,b
1,b
2∈C∞
c(E), i ollows by Assump ion 3.4 (b) ha
1, 2,b
1,b
2∈ D(H). By compa ibili y, c . Assump ion 3.5, we ha e V◦sz,Ξ◦sz∈ D(H).
By Assump ion 3.4 (e) and he ac ha Vhas compac suble el se s, c . De ini ion
2.12, we hus ha e (1 −φ)V◦sz+φΞ◦sz∈ D(H+). Consequen ly, b
†, †∈ D(H+)and
b
‡, ‡∈ D(H−)by Assump ion 3.4 ( ).□
P oo o P oposi ion 6.3.Le ube a subsolu ion o −λH+ =h1and a supe solu ion
o −λH− =h2. Conside he cons uc ions in P oposi ions 5.1 and 5.3 o he
subsolu ion u, supe solu ion and ε∈(0,1) and φ∈(0,1].
By Lemma 6.2, we ha e †∈ D(H+)and ‡∈ D(H−)and, by P oposi ion 5.3 (c),
we ind ha (xα, x′
α)a e he unique op imize s in
u(xα)− †(xα) = ⌈u− †⌉,
(x′
α)− ‡(x′
α) = ⌈ − ‡⌉,
which, by he sub- and supe solu ion p ope ies o H+and H−, espec i ely, and
Lemma D.1, implies ha
u(xα)−λH+ †(xα)≤h1(xα),
(x′
α)−λH− ‡(x′
α)≥h2(x′
α).(6.10)
By P oposi ion 5.1 (i), we ind
⌈u− ⌉K≤1
1−εu(xα)−1
1 + ε (x′
α) + ε(cε,φ +o(1)) ,
whe e
cε,φ := 2
1−ε2(1 −φ)⌈V⌉K−1
1−εu−1
1 + ε K
,(6.11)
and o(1) is in e ms o α→ ∞. Using (6.10), we es ima e
⌈u− ⌉K≤1
1−εu(xα)−1
1 + ε (x′
α) + ε(cε,φ +o(1))
≤1
1−εh1(xα)−1
1 + εh2(x′
α) + λH+ †(xα)
1−ε−H− ‡(x′
α)
1 + ε+ε(cε,φ +o(1))
≤h1(xα)−h2(x′
α) + λH+ †(xα)
1−ε−H− ‡(x′
α)
1 + ε
+ε
1−ε||h1||+ε
1 + ε||h2||+ε(cε,φ +o(1)) .
34 SERENA DELLA CORTE, FABIAN FUCHS, RICHARD C. KRAAIJ, AND MAX NENDEL
We nex expand cε,φ om (6.11). Fu he mo e, aking lim in α→∞ on he igh -hand
side, using P oposi ion 5.1 (j) o ea he di e ence h1−h2, and (6.4) o ea he
di e ence o Hamil onians, we ind
⌈u− ⌉K≤ ⌈h1−h2⌉b
K+λ(εC0+Cε,φ) + ε
1−ε||h1||+ε
1 + ε||h2||
+ε2
1−ε2(1 −φ)⌈V⌉K−1
1−εu−1
1 + ε K.
As φ∈(0,1] was a bi a y, we can ake he limi o φ↓0, which leads o
⌈u− ⌉K≤ ⌈h1−h2⌉b
K
+ελC0
ε+2
1−ε2⌈V⌉K+1
1−ε||h1||+1
1 + ε||h2||−1
1−εu−1
1 + ε K,
es ablishing he claim. □
6.3. P oo o Theo em 3.1.We s a wi h an auxilia y lemma ha p o ides a de ailed
decomposi ion o he ope a o s Aand Be alua ed in he es unc ions.
Lemma 6.4. Le Aand Bbo h sa is y Assump ion 3.4 and Assump ion 3.5 (a) and
(b), espec i ely. Fix z0, z1∈Rqand p∈Rq. Le Ξ = Ξz0,p,z1as in De ini ion 2.13 and,
o b
∈C∞
c(E),ε∈(0,1), and φ∈(0,1], se
b
†:= (1 −ε)b
+ε(1 −φ)V+εφΞ,
b
‡:= (1 + ε)b
−ε(1 −φ)V−εφΞ.
Fo z∈E, se †=b
†◦sz, and ‡=b
‡◦sz. Then, he ollowing s a emen s hold:
(a) †∈ D(A+)and ‡∈ D(A−). Suppose u he mo e ha Ais linea on i s
domain, hen
A+ †
1−ε=A(b
◦sz) + ε
1−ε(1 −φ)A+(V◦sz) + ε
1−εφA(Ξ ◦sz),(6.12)
A− ‡
1 + ε=A(b
◦sz)−ε
1 + ε(1 −φ)A+(V◦sz)−ε
1 + εφA(Ξ ◦sz),
(b) †,b
†∈ D(B+)and ‡,b
‡∈ D(B−). Suppose u he mo e ha Bis con ex, hen
o any x, y such ha z=x−y, we ha e
B+ †
1−ε(x)≤1
1−εB+ †(x)−B+b
†(y)+Bb
(y)(6.13)
+ε
1−ε(1 −φ)B+V(y) + ε
1−εφB+Ξ(y),
B− ‡
1 + ε(x)≥1
1 + εB− ‡(x)−B−b
‡(y)+Bb
(y)
−ε
1 + ε(1 −φ)B−V(y)−ε
1 + εφB−Ξ(y).
P oo . The domain s a emen s †∈ D(A+), ‡∈ D(A−), †,b
†∈ D(B+)and ‡,b
‡∈
D(B−) ollow by Lemma 6.2. The ou s a emen s in (6.12) and (6.13) ollow om
linea i y o A+and con exi y o B+.□
P oo o Theo em 3.1.To p o e inequali y (3.3), and consequen ly he s ong compa i-
son p inciple o he Hamil on–Jacobi equa ion in e ms o H, i su ices by Lemma 6.1
and P oposi ion 6.3 o es ablish (6.4), which we epea o eadabili y:
lim in
α→∞
H+ †(xα)
1−ε−H− ‡(x′
α)
1 + ε≤εC0
ε+Cε,φ.(6.14)
A COMPARISON PRINCIPLE BASED ON COUPLINGS 35
Le θ∗
1,α ∈Θ1be such ha
H+ †(xα) = sup
θ1∈Θ1
in
θ2∈Θ2{Aθ1,θ2 †(xα) + Bθ1,θ2 †(xα)−I(xα, θ1, θ2)}
= in
θ2∈Θ2nAθ∗
1,α,θ2 †(xα) + Bθ∗
1,α,θ2 †(xα)−I(xα, θ∗
1,α, θ2)o.
Such op imize exis s by he compac ness o Θ1and he lowe semi-con inui y o Iin
θ1assumed in (d). By Isaacs’ condi ion (a), we can w i e
H− ‡(x′
α) = in
θ2∈Θ2
sup
θ1∈Θ1Aθ1,θ2 ‡(x′
α) + Bθ1,θ2 ‡(xα)−I(x′
α, θ1, θ2).
Then, by compac ness o Θ2and he uppe semi-con inui y o Iin θ2assumed in (d),
we can ind θ∗
2,α ∈Θ2such ha
H− ‡(x′
α) = sup
θ1∈Θ1nAθ1,θ∗
2,α ‡(x′
α) + Bθ1,θ∗
2,α ‡(x′
α)−I(x′
α, θ1, θ∗
2,α)o.
Consequen ly, we can es ima e
1
1−εH+ †(xα)−1
1 + εH− ‡(x′
α)≤1
1−εAθ∗
1,α,θ∗
2,α †(xα)−1
1 + εAθ∗
1,α,θ∗
2,α ‡(x′
α)
|{z }
(1)
+1
1−εBθ∗
1,α,θ∗
2,α †(xα)−1
1 + εBθ∗
1,α,θ∗
2,α ‡(x′
α)
| {z }
(2)
+1
1 + εI(x′
α, θ∗
1,α, θ∗
2,α)−1
1−εI(xα, θ∗
1,α, θ∗
2,α)
| {z }
(3)
.
We ea (1),(2), and (3) sepa a ely. No e, ha due he compac ness o Θ1and Θ2,
he sequences o op imize s θ∗
1,α and θ∗
2,α con e ge o some θ∗
1and θ∗
2, espec i ely.
Es ima e (1):Using he expansions o A+ †and A− ‡ob ained in Lemma 6.4 we
ind
Aθ∗
1,α,θ∗
2,α †(xα)
1−ε−
Aθ∗
1,α,θ∗
2,α ‡(x′
α)
1 + ε=Aθ∗
1,α,θ∗
2,α,+ †(xα)
1−ε−Aθ∗
1,α,θ∗
2,α,− ‡(x′
α)
1 + ε
≤Aθ∗
1,α,θ∗
2,α 1(xα)−Aθ∗
1,α,θ∗
2,α 2(x′
α)
+ε
1−ε(1 −φ)Aθ∗
1,α,θ∗
2,α,+(V◦sxα−yα) (xα)
+ε
1 + ε(1 −φ)Aθ∗
1,α,θ∗
2,α,+V◦sx′
α−y′
α(x′
α)
+ε
1−εφAθ∗
1,α,θ∗
2,α (Ξ1◦sxα−yα) (xα)
+ε
1 + εφAθ∗
1,α,θ∗
2,α Ξ2◦sx′
α−y′
α(x′
α).(6.15)
We i s conside he e ms in ol ing Vand Ξ. By P oposi ion 5.1 (j), we ha e ha ,
along subsequences, he op imize s (xα, yα, yα,0, y′
α,0, y′
α, x′
α)con e ge o (z, z, z, z, z, z)
wi h z∈b
Kand pα, p′
α∈B1/α(0). Then, using he compa ibili y o Aθ1,θ2, c . Assump-
ion 3.5, we ind

36 SERENA DELLA CORTE, FABIAN FUCHS, RICHARD C. KRAAIJ, AND MAX NENDEL
lim in
α→∞
ε
1−ε(1 −φ)Aθ∗
1,α,θ∗
2,α,+(V◦sxα−yα) (xα)
+ε
1 + ε(1 −φ)Aθ∗
1,α,θ∗
2,α,+V◦sx′
α−y′
α(x′
α)
+ε
1−εφAθ∗
1,α,θ∗
2,α (Ξ1◦sxα−yα) (xα) + ε
1 + εφAθ∗
1,α,θ∗
2,α Ξ2◦sx′
α−y′
α(x′
α)
≤2ε
1−ε2(1 −φ)Aθ∗
1,θ∗
2,+(V)(z) + φAθ∗
1,θ∗
2(Ξz,0,z)(z).(6.16)
Nex , we conside he second line in (6.15). Using ha , o all θ1,θ2,Aθ1,θ2has a
con olled g ow h coupling b
Aθ1,θ2wi h a modulus uni o m in θ1and θ2sa is ying he
maximum p inciple and P oposi ion 5.3 (a), we ind
Aθ∗
1,α,θ∗
2,α 1(xα)−Aθ∗
1,α,θ∗
2,α 2(x′
α) = b
Aθ∗
1,α,θ∗
2,α ( 1⊖ 2) (xα, x′
α)
≤b
Aθ∗
1,α,θ∗
2,α α
2d2
xα−yα,x′
α−y′
α(xα, x′
α)
≤ωb
A,
b
Kαd(xα, yα) + d(yα, y′
α) + d(y′
α, x′
α)2
+d(xα, yα) + d(yα, y′
α) + d(y′
α, x′
α),(6.17)
which con e ges o 0as α→ ∞ by P oposi ion 5.1 (g).
Es ima e (2):By using he expansions o B+ †and B− ‡ob ained in Lemma 6.4,
we ind
Bθ∗
1,α,θ∗
2,α †(xα)
1−ε−Bθ∗
1,α,θ∗
2,α ‡(x′
α)
1 + ε=Bθ∗
1,α,θ∗
2,α,+ †(xα)
1−ε−Bθ∗
1,α,θ∗
2,α,− ‡(x′
α)
1 + ε
≤Bθ∗
1,α,θ∗
2,α b
1(yα)−Bθ∗
1,α,θ∗
2,α b
2(y′
α)
+1
1−εBθ∗
1,α,θ∗
2,α,+ †(xα)−Bθ∗
1,α,θ∗
2,α,+b
†(yα)
+1
1 + εBθ∗
1,α,θ∗
2,α,−b
‡(y′
α)−Bθ∗
1,α,θ∗
2,α,− ‡(x′
α)
+ε
1−ε(1 −φ)Bθ∗
1,α,θ∗
2,α,+V(yα) + ε
1 + ε(1 −φ)Bθ∗
1,α,θ∗
2,α,+V(y′
α)
+ε
1−εφBθ∗
1,α,θ∗
2,α Ξ1(yα) + ε
1 + εφBθ∗
1,α,θ∗
2,α Ξ2(y′
α).
Again, by sending α→ ∞, using P oposi ion 5.1 (j), and he compa ibili y o Bθ1,θ2, c .
Assump ion 3.5, we ob ain ha
lim in
α→∞
ε
1−ε(1 −φ)Bθ∗
1,α,θ∗
2,α,+V(yα) + ε
1 + ε(1 −φ)Bθ∗
1,α,θ∗
2,α,+V(y′
α)(6.18)
+ε
1−εφBθ∗
1,α,θ∗
2,α Ξ1(yα) + ε
1 + εφBθ∗
1,α,θ∗
2,α Ξ2(y′
α)
≤2ε
1−ε2(1 −φ)Bθ∗
1,θ∗
2,+(V)(z) + φBθ∗
1,θ∗
2(Ξz,0,z)(z).
A COMPARISON PRINCIPLE BASED ON COUPLINGS 37
Using ha , o all θ1,θ2,Bθ1,θ2is semi-mono one wi h Bθ1,θ2and he exp essions o
he g adien s ob ained in P oposi ion 5.3, we ind ha
1
1−εBθ∗
1,α,θ∗
2,α,+ †(xα)−Bθ∗
1,α,θ∗
2,α,+b
†(yα)
+Bθ∗
1,α,θ∗
2,α b
1(yα)−Bθ∗
1,α,θ∗
2,α b
2(y′
α)
+1
1 + εBθ∗
1,α,θ∗
2,α,−b
‡(y′
α)−Bθ∗
1,α,θ∗
2,α,− ‡(x′
α)
=1
1−εBθ∗
1,α,θ∗
2,α (xα, α(xα−yα)) −Bθ∗
1,α,θ∗
2,α (yα, α(xα−yα))
+Bθ∗
1,α,θ∗
2,α (yα, α(yα−y′
α)) −Bθ∗
1,α,θ∗
2,α (y′
α, α(yα−y′
α))
+1
1 + εBθ∗
1,α,θ∗
2,α (yα, α(y′
α−x′
α)) −Bθ∗
1,α,θ∗
2,α (x′
α, α(y′
α−y′
α)).(6.19)
By he semi-mono onici y p ope y o Bθ1,θ2, (6.19) is bounded by
1
1−εωB,
b
K(d(xα, yα) + αd2(xα, yα)) + ωB,
b
K(d(yα, y′
α) + αd2(yα, y′
α))
+1
1 + εωB,
b
K(d(y′
α, x′
α) + αd2(y′
α, x′
α)).(6.20)
Thus, aking he lim in α→∞ gi es 0by P oposi ion 5.1 (g).
Es ima e (3):We ha e
1
1 + εI(x′
α, θ∗
1,α, θ∗
2,α)−1
1−εI(xα, θ∗
1,α, θ∗
2,α)
=I(x′
α, θ∗
1,α, θ∗
2,α)−I(xα, θ∗
1,α, θ∗
2,α)−ε
1−εI(xα, θ∗
1,α, θ∗
2,α)−ε
1 + εI(x′
α, θ∗
1,α, θ∗
2,α).
By assump ion, Iadmi s a modulus o con inui y ωI,K, uni o m in θ1,θ2, implying
1
1 + εI(x′
α, θ∗
1,α, θ∗
2,α)−1
1−εI(xα, θ∗
1,α, θ∗
2,α)
≤ωI,
b
K(d(xα, x′
α)) −ε
1−ε(1 −φ)I(xα, θ∗
1,α, θ∗
2,α)−ε
1 + ε(1 −φ)I(x′
α, θ∗
1,α, θ∗
2,α).
Sending α→ ∞, using he lowe semi-con inui y o I, and using P oposi ion 5.1 (j), we
ind
lim in
α→∞
1
1 + εI(x′
α, θ∗
1,α, θ∗
2,α)−1
1−εI(xα, θ∗
1,α, θ∗
2,α)
≤lim in
α→∞ ωI,
b
K(d(xα, x′
α))
+ lim sup
α→∞ −ε
1−ε(1 −φ)I(xα, θ∗
1,α, θ∗
2,α)−ε
1 + ε(1 −φ)I(x′
α, θ∗
1,α, θ∗
2,α)
≤ − 2ε
1−ε2(1 −φ)I(z, θ∗
1, θ∗
2).
(6.21)
38 SERENA DELLA CORTE, FABIAN FUCHS, RICHARD C. KRAAIJ, AND MAX NENDEL
Conclusion: Pu ing oge he (6.16), (6.17), (6.18), (6.20), and (6.21), we can con-
clude ha
lim in
α→∞
H+ †(xα)
1−ε−H− ‡(x′
α)
1 + ε≤2ε
1−ε2(1 −φ)Aθ∗
1,θ∗
2,+V(z) + φAθ∗
1,θ∗
2(Ξz,0,z)(z)
+2ε
1−ε2(1 −φ)Bθ∗
1,θ∗
2,+V(z) + φBθ∗
1,θ∗
2(Ξz,0,z)(z)
−2ε
1−ε2(1 −φ)I(z, θ∗
1, θ∗
2)
≤2ε
1−ε2(1 −φ)(Aθ∗
1,θ∗
2,++Bθ∗
1,θ∗
2,+)(V)−I(·, θ∗
1, θ∗
2)
+2ε
1−ε2φ(Aθ∗
1,θ∗
2+Bθ∗
1,θ∗
2)(Ξ·,0,·)b
Kε
≤ε2
1−ε2cV+2
1−ε2φ(Aθ∗
1,θ∗
2+Bθ∗
1,θ∗
2)(Ξ·,0,·)b
Kε
≤εC0
ε+Cε,φ
wi h cVgi en by (3.1), and C0
εand Cε,φ de ined ia he las wo lines. The es ima e on
he di e ence o Hamil onians (6.14) and hus (6.4) o P oposi ion 6.3 a e sa is ied. As a
consequence ou inal es ima e (3.3) and, consequen ly, he s ong compa ison p inciple
ollow. □
Appendix A. The Jensen pe u ba ion
The main esul o his sec ion is P oposi ion A.1 ha allows us o pe u b a semi-
con ex unc ion wi h a unique ex eme poin such ha we ge a new ex eme poin
close by, in which he unc ion is wice di e en iable. The esul is a a ian o he
well-known pe u ba ion esul by Jensen, see e.g. [16, Lemma A.3].
P oposi ion A.1. Fix η > 0. Le ϕ:E×E→Rbe bounded abo e and semi-con ex
wi h con exi y cons an κ≥1. Suppose ha (x0, y0)is an op imize o
ϕ(x0, y0) = ⌈ϕ⌉.
Le R > 0,{ζz,p}z∈E,p∈Rq⊂C(E)and {ξz}z∈E⊂C1(E)and semi-conca i y cons an
κξbe as in De ini ion 2.13.
Fix ε1, ε2>0such ha 1−(ε1+ε2)κξ>0. Fu he mo e, de ine o p= (p1, p2)∈
Rq×Rq he pe u bed unc ions
ϕp(x, y) := ϕ(x, y)−ε1(ξx0(x) + ζx0,p1(x)) −ε2(ξy0(y) + ζy0,p2(y)) .(A.1)
Then he e exis p1, p2∈Bη(0), and a pai (x1, y1)∈Bη(x0)×Bη(y0)globally maxi-
mizing ϕpa which ϕpis wice di e en iable.
Co olla y A.2. Fo η > 0,pand (x1, y1)as in P oposi ion A.1, we ha e
0≤ −ε1(ξx0(x1) + ζx0,p1(x1)) −ε2(ξy0(y1) + ζy0,p2(y1)) ≤ε1η+εη2,(A.2)
and
⌈ϕ⌉ ≤ ϕp,ε(x1, y1)≤ ⌈ϕ⌉+ε1η+εη2.(A.3)
The p oo o he pe u ba ion p oposi ion is based pa ly on esul s om se - alued
analysis. To acili a e he p oo , we i s in oduce he necessa y auxilia y de ini ions
and esul s.
A COMPARISON PRINCIPLE BASED ON COUPLINGS 39
De ini ion A.3. A se - alued unc ion Γ : A⇒Bis called uppe hemi-con inuous a
a∈A, i , o all open neighbou hoods V⊆Bo Γ(a)(meaning ha Γ(a)⊆V), he e
exis s a neighbou hood Uo asuch ha , o all x∈U, we ha e Γ(x)⊆V.
I A, B a e me ic, his can equi alen ly o mula ed in e ms o sequences: A se -
alued map Γ : A⇒B, which akes closed alues, is uppe hemi-con inuous a a, i , o
any sequence an→aand bn∈Γ(an)sa is ying bn→b, we ha e b∈Γ(a).
We say ha Γis uppe hemi-con inuous, i i is uppe hemi-con inuous a all poin s.
Lemma A.4. Le Kbe a compac me ic space and le Ξbe a me ic space.
Fo any ξ∈Ξ, le ϕξ∈C(K)and suppose ha he map ξ7→ ϕξis con inuous
om Ξ o C(K), endowed wi h he sup emum no m on K. Then he se - alued map
Op : Ξ ⇒Kde ined by
Op (ξ) := {x∈K|ϕξhas a maximum a x}
is uppe hemi-con inuous.
P oo . The esul ollows immedia ely om Be ge’s Maximum Theo em [1, Theo em
17.31] wi h ξ7→ imageϕξ(K)being he ele an se - alued map. □
Rema k A.5. In he p oo below, we will make use o he no ion o a lim sup o se s.
Fo a sequence o se s (An)n∈Ndeno e
lim sup
n→∞ An=
n∈N[
m≥n
Am
o be in e p e ed as x∈lim supn→∞ Ani and only i he e a e in ini ely many n∈N
such ha x∈An.
The ollowing p oo is a a ian o he p oo o [16, Lemma A.3] and [11, Theo em
2.3.3].
P oo o P oposi ion A.1.Fo no a ional con enience, we will w i e w= (x, y)and w0=
(x0, y0). Le R > 0and {ζz,p}z∈E,p∈Rq⊂C(E)and {ξz}z∈E⊂C(E)be wo collec ions
o unc ions as in De ini ion 2.13. Wi hou loss o gene ali y, we can assume ha R≥η.
We s a ou by making z0 he unique op imize by eplacing ϕby
b
ϕ(w) = ϕ(w)−ε1ξx0(x)−ε2ξy0(y).
No e ha as 1−(ε1+ε2)κξ>0 he map b
ϕis semi-con ex and bounded om abo e
wi h a unique op imize w0.
Ou nex s ep is o locally, linea ly pe u b b
ϕ o ob ain ϕpas in equa ion (A.1). This
p ocedu e p oduces a new op imize close o w0in which he pe u bed unc ion ϕpis
wice di e en iable.
To u he acili a e he analysis o op imize s, we smoo hen ou ϕ. To ha end, le
Cδ:Cb(E)→C2
b(E)be a molli ie wi h supδ>0||Cδ || <∞and Cδ → uni o mly on
compac s as δ↓0. De ine
ϕp,δ(w):= (Cδϕ)(w)−ε1(ξx0(x) + ζx0,p1(x)) −ε2(ξy0(y) + ζy0,p2(y)) ,
whe e we will ead C0=
1
such ha ϕp,0=ϕpand ϕ0,0=b
ϕ.
We nex s udy he op imize s o he map (p, δ)7→ ϕp,δ on Ξ=(B1(0) ×B1(0))×[0,1]
using Be ge’s Maximum Theo em wi h K=BR(w0). Se
Op (p, δ):=nw∈BR(w0)ϕp,δ has a local maximum a w∈BR(w0)o.
Fi s no e ha he local na u e o he p oblem can be emo ed due o he ac ha he
pe u ba ions all anish in w0, whe eas hey add up o some hing nega i e ou side he
46 REFERENCES
[27] H. Ishii, “Uniqueness o unbounded iscosi y solu ion o Hamil on-Jacobi equa-
ions,” Indiana Uni . Ma h. J., ol. 33, no. 5, pp. 721–748, 1984.
[28] R. Jensen, P.-L. Lions, and P. E. Souganidis, “A uniqueness esul o iscosi y
solu ions o second o de ully nonlinea pa ial di e en ial equa ions,” P oceedings
o he Ame ican Ma hema ical Socie y, ol. 102, no. 4, pp. 975–978, 1988.
[29] R. Jensen, “The maximum p inciple o iscosi y solu ions o ully nonlinea second
o de pa ial di e en ial equa ions,” A chi e o Ra ional Mechanics and Analysis,
ol. 101, no. 1, pp. 1–27, 1988.
[30] T. Lind all, Lec u es on he coupling me hod (Wiley Se ies in P obabili y and
Ma hema ical S a is ics: P obabili y and Ma hema ical S a is ics). John Wiley &
Sons, Inc., New Yo k, 1992, pp. xi +257, A Wiley-In e science Publica ion.
[31] P.-L. Lions, “Solu ions de iscosi é des équa ions de Hamil on-Jacobi du p emie
o d e e applica ions,” in Goulaouic-Meye -Schwa z semina , 1983–1984, École
Poly ech., Palaiseau, 1984, Exp. No. 6, 13.
[32] M. Nendel, “Lowe semicon inui y o mono one unc ionals in he mixed opology
on Cb,” 2024. a Xi : 2210.09133 [q- in.MF].
[33] F. D. Sen illes, “Bounded con inuous unc ions on a comple ely egula space,”
T ans. Ame . Ma h. Soc., ol. 168, pp. 311–336, 1972.
[34] H. M. Sone , “Op imal con ol wi h s a e-space cons ain . II,” SIAM J. Con ol
Op im., ol. 24, no. 6, pp. 1110–1122, 1986.
[35] M. Spi ak, A comp ehensi e in oduc ion o di e en ial geome y. Vol. One. M.
Spi ak, B andeis Uni ., Wal ham, MA, 1970, iii+656 pp. (no consecu i ely paged)
pape bound, Published by M. Spi ak.
[36] D. W. S oock and S. R. S. Va adhan, Mul idimensional di usion p ocesses (G und-
leh en de Ma hema ischen Wissenscha en [Fundamen al P inciples o Ma hema -
ical Sciences]). Sp inge -Ve lag, Be lin-New Yo k, 1979, ol. 233, pp. xii+338.
[37] H. Tho isson, Coupling, s a iona i y, and egene a ion. Sp inge -Ve lag, 2000.
[38] H. Whi ney, “Analy ic ex ensions o di e en iable unc ions de ined in closed se s,”
T ans. Ame . Ma h. Soc., ol. 36, no. 1, pp. 63–89, 1934.
[39] J. Yong and X. Y. Zhou, S ochas ic con ols (Applica ions o Ma hema ics (New
Yo k)). Sp inge -Ve lag, New Yo k, 1999, ol. 43, pp. xxii+438, Hamil onian sys-
ems and HJB equa ions.
Del Ins i u e o Applied Ma hema ics, Del Uni e si y o Technology, The Ne he -
lands
Email add ess:[email p o ec ed]
Cen e o Ma hema ical Economics, Biele eld Uni e si y, Ge many
Email add ess:[email p o ec ed]
Del Ins i u e o Applied Ma hema ics, Del Uni e si y o Technology, The Ne he -
lands
Email add ess:[email p o ec ed]
Cen e o Ma hema ical Economics, Biele eld Uni e si y, Ge many
Email add ess:[email p o ec ed]