The pe sis ence o synch oniza ion unde
en i onmen al noise
By Tom´
as Ca aballo aand Pe e E. Kloeden b
aDepa amen o de Ecuaciones Di e enciales y An´alisis Num´e ico,
Uni e sidad de Se illa, Apdo. de Co eos 1160,
41080-Se illa, Spain
E-mail: ca abal[email p o ec ed]
bFachbe eich Ma hema ik, Johann Wol gang Goe he Uni e si ¨a
D-60054 F ank u am Main, Ge many
E-mail: kloe[email p o ec ed]ank u .de
I is shown ha he synch oniza ion o dissipa i e sys ems pe sis s when hey a e
dis u bed by addi i e noise no ma e how la ge he in ensi y o he noise p o ided
asymp o ically s able s a iona y s ochas ic solu ions a e used ins ead o asymp o -
ically s able equilib ia.
Keywo ds: Synch oniza ion, addi i e noise, andom a ac o , s a iona y
s ochas ic p ocess, one-sided Lipschi z dissipa i e condi ion
1. In oduc ion
Synch oniza ion o coupled sys ems is a e y well known phenomenon in biology
and physics, and also in he social sciences. A eadable desc ip i e accoun o i s
di e si y o occu ence can be ound in he ecen book o S oga z (2003), which
con ains an ex ensi e lis o e e ences. The synch oniza ion o coupled dissipa i e
sys ems has been in es iga ed ma hema ically in he case o au onomous sys ems
by A aimo ich and Rod igues (1998), Ca alho e al. (1998) and Rod igues (1996),
bo h o asymp o ically s able equilib ia and gene al a ac o s, such as chao ic a -
ac o s. Analogous esul s also hold o nonau onomous sys ems (Kloeden (2003)),
bu equi e a new concep o a nonau onomous a ac o .
In his no e we in es iga e he e ec o addi i e noise on he synch oniza ion
o coupled dissipa i e sys ems wi h asymp o ically s able equilib ia, which esul s
in a coupled sys em o I o s ochas ic di e en ial equa ions. Such noise is o en
conside ed as modelling backg ound en i onmen al e ec s. We show ha synch o-
niza ion pe sis s independen ly o noise in ensi y in e ms o asymp o ically s able
s a iona y s ochas ic solu ions a he han equilib ia. Such asymp o ically s able
s a iona y s ochas ic solu ions a e a special case o andom a ac o s (see C auel
& Flandoli (1994), C auel e al. (1997), Schmal uss (1992)) which a e a p obabilis ic
coun e pa o he de e minis ic nonau onomous a ac o used in Kloeden (2003).
We p esen some equi ed backg ound ma e ial in Sec ion 2 and hen o mula e
he p oblem o synch oniza ion o dissipa i e sys ems o bo h he de e minis ic
and s ochas ic cases in Sec ion 3. In Sec ion 4 we show ha he uncoupled sys ems
wi h addi i e noise ha e andom a ac o s consis ing o single s a iona y s ochas ic
A icle submi ed o Royal Socie y T
EX Pape
2T. Ca aballo & P.E. Kloeden
p ocesses and hen es ablish he co esponding esul o coupled sys ems in Sec ion
5. Finally, he asymp o ic beha iou o he coupled sys ems in he limi as he
coupling pa ame e becomes a bi a ily la ge a e p esen ed in Sec ion 6.
2. Backg ound ma e ial
We need some backg ound no a ion and esul s. Le (Ω,F,P) be a p obabili y space.
Following A nold (1998) a andom dynamical sys em (RDS) (θ, φ) on Ω ×Rd
consis s o a me ic dynamical sys em θon Ω and a cocycle mapping φ:R+×Ω×Rd
→Rd. Essen ially (and su icien o ou pu poses he e), θ ep esen s he d i ing
noise p ocess and φ he s a e space e olu ion o he sys em. Fo example, o a
s ochas ic di e en ial equa ion on Rdwi h a wo-sided scala Wiene p ocess W ,
i.e. de ined o all ∈R,θis de ined by θ ω(·) = ω( +·)−ω(·) on a canonical
sample space Ω = C0(R,R) whe e, by de ini ion, W (ω) := ω( ), ∈R, and φis
de ined by φ( , ω, x0) = Xx0
(ω), he solu ion o he SDE s a ing a Xx0
0(ω) = x0.
See he exposi ions in A nold (1998) and Kloeden e al. (1999) o mo e de ails.
A amily b
A={A(ω), ω ∈Ω}o nonemp y measu able compac subse s A(ω)
o Rdis called φ-in a ian i φ( , ω, A(ω)) = A(θ ω) o all ≥0 and is called a
andom a ac o i in addi ion i is pa hwise pullback a ac ing in he sense ha
H∗
d(φ( , θ− ω, D(θ− ω)), A(ω)) →0 as → −∞
o all sui able (i.e. in a gi en a ac ing uni e se as, o ins ance, in Kloeden e al.
(1999)) amilies o b
D={D(ω), ω ∈Ω}o nonemp y measu able bounded subse s
D(ω) o Rd. He e H∗
dis he Hausdo semi-dis ance on Rd. The ollowing esul
(c . A nold (1998), Kloeden e al. (1999), Schmal uss (1992)) ensu es he exis ence
o a andom a ac o .
Theo em 2.1. Le (θ, φ)be an RDS on Ω×Rd. I he e exis s a amily b
B=
{B(ω), ω ∈Ω}o nonemp y measu able compac subse s B(ω)o Rdand a Tb
D,ω ≥
0such ha
φ( , θ− ω, D(θ− ω)) ⊂B(ω),∀ ≥Tb
D,ω
o all amilies b
D={D(ω), ω ∈Ω}in he gi en a ac ing uni e se, hen he RDS
(θ, φ)has a andom a ac o b
A={A(ω), ω ∈Ω}wi h he componen subse s de ined
o each ω∈Ωby
A(ω) =
s>0[
≥s
φ( , θ− ω, B(θ− ω)).
No e ha i he andom a ac o consis s o single on se s, i.e A(ω) = {X∗(ω)}
o some andom a iable X∗, hen X∗
(ω) := X∗(θ ω) is a s a iona y s ochas ic
p ocess.
We also need he ollowing lemma a.
Lemma 2.2. Le {xn}be a sequence in a comple e me ic space (X, d)such ha
e e y subsequence {xni}has a subsequence {xnij}con e ging o a common limi
x∗. Then he sequence {xn}con e ges o x∗.
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Pe sis ence o synch oniza ion unde en i onmen al noise 3
P oo . I no , hen he e would exis an η > 0 and a subsequence {xni}such ha
d(xni, x∗)≥η o all i, which is no possible because {xni}has a subsequence
{xnij}con e ging o he x∗.
Lemma 2.3. Le W be a wo-sided Wiene p ocess, i.e. de ined o all ∈R.
Then he in eg als νR
−∞ e−ν( −s)dW (ω)a e pa hwise uni o mly bounded in ν > 0
on ini e ime in e als [T1, T2]o Rand he in eg als
Z
T1
e−ν( −s)dW (ω)→0as ν→ ∞
pa hwise on ini e ime in e als [T1, T2]o R.
P oo . In eg a ing by pa s (see Mao (1997)), we ha e
Z
−∞
e−ν( −s)dWs(ω) = W (ω)−νZ
−∞
e−ν( −s)Ws(ω)ds.
Then we use he sub-exponen ial g ow h o he pa hs o Wiene p ocesses. Fo he
second pa we i s no ice ha
νZ
T1
e−ν( −s)W (ω)ds =W (ω)−e−ν( −T1)W (ω)
o any ω∈Ω. In eg a ing again by pa s i ollows
Z
T1
e−ν( −s)dWs(ω) = W (ω)−e−ν( −T1)WT1(ω)−νZ
T1
e−ν( −s)Ws(ω)ds
=e−ν( −T1)(W (ω)−WT1(ω))
+νZ
T1
e−ν( −s)(W (ω)−Ws(ω)) ds
om which he esul ollows.
3. Fo mula ion o he p oblem
Suppose we ha e wo au onomous o dina y di e en ial equa ions in Rd,
dx
d = (x),dy
d =g(y),(3.1)
which a e su icien ly egula o ensu e he o wa ds exis ence and uniqueness o
solu ions and sa is y one-sided dissipa i e Lipschi z condi ions
hx1−x2, (x1)− (x2)i ≤ −L|x1−x2|2,
hy1−y2, g(y1)−g(y2)i ≤ −L|y1−y2|2,(3.2)
on Rd o some L > 0, and hus ha e unique equilib ia ¯xand ¯y, espec i ely, which
a e globally asymp o ically s able (c . Kloeden (2004)). No ice ha he con inui y o
and g, and he one-sided dissipa i e Lipschi z condi ions (3.2) ensu e he o wa ds
exis ence and uniqueness o solu ions o (3.1).
A icle submi ed o Royal Socie y
4T. Ca aballo & P.E. Kloeden
Conside now he dissipa i ely coupled sys em
dx
d = (x) + ν(y−x),dy
d =g(y) + ν(x−y) (3.3)
wi h ν > 0. I can be shown (see A aimo ich and Rod igues (1998), and Ca alho
e al. (1998)) ha his also has a unique equilib ium (¯xν,¯yν), which is globally
asymp o ically s able. Mo eo e , (¯xν,¯yν)→(¯z, ¯z) as ν→ ∞, whe e ¯zis he unique
globally asymp o ically s able equilib ium o he “a e aged” sys em
dz
d =1
2( (z) + g(z)) .(3.4)
This phenomena is known as synch oniza ion. Analogous esul s hold o mo e gen-
e al au onomous a ac o s (c . A aimo ich and Rod igues (1998), Ca alho e al.
(1998)) as well as o nonau onomous sys ems (Kloeden (2003)) wi h app op ia ely
de ined nonau onomous a ac o s.
The aim o his no e is o show ha his synch oniza ion e ec is p ese ed un-
de addi i e noise p o ided equilib ia a e eplaced by s a iona y andom solu ions.
Speci ically, we conside wo I o s ochas ic di e en ial equa ions in Rd,
dX = (X )d +α dW1
, dY =g(Y )d +β dW 2
,(3.5)
whe e α,β∈Rd
+a e cons an ec o s wi h no componen s equal o ze o, W1
,
W2
a e independen wo-sided scala Wiene p ocesses†, and ,ga e as abo e,
in pa icula , sa is ying he one-sided dissipa i e Lipschi z condi ions (3.2). We
will show in he nex sec ion ha each o hese s ochas ic sys ems has a pa hwise
asymp o ically s able andom a ac o consis ing o a s a iona y andom a iable.
Fo example, o linea d i e ms, i.e he SDEs
dX =−X d +α dW1
, dY =−Y d +β dW 2
,(3.6)
hese andom a iables a e gi en explici ly by
¯
X =αe− Z
−∞
esdW1
s,¯
Y =βe− Z
−∞
esdW2
s.(3.7)
The synch onized sys em co esponding o SDEs (3.5) eads
dX = ( (X ) + ν(Y −X )) d +α dW1
,
dY = (g(Y ) + ν(X −Y )) d +β dW 2
.(3.8)
I will be shown ha his sys em is dissipa i e and has a unique s a iona y solu ion
(¯
Xν
,¯
Yν
), which is pa hwise globally asymp o ically s able wi h
(¯
Xν
,¯
Yν
)→(¯
Z∞
,¯
Z∞
) as ν→ ∞,
pa hwise on ini e ime in e als [T1, T2] o R, whe e ¯
Z∞
is he unique pa hwise
globally asymp o ically s able s a iona y solu ion o he “a e aged” SDE
dZ =1
2( (Z ) + g(Z )) d +1
2α dW 1
+1
2β dW2
.(3.9)
†al e na i ely, one could ake α,βscala alued and W1
,W2
ec o alued
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Pe sis ence o synch oniza ion unde en i onmen al noise 5
Fo example, o he linea SDEs (3.6) we ha e
dZ =−Z d +1
2α dW1
+1
2β dW2
wi h
¯
Z∞
=1
2e− µαZ
−∞
esdW1
s+βZ
−∞
esdW2
s¶=1
2¡¯
X +¯
Y ¢,
his a e aged exp ession being due o he special linea s uc u e.
In addi ion o he one-sided Lipschi z dissipa i e condi ion on he unc ions
and gwe assume he ollowing in eg abili y condi ion:
The e exis s m0>0 such ha o any m∈(0, m0], and any con inuous unc ion
u:R→Rdwi h sub-exponen ial g ow h i ollows
Z
−∞
ems| (u(s))|2ds < +∞,Z
−∞
ems|g(u(s))|2ds < +∞.(3.10)
Wi hou loss o gene ali y, we can assume ha L≤m0.
4. The uncoupled sys ems wi h addi i e noise
We conside he i s o he uncoupled equa ions in (3.5),
dX = (X )d +α dW 1
.(4.1)
I s solu ion pa hs a e gene ally no di e en iable, so in o de o use he one-sided
dissipa i e Lipschi z condi ion (3.2) we conside he di e ence X −¯
X whe e ¯
X
is he O ns ein-Uhlenbeck s a iona y p ocess (3.7) sa is ying he i s o he linea
equa ions (3.6). This di e ence is pa hwise di e en iable since he pa hs X and ¯
X
a e con inuous and sa is y he in eg al equa ion
X −¯
X =X0−¯
X0+Z
0¡ (Xs) + ¯
Xs¢ds,
which, by he undamen al heo em o calculus, is hus equi alen o he di e en ial
exp ession
d
d ¡X −¯
X ¢= (X ) + ¯
X ,
om which i ollows ha
d
d ¯¯X −¯
X ¯¯2= 2 X −¯
X , (X )− (¯
X )®+ 2 X −¯
X , (¯
X ) + ¯
X )®
≤ −2L¯¯X −¯
X ¯¯2+L¯¯X −¯
X ¯¯2+1
L¯¯ (¯
X ) + ¯
X )¯¯2
and hus
¯¯X −¯
X ¯¯2≤¯¯X 0−¯
X 0¯¯2e−L( − 0)+e−L
LZ
0
eLs ¯¯ (¯
Xs) + ¯
Xs¯¯2ds.
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6T. Ca aballo & P.E. Kloeden
Pa hwise pullback con e gence (i.e. as 0→ −∞) gi es pullback abso p ion (c .
Theo em 2.1)
¯¯X −¯
X ¯¯2≤R2
X(θ ω) := 1 + e−L
LZ
−∞
eLs ¯¯ (¯
X(θsω)) + ¯
X(θsω)¯¯2ds
o all ≥Tb
D(ω) o app op ia e amilies b
D(ω) o bounded se s {D(θ ω), ∈R}o
ini ial condi ions.
The in eg als he e exis due o assump ion (3.10) and he ac ha O ns ein-
Uhlenbeck p ocesses inhe i he sub-exponen ial g ow h o hei Wiene p ocesses.
Thus ¯¯X (ω)−¯
X (ω)¯¯≤RX(θ ω),∀ ≥Tb
D(ω),
and so
|X (ω)| ≤ ¯¯¯
X (ω)¯¯+RX(θ ω),∀ ≥Tb
D(ω),
which means ha his sys em has a andom a ac o b
A={A(ω), ω ∈Ω}. Bu he
di e ence o any wo solu ions sa is ies he di e en ial inequali y
d
d ¯¯X1
−X2
¯¯2≤ −2L¯¯X1
−X2
¯¯2,
which means all solu ions con e ge pa hwise o each o he and hus he andom a -
ac o se s a e single on se s A(ω) = {X∗(ω)}, i.e. he andom a ac o is o med
by a s a iona y andom p ocess X∗
(ω) = X∗(θ ω) which pa hwise a ac s all o he
solu ions.
An analogous si ua ion holds o he second equa ion o he uncoupled equa ions
in (3.5).
5. The synch onized sys em wi h addi i e noise
To show ha he synch onized sys em (3.8) is s ongly dissipa i e, we use he OU
p ocesses
X∗,ν
=αe−ν Z
−∞
eνs dW 1
s, Y ∗,ν
=βe−ν Z
−∞
eνs dW 2
s,(5.1)
which a e he s a iona y solu ions o he linea equa ions
dX =−νX d +α dW 1
, dY =−νY d +β dW 2
.(5.2)
The di e ences o he solu ions o (3.8) and hese s a iona y solu ions a e pa hwise
di e en iable and sa is y he sys em o andom di e en ial equa ions
d
d ¡X −X∗,ν
¢= (X ) + ν(Y −X ) + νX∗,ν
d
d ¡Y −Y∗,ν
¢=g(Y ) + ν(X −Y ) + νY ∗,ν
,
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Pe sis ence o synch oniza ion unde en i onmen al noise 7
which, wi h Uν
:= X −X∗,ν
and Vν
:= Y −Y∗,ν
, is equi alen o
d
d Uν
= (Uν
+X∗,ν
)+ν(Vν−Uν)+νY ∗,ν
,d
d Vν=g(Y )+ν(Uν−Vν)+νX∗,ν
.
Thus
1
2
d
d ¡|Uν
|2+|Vν
|2¢=Uν
, (Uν
+X∗,ν
)− (X∗,ν
)®+Vν
, g(Vν
+Y∗,ν
)−g(Y∗,ν
)®
+Uν
, (X∗,ν
) + νY ∗,ν
®+Vν
, g(Y∗,ν
) + νX∗,ν
®
−νhUν
−Vν
, Uν
−Vν
i
≤ −L¡|Uν
|2+|Vν
|2¢+L
2¡|Uν
|2+|Vν
|2¢
+2
L¯¯ (X∗,ν
) + νY ∗,ν
¯¯2+2
L¯¯g(Y∗,ν
) + νX∗,ν
¯¯2,
and hence
d
d ¡|Uν
|2+|Vν
|2¢≤ −L¡|Uν
|2+|Vν
|2¢+4
L¯¯ (X∗,ν
) + νY ∗,ν
¯¯2+4
L¯¯g(Y∗,ν
) + νX∗,ν
¯¯2.
This means ha |Uν
(ω)|2+|Vν
(ω)|2is pa hwise abso bed by he amily b
Bν=
{Bν(ω), ω ∈Ω}o closed balls in R2dcen e ed on he o igin and o adius Rν(ω),
whe e R2
ν(ω) is de ined by
1+ 4
LZ0
−∞
eLs ³| (X∗,ν (θsω) + νY ∗,ν (θsω)|2ds +|g(Y∗,ν (θsω) + νX∗,ν (θsω)|2´ds.
Hence, by Theo em 2.1, he synch onized sys em has a andom a ac o b
Aν=
{Aν(ω), ω ∈Ω}wi h Aν(ω)⊂Bν(ω). Bu he di e ence (∆X ,∆Y ) = (X1
−
X2
, Y 1
−Y2
) o any pai o solu ions sa is ies he sys em o andom di e en ial
equa ions
d
d ∆X = (X1
)− (X2
)+ν(∆Y −∆X ),d
d ∆Y =g(Y1
)−g(Y2
)+ν(∆X −∆Y ),
so
1
2
d
d ¡|∆X |2+|∆Y |2¢=∆X , (X1
)− (X2
)®+∆Y , g(Y1
)−g(Y2
)®
−νh∆X −∆Y ,∆X −∆Y i
≤ −L¡|∆X |2+|∆Y |2¢,
om which we ob ain
|∆X (ω)|2+|∆Y (ω)|2≤¡|∆X0(ω)|2+|∆Y0(ω)|2¢e−2L ,
which means all solu ions con e ge pa hwise o each o he as → ∞. Thus he
andom a ac o consis s o single on se s o med by an o de ed pai o s a iona y
p ocesses ( ¯
Xν
(ω),¯
Yν
(ω)).
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8T. Ca aballo & P.E. Kloeden
6. The synch onized s a iona y solu ions as ν→ ∞
Lemma 6.1. ¯
Xν
(ω)−¯
Yν
(ω)→0as ν→ ∞ pa hwise on any bounded ime in e al
[T1, T2]o R.
P oo . Sub ac ing he second om he i s equa ion o (3.8) gi es
d¡¯
Xν
−¯
Yν
¢=¡−2ν¡¯
Xν
−¯
Yν
¢+ (¯
Xν
)−g(¯
Yν
)¢d
+α dW1
−β dW 2
,
o , wi h Dν
=¯
Xν
−¯
Yν
,
d¡Dν
e2ν ¢=e2ν ¡ (¯
Xν
)−g(¯
Yν
)¢d +αe2ν dW1
−βe2ν dW 2
,
so pa hwise
|Dν
| ≤ e−2νT1|Dν
T1|+Z
T1
e−2ν( −s)¡¯¯ (¯
Xν
s)¯¯+¯¯g(¯
Yν
s)¯¯¢ds
+|α|¯¯¯¯Z
T1
e−2ν( −s)dW1
¯¯¯¯+|β|¯¯¯¯Z
T1
e−2ν( −s)dW2
¯¯¯¯.(6.1)
By Lemma 2.3 we see ha he adius Rν(θ ω) is pa hwise uni o mly bounded on
each bounded ime in e al [T1, T2], so |Xν
(ω)|,|Yν
(ω))|and |Dν
0(ω))|a e pa hwise
uni o mly bounded on each bounded ime in e al [T1, T2]. Then by Lemma 2.3 and
condi ion (3.10) we see ha all o he in eg als in (6.1) con e ge o ze o as ν→ ∞
pa hwise on he bounded ime in e al [T1, T2].
Theo em 6.2. (¯
Xνn
,¯
Yνn
)→(Z∞
, Z∞
)pa hwise uni o mly on bounded ime in-
e als [T1, T2] o any sequence νn→ ∞, whe e Z∞
is he s a iona y s ochas ic
solu ion o he a e aged SDE
dZ =1
2( (Z ) + g(Z )) d +1
2αdW 1
+1
2βdW2
.(6.2)
P oo . De ine
Zν
:= 1
2¡¯
Xν
+¯
Yν
¢∀ ∈R
and obse e ha Zν
sa is ies he equa ion
dZν
=1
2¡ (¯
Xν
) + g(¯
Yν
)¢d +1
2α dW1
+1
2β dW2
.
Also de ine
¯
Z := 1
2¡¯
X +¯
Y ¢∀ ∈R
whe e ¯
X and ¯
Y a e he OU p ocesses sa is ying he linea SDEs (3.6).
The di e ence Zν
−¯
Z is pa hwise con inuous and hus by Lemma 2.3 and
he de ini ions o he espec i e abso bing se s, equi-bounded w. . . ν > 0 on he
bounded ime in e al [T1, T2]. Mo eo e Zν
−¯
Z is pa hwise di e en iable and
sa is ies he andom di e en ial exp ession
d
d ¡Zν
−¯
Z ¢=1
2 (¯
Xν
) + 1
2g(¯
Yν
) + 1
2¯
X +1
2¯
Y ,
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Pe sis ence o synch oniza ion unde en i onmen al noise 9
Since
¯¯¯¯
d
d ¡Zν
(ω)−¯
Z (ω)¢¯¯¯¯≤1
2¯¯ (¯
Xν
(ω))¯¯+1
2¯¯g(¯
Yν
(ω))¯¯+1
2¯¯¯
X (ω)¯¯+1
2¯¯¯
Y (ω)¯¯
≤MT1,T2(ω)<∞
by Lemma 2.3, we can use he Ascoli Theo em o conclude ha o any sequence νn
→ ∞, he e is a (possibly) andom subsequence νnj(ω)→ ∞ such ha Zνnj
(ω)−
¯
Z (ω)→Z∞
(ω)−¯
Z (ω) as j→ ∞, and hus ha Zνnj
(ω)→Z∞
(ω) as j→ ∞.
Now
Zνnj
(ω)−¯
Yνnj
(ω) = 1
2³¯
Xνnj
(ω)−¯
Yνnj
(ω)´→0,
Zνnj
(ω)−¯
Xνnj
(ω) = 1
2³¯
Yνnj
(ω)−¯
Xνnj
(ω)´→0
as νnj→ ∞, so
¯
Xνnj
(ω)=2Zνnj
(ω)−¯
Yνnj
(ω)→Z∞
(ω),
¯
Yνnj
(ω)=2Zνnj
(ω)−¯
Xνnj
(ω)→Z∞
(ω)
as νnj→ ∞. Mo eo e ,
Zν
−¯
Z =Zν
−¯
Z +1
2Z
T1
(¯
Xν
s)ds +1
2Z
T1
g(¯
Yν
s)ds
+1
2Z
T1
¯
Xsds +1
2Z
T1
¯
Ysds
which con e ges pa hwise o
Z∞
=Z∞
T1+1
2Z
T1
(Z∞
s)ds +1
2Z
T1
g(Z∞
s)ds
+¯
Z −¯
ZT1+1
2Z
T1
¯
Xsds +1
2Z
T1
¯
Ysds
=Z∞
T1+1
2Z
T1
(Z∞
s)ds +1
2Z
T1
g(Z∞
s)ds
+1
2αZ
T1
dW1
s+1
2βZ
T1
dW2
s,
on he in e al [T1, T2], so Z∞
is a solu ion o he SDE (6.2) o all ∈R. The d i
o his SDE sa is ies he s ongly dissipa i e one-sided Lipschi z condi ion (3.2),
so i has a andom a ac o consis ing o a single on se o med by a s a iona y
s ochas ic p ocess which hus mus be equal o Z∞
.
Finally, we no e ha pa hwise all possible subsequences he e ha e he same
limi , so by Lemma 2.2 e e y ull sequence Zνn
ac ually con e ges o Z∞
as νn→
∞.
As a s aigh o wa d consequence o he a gumen s in he p e ious p oo we
ha e
A icle submi ed o Royal Socie y