EXPONENTIALLY STABLE STATIONARY SOLUTIONS FOR
STOCHASTIC EVOLUTION EQUATIONS AND THEIR
PERTURBATION
TOM´
AS CARABALLO, PETER E. KLOEDEN, AND BJ¨
ORN SCHMALFUSS
Abs ac . We conside he exponen ial s abili y o s ochas ic e olu ion equa-
ions wi h Lipschi z con inuous non-linea i ies when ze o is no a solu ion o
hese equa ions. We p o e he exis ence o a non- i ial s a iona y solu ion
which is exponen ially s able, whe e he s a iona y solu ion is gene a ed by
he composi ion o a andom a iable and he Wiene shi . We also cons uc
s a iona y solu ions wi h he s onge p ope y o a ac ing bounded se s uni-
o mly. The exis ence o hese s a iona y solu ions ollows om he heo y o
andom dynamical sys ems and hei a ac o s. In addi ion, we p o e some
pe u ba ion esul s and o mula e condi ions o he exis ence o s a iona y
solu ions o semi-linea s ochas ic pa ial di e en ial equa ions wi h Lipschi z
con inuous non-linea i ies.
1. In oduc ion
The exponen ial s abili y o s ochas ic pa ial di e en ial equa ions is an impo an
p oblem, and i has ecei ed conside able a en ion du ing he ecen decades as
he as li e a u e on his opic shows. Ou aim he e is o s udy he exponen ial
s abili y o non- i ial s a iona y solu ions o hese equa ions.
The in es iga ion o s abili y o cons an s a iona y solu ions o ini e dimensional
s ochas ic di e en ial equa ions goes back o Has ´minski˘ı [14] using Lyapuno unc-
ions o he gene a o o he Ma ko semi-g oup. These ideas ha e been ex ended
by Mao [20] in he ini e dimensional con ex , while non-cons an s a iona y solu-
ions ha e been ea ed in Schmal uß [24]. He e we will gene alize some echniques
om hese las wo publica ions.
Fo in ini e dimensional (pa abolic) s ochas ic di e en ial equa ions he p oblem o
exponen ially s able cons an s a iona y solu ions has been conside ed by Ca aballo
and Real [5] (see also [3], [4]), Liu and Mao [19], Chow [7], Haussmann [15] and
Ichikawa [16] among o he s.
In con as o hese cons an s a iona y solu ions we will in es iga e he asymp-
o ic exponen ial s abili y o non- i ial s a iona y solu ions. In his espec , we
will conside semilinea s ochas ic e olu ion equa ions wi h Lipschi z con inuous
non-linea i ies. Unde sui able assump ions we p o e he exis ence o a unique s a-
iona y solu ion by using a ixed poin a gumen based on he pullback echnique.
This s a iona y solu ion u ns o be exponen ially s able in mean squa e, and also
Da e: June, 2004.
Key wo ds and ph ases. andom dynamical sys ems, s a iona y solu ions, exponen ial s abili y,
s abiliza ion.
1
2 TOM´
AS CARABALLO, PETER E. KLOEDEN, AND BJ ¨
ORN SCHMALFUSS
in he almos su e sense.
Al hough we p o e almos su e con e gence o he s a iona y solu ion, he excep-
ional se s depend on he ini ial condi ion, so i is no possible o conside uni o m
con e gence wi h espec o a bounded se o ini ial condi ions. Howe e , his p ob-
lem will be o e come by using a echnique om andom dynamical sys ems which
allows us o s udy he p oblem o excep ional se s independen o he se o ini ial
condi ions. I is unknown in gene al i s ochas ic pa ial di e en ial equa ions wi h
gene al di usion coe icien s gene a e andom dynamical sys ems. Howe e , i we
suppose some kind o commu a i i y on hese coe icien s, hen we a e able o p o e
he exis ence o such a andom dynamical sys em.
We p o e he exis ence o a andom ixed poin , which is in ac a andom a iable.
This andom a iable gene a es he exponen ially s able s a iona y solu ion o he
s ochas ic pa ial di e en ial equa ion. Mo eo e , his s a iona y solu ion a ac s
bounded se s o ini ial condi ions.
S a iona y solu ions in his in e p e a ion co espond wi h single poin andom
a ac o s, an impo an objec in he heo y o andom dynamical sys ems, see
C auel, Debussche and Flandoli [8], Flandoli and Schmal uß [12] o [23].
Ano he aim o his pape is o analyze pe u ba ions o s ochas ic pa ial di e -
en ial equa ions and he ela ion be ween hei s a iona y solu ions. These pe -
u ba ions will be gi en by modi ying he non-linea pa o he equa ion. Unde
he assump ion ha he pe u ba ions app oach he non-linea pa o he o iginal
equa ion and ha he Lipschi z cons an s o he pe u bed ope a o s a e uni o mly
bounded and no oo la ge, we ob ain he exis ence o s a iona y solu ions which
con e ge o he s a iona y solu ion o ou o iginal p oblem in he mean squa e
sense.
Fo omega-wise con e gence we o mula e a heo em on he con inuous dependence
o andom ixed poin s on a pa ame e .
The pape is o ganized as ollows. In Sec ion 2 we in oduce basic concep s o
s ochas ic e olu ion equa ions and andom dynamical sys ems. The hi d sec ion
deals wi h he exponen ial s abili y o s a iona y solu ions o s ochas ic e olu ion
equa ions in he mean squa e sense and almos su ely. In he nex sec ion, his
exponen ial s abili y is analyzed om he poin o iew o andom dynamical sys-
ems. Then in Sec ion 5, we conside he pe u ba ion p oblems, and illus a e he
esul s wi h an example in he inal sec ion.
2. Random dynamical sys ems and s ochas ic e olu ion equa ions
In his sec ion we will desc ibe he concep o exponen ially s able s a iona y so-
lu ions o s ochas ic non-linea e olu ion equa ions gene a ed by andom ixed
poin s. To do his we s a by desc ibing he noise d i ing he di e en ial equa ion.
Le (Ω,F,{F } ∈R,P) be a il e ed p obabili y space such ha
Fs⊂ F ⊂ F o s≤ .
In wha ollows we will conside a wo-sided Wiene p ocess Wwi h alues in some
sepa able Hilbe space Uwhe e he co a iance Qis a symme ic ope a o on Uo
ace class. Fo ins ance, o he abo e p obabili y space we will choose o Ω he
3
se o con inuous pa hs C0(R, U) which a e ze o a ze o equipped wi h he compac
open opology. Fis supposed o be he associa ed Bo el-σ-algeb a and Pis de ined
o be he Wiene measu e wi h espec o he co a iance Q. Fo F we se
σ{ω(u)−ω( ) : , u ≤ }.
Wha we ha e in oduced is he B ownian mo ion me ic dynamical sys em which
is he s anda d noise o andom dynamical sys ems gene a ed by s ochas ic di e -
en ial equa ions.
No e ha he abo e p obabili y space is no comple ed. The comple ion o his
p obabili y space is deno ed by (Ω,¯
F,{¯
F } ∈R,P) whe e{¯
F } ∈Rhas o be a no -
mal il a ion, see Da P a o and Zabczyk [10] Page 75.
We now in oduce on he abo e non-comple ed p obabili y space a measu able low
θ={θ } ∈Ron Ω:
(1) θ: (R×Ω,F ⊗ B(R)) →(Ω,F), θ +τ=θ ◦θτ, , τ ∈R, θ0= idΩ.
The Wiene shi ope a o s which o m he low θ
θ ω(·) = ω(·+ )−ω( ), ∈R, ω ∈Ω
lea e he Wiene measu e Pin a ian . Mo e p ecisely, Pis e godic wi h espec o
θ. In addi ion, wi h espec o he il a ion we ha e ha
(2) θ−1
uF =F +u
o any , u ∈R, see A nold [1] Page 72.
Since he abo e p obabili y space is canonical we ha e o a Wiene p ocess and i s
shi ope a o
W( , ω) = ω( ), W ( , θsω) = ω( +s)−ω(s) = W( +s, ω)−W(s, ω).
I is impo an o no e ha he measu abili y in (1) is no ue i we eplace F
by i s comple ion, see A nold [1] Appendix A3. Howe e , o ixed we ha e he
measu abili y o
θ : (Ω,¯
F)→(Ω,¯
F).
Mo eo e , he mapping R3 →θ ω∈C(R, U) is con inuous o a ixed ω∈Ω.
We will s udy he quali a i e beha iou o s ochas ic e olu ion equa ions on some
sepa able Hilbe space Hwhich ha e he o m
(3) dX =AXd + (X)d +B(X)dW, X(0) = u0,
whe e Wis he Wiene p ocess on he p obabili y space (Ω,¯
F,{¯
F } ∈R,P) in o-
duced abo e. Assume ha he e exis s a Gel and iple V⊂H⊂V0o sepa able
Hilbe spaces, whe e V0deno es he dual o V(see Temam [25] Page 55 o mo e
de ails). We deno e by k · k,k · kV he no ms in Hand V espec i ely. The inne
p oduc in Hwill be deno ed by (·,·), and he duali y mapping be ween V0and V
by h·,·i. The andom a iable u0is supposed o be ( ¯
F0,B(H)) measu able. Le us
deno e by a1>0 he cons an o he injec ion V⊂H, i.e.
a1kuk2≤ kuk2
V, o ∈V,
4 TOM´
AS CARABALLO, PETER E. KLOEDEN, AND BJ ¨
ORN SCHMALFUSS
and le −A:V→V0be a posi i e, linea and con inuous ope a o o which he e
exis s a a2<0 such ha
h−Au, ui ≥ −a2kuk2
V, o all u∈V.
Then, i is well known (see, o ins ance, Dau ay and Lions [9]) ha Ais he
gene a o o a s ongly con inuous semig oup {S( )} ≥0in Hsa is ying ha
(4) kS( )kL(H)≤ea ,
whe e a=a1a2<0.
The ope a o is supposed o be Lipschi z con inuous om H o H:
k (u1)− (u2)k ≤ L ku1−u2k, u1, u2∈H,
and Bis supposed o be Lipschi z con inuous wi h espec o he Hilbe -Schmid
no m LQ
2(U, H) o linea ope a o s om U o H:
H((B(u1)−B(u2))Q(B(u1)−B(u2))∗=: kB(u1)−B(u2)k2
LQ
2
≤LBku1−u2k2
o u1, u2∈H.
We now need he spaces L2,s := L2(Ω,¯
Fs,P;H), s ∈R. We ha e he ollowing
heo em abou he exis ence, uniqueness and egula i y o (3).
Theo em 2.1. Suppose ha u0∈L2,0. Then (3) has a unique (up o equi alence)
mild solu ion X(·)on [0,∞)which has a con inuous e sion. In addi ion,
EZT
0
kX( )k2
Vd < ∞,
and
Esup
∈[0,T ]
kX( )k2<∞,
o any T≥0.
Fo he exis ence o a mild solu ion see Da P a o and Zabczyk [10] Theo em 7.4.
The egula i y asse ion can be ound in K ylo and Rozo skii [17] Chap e 2.
We a e in e es ed in s a iona y solu ions ha a e exponen ially a ac ing in he
L2sense o almos su ely. S a iona y solu ion means ha he ini e dimensional
dis ibu ions o he solu ion Xa e independen o shi s wi h espec o . These
exponen ially s able s a iona y solu ions will be gene a ed by exponen ially a -
ac ing ixed poin s gi en by an ( ¯
F0,B(H))-measu able andom a iable X∗wi h
alues in Hsuch ha i we choose he ini ial condi ion u0(ω) = X∗(ω) we ha e
X( , ω) = X∗(θ ω) almos su ely o all ≥0,whe e he excep ional se may de-
pend on . This ixed poin is said o be exponen ially a ac ing i he p ocess
( , ω)→X∗(θ ω) (o a e sion o his p ocess) a ac s he solu ion o (3) o any
(app op ia e) ini ial condi ion exponen ially as in he L2-sense o almos su ely.
By he θ in a iance o Pwe ha e ha
P(X∗(θ 1ω)∈B1,· · · , X∗(θ nω)∈Bn)
=P(X∗(θ 1+ ω)∈B1,· · · , X∗(θ n+ ω)∈Bn)
o ≥0,0≤ 1< 2<· · · , nand B1,· · · , Bn∈ B(H).
5
Ano he ool ha can be used o desc ibe he s abili y beha iou o a s ochas ic
e olu ion equa ion a e andom dynamical sys ems. A comp ehensi e p esen a ion
can be ound in A nold [1]. A andom dynamical sys em is gi en by a measu able
mapping
φ: (R+×Ω×H, B(R+)⊗ F ⊗ B(H)) →(H, B(H)),
sa is ying he cocycle p ope y:
φ( +τ, ω, x) = φ( , θτω, φ(τ, ω, x)), , τ ∈R+, ω ∈Ω, x ∈H,
φ(0, ω, x) = x,
(5)
whe e θis he low o shi ope a o s (Wiene shi ) in oduced abo e. We emphasize
ha (5) has o be sa is ied o any ω∈Ω. Howe e i is su icien o eplace Ω
by a {θ } ∈R-in a ian se o ull measu e. Ou side o his in a ian se we can
ede ine φby he iden i y mapping on H. La e on we will eplace Ω by a smalle
{θ } ∈R-in a ian se Ω0∈ F. The measu abili y o (1) emains ue i we eplace
Fby i s ace σ-algeb a.
The mapping φis ela ed o he solu ion o a s ochas ic o andom di e en ial
equa ion. Fo he ollowing we will always suppose ha he mapping
H3x7→ φ( , ω, x)∈H
is con inuous o any , ω.
Al hough i is known ha ini e dimensional s ochas ic di e en ial equa ions gen-
e a e andom dynamical sys ems (see A nold [1] Chap e 1), his is no ue in
gene al o in ini e dimensional equa ions. Howe e , o pa icula kinds o noise
we can apply he ollowing simple lemma o ob ain a andom dynamical sys em.
Lemma 2.2. Le φbe a andom dynamical sys em. Suppose ha he mapping
T: Ω ×H→Hhas he ollowing p ope ies: Fo ixed ω∈Ω he mapping T(ω, ·)
is a homeomo phism on H. Fo ixed x∈H he mappings T(·, x), T−1(·, x)a e
measu able. Then he mapping
(6) ( , ω, x)→T−1(θ ω, φ( , ω, T(ω, x))) =: ψ( , ω, x)
sa is ies (5). Hence ψis a andom dynamical sys em.
The measu abili y o ψ ollows because o he p ope ies o T. La e on we will
ans o m a s ochas ic e olu ion equa ion con aining a noise e m in o an e olu ion
equa ion wi hou noise bu wi h andom coe icien s.
A andom a iable Yon (Ω,F,P) wi h alues in His called empe ed i
(7) lim
→±∞
log+kY(θ ω)k
| |= 0,
o equi alen ly i → kY(θ ω)khas a sub-exponen ial g ow h o → ±∞, in o he
wo ds, o ε > 0 and ω∈Ω he e exis s a 0(ε, ω)≥0 such ha o | | ≥ 0(ε, ω) i
holds
(8) kY(θ ω)k ≤ eε| |,
which means ha he exponen ial g ow h a e o → kY(θ ω)kis ze o.

6 TOM´
AS CARABALLO, PETER E. KLOEDEN, AND BJ ¨
ORN SCHMALFUSS
Le ω→G(ω) be a se alued mapping om Ω in o he space o non-emp y closed
subse s om H. Such a mapping is called a andom se i o any y∈H he
mapping
ω→in
x∈G(ω)kx−yk
is a andom a iable. I Gis a andom se hen he e exis s a andom a iable g
wi h g(ω)∈G(ω) (see Cas aing and Valadie [6] Chap e 3). A andom se Gis
called empe ed i he andom a iable
ω→sup
x∈G(ω)
kxk
is empe ed. I is easily seen ha he se o ω o which (7), (8) a e sa is ied is
{θ } ∈T-in a ian .
An (F,B(H))-measu able andom a iable X∗is called a andom ixed poin in he
sense o andom dynamical sys ems i
φ( , ω, X∗(ω)) = X∗(θ ω)
o > 0, whe e ωis con ained in a {θ } ∈T-in a ian se o ull measu e. X∗is
an exponen ially s able andom ixed poin wi h espec o a closed andom se G
con aining X∗i o any andom a iable g∈Gwe ha e
lim
→∞ kφ( , ω, g(ω)) −X∗(θ ω)k= 0
o all ωin he abo e {θ } ∈T-in a ian se o ull measu e wi h exponen ial speed,
such ha he excep ional se is independen o . I φis de ined by he solu ion
mapping o a s ochas ic/ andom di e en ial equa ion hen ( , ω)7→ X∗(θ ω) is a
s a iona y solu ion o a s ochas ic/ andom di e en ial equa ion.
Ou s a egy will be o p o e ha , unde ce ain assump ions, a andom dynamical
sys em possesses a andom a ac o which is a single ( andom) poin . The ollowing
de ini ion can be ound in Flandoli and Schmal uß [12].
De ini ion 2.3. Le Dbe he se o all closed empe ed andom se s in H. A
compac andom se A ∈ D is called a andom a ac o i he in a iance p ope y
(9) φ( , ω, A(ω)) = A(θ ω)
is sa is ied o ω∈Ω, ≥0and i , in addi ion, he pullback con e gence
(10) lim
→∞ dis H(φ( , θ− ω, D(θ− ω),A(ω))) = 0
holds o D∈ D and ω∈Ω.
We no e ha om his con e gence i ollows
(l.i.p.) lim
→∞ dis H(φ( , ω, D(ω),A(θ ω)) = 0,
whe e by (l.i.p.) we deno e limi in p obabili y. Howe e , in gene al i does no
imply ωwise almos su e con e gence. Su icien condi ions o he exis ence o a
andom a ac o can be ound in [12].
Theo em 2.4. Suppose ha he mapping x→φ( , ω, x)is con inuous o ≥0,
and comple ely con inuous o > 0(which means ha he image o e e y bounded
7
se by he mapping x→φ( , ω, x)is ela i ely compac ). In addi ion, suppose he e
is a G∈ D such ha o any D∈ D and ω∈Ω he e exis s T(D, ω)>0such ha
(11) φ( , θ− ω, D(θ− ω)) ⊂G(ω), o all ≥T(D, ω).
Then he e exis s a unique andom a ac o A(in D).
I he andom a ac o A(ω), ω ∈Ω,consis s o a single poin hen Ade ines a
andom ixed poin which a ac s empe ed andom se s.
3. Exponen ial s abili y s ochas ic e olu ion equa ions
In his sec ion we will p o e he exis ence o exponen ially s able (bo h in he mean
squa e sense and almos su ely) non- i ial s a iona y solu ions o ou s ochas ic
semi-linea pa ial di e en ial equa ion (3). The exponen ial s abili y o i ial
s a iona y solu ions (in pa icula , he null solu ion) o s ochas ic PDEs has been
ex ensi ely analyzed (see, o ins ance, [4], [19], [16],... and he li e a u e ci ed
he ein). Howe e , when ze o is no a solu ion o he equa ion, i is in e es ing o
ind ou whe he o no he e exis o he s a iona y solu ions, which a e gene a ed
by andom a iables chosen as ini ial alues in ou p oblem, and o analyze hei
s abili y p ope ies. This ac , can also be conside ed as a connec ion be ween
he classical me hod o he local analysis o he long- ime beha iou o s ochas ic
pa ial di e en ial equa ions and he global one p o ided by he heo y o andom
dynamical sys ems.
In he sequel we conside he p ocess θsW(·, ω) = W(·, θsω) = W(·+s, ω)−W(s, ω),
o s∈R, which is also a Wiene p ocess wi h co a iance Q. Fo ≥0 his p ocess
is adap ed o he il a ion {¯
Fs+ } ≥0which ollows om (2). We will deno e by
Φ(·, s, u0) he solu ion o (3), co esponding o he ini ial alue u0∈L2,s, which is
d i en by θsW, and sa is ying he asse ions o Theo em 2.1.
The ollowing equali y holds o u0∈L2,0:
(12) Φ(·,0, u0)(θs·) = Φ(·, s, us)(·),almos su ely,
whe e us(·) := u0(θs·). Bo h sides o (12) a e d i en by he same Wiene p ocess
and he same ini ial condi ion and he ac ha solu ions o (3) a e unique. Indeed,
by (2) u0(θs·) is ¯
Fs-measu able.
Lemma 3.1. i) Fo s∈R, τ ≥0, u0∈L2,s
Φ(·, s +τ, Φ(τ, s, u0)) = Φ(·+τ, s, u0),almos su ely.
ii) Se
(13) µ:= 2a+ 2L +LB.
Then
EkΦ( , s, u1
0)−Φ( , s, u2
0)k2≤eµ Eku1
0−u2
0k2
o ≥0, s ∈R, u1
0, u2
0∈L2,s.
P oo . i) Conside he p ocess
Y( , ω) = ½Φ( , s, u0)(ω) : 0 ≤ ≤τ
Φ( −τ, s +τ, Φ(τ, s, u0))(ω) : > τ ,
8 TOM´
AS CARABALLO, PETER E. KLOEDEN, AND BJ ¨
ORN SCHMALFUSS
which is con inuous (almos su ely) by he con inui y o Φ. Fo > τ we ha e
Y( ) =S( −τ)Φ(τ, s, u0)
+Z −τ
0
S( −τ−q) (Φ(q, s +τ, Φ(τ, s, u0)))dq
+Z −τ
0
S( −τ−q)B(Φ(q, s +τ, Φ(τ, s, u0))d(W(q+s+τ)−W(s+τ))
=S( )u0
+Z
τ
S( −q) (Y(q))dq +S( −τ)Zτ
0
S(τ−q) (Y(q))dq
+Z
τ
S( −q)B(Y(q))d(W(q+s)−W(s))
+S( −τ)Zτ
0
S(τ−q)B(Y(q))d(W(q+s)−W(s)).
The i s conclusion ollows i we conca ena e he in eg als and use he ac ha o
he inc emen s o he Wiene p ocess we ha e
W(q1+s+τ)−W(s+τ)−(W(q2+s+τ)−W(s+τ))
=W(q1+s+τ)−W(s)−(W(q2+s+τ)−W(s)).
ii) I is no ha d o see by he p ope ies o he coe icien s ha (3) has he ajec-
o ies in L2(0, T;V) (see K ylo and Rozo skii [17], o G ecksch and Tudo [13])
which allows us o apply he I o o mula o he p ocess e−µ kX1( )−X2( )k2whe e
we ha e deno ed Xi( ) := Φ( , s, ui
0), i = 1,2.Thus
e−µ kX1( )−X2( )k2
=ku1
0−u2
0k2−µZ
0
e−µqkX1(q)−X2(q)k2dq
+ 2 Z
0
e−µqhA(X1(q)−X2(q), X1(q)−X2(q)idq
+ 2 Z
0
e−µq( (X1(q)) − (X2(q)), X1(q)−X2(q))dq
+Z
0
e−µqkB(X1(q)) −B(X2(q))k2
LQ
2
dq
+ 2 Z
0
e−µq(X1(q)−X2(q),(B(X1(q)) −B(X2(q)))dW (q, θsω))
≤ku1
0−u2
0k2+Z
0
e−µq(−µ+ 2a+ 2L +LB)kX1(q)−X2(q)k2dq
+ 2 Z
0
e−µq(X1(q)−X2(q),(B(X1(q)) −B(X2(q)))dW (q, θsω)).
≤ku1
0−u2
0k2+ 2 Z
0
e−µq(X1(q)−X2(q),(B(X1(q)) −B(X2(q)))dW (q, θsω)).
(14)
9
The esul ollows easily by calcula ing he expec a ion i we eplace by ∧TN,
whe e TNis a amily o s opping imes
TN(ω) = in { ≥0 : kX1( , ω)k2+kX2( , ω)k2≥N}
such ha
lim
N→∞(TN∧ ) = , almos su ely,
since X1, X2ha e con inuous pa hs. ¤
Co olla y 3.2. Φ( , s, ·)maps L2,s in o L2,s+ con inuously.
We shall now show he exis ence o an exponen ially s able solu ion o (3).
Theo em 3.3. Suppose ha he cons an µappea ing in (13) is nega i e. Then
he e exis s an exponen ially a ac ing ixed poin X∗∈L2,0gene a ing an exponen-
ially s able s a iona y solu ion o (3). In pa icula , he p ocess ( , ω)→X∗(θ ω)
has a con inuous e sion gi en by Φ(·,0, X∗).
P oo . We show ha (Φ(k, −k, u0(θ−k·)))k∈Nis a Cauchy sequence in L2,0 o u0∈
L2,0. No ice ha o his u0we ha e ha u0(θ−k·)∈L2,−k. Indeed,
EkΦ(k, −k, u0(θ−k·)) −Φ(k−1,1−k, u0(θ1−k·))k2
=EkΦ(k−1,1−k, Φ(1,−k, u0(θ−k·))) −Φ(k−1,1−k, u0(θ1−k·))k2
≤eµ(k−1)EkΦ(1,−k, u0(θ−k·)) −u0(θ1−k·)k2
=eµ(k−1)EkΦ(1,0, u0(·)) −u0(θ1·)k2
He e we ha e applied Lemma 3.1 i), he {θ } ∈R-in a iance o Pand (12). The
Cauchy sequence p ope y ollows since µ < 0. Le he limi o his sequence be
deno ed by X∗∈L2,0.
X∗as an elemen in L2,0is independen o he choice o u0∈L2,0. Indeed, o
0∈L2,0we ha e ha
EkΦ(k, −k, u0(θ−k·)) −Φ(k, −k, 0(θ−k·))k2
=EkΦ(k, 0, u0(·)) −Φ(k, 0, 0(·))k2≤eµkEku0− 0k2,
which goes o ze o o k→ ∞ such ha he limi o he abo e Cauchy sequence
in L2,0is independen o u0. Since X∗∈L2,0 he p ocess Φ(·,0, X∗) sa is ies all o
he conclusions o Theo em 2.1. We now show ha
Φ( , 0, X∗)(·) = X∗(θ ·) almos su ely o any ∈R+.
By he de ini ion o X∗(ω) he andom a iable X∗(θ ·) is gi en by
(L2) lim
k→∞ Φ(k, −k, u0(θ−k·))(θ ω)
which is equal o
(L2) lim
k→∞ Φ(k, −k, u0(θ −k·))(ω)
16 TOM´
AS CARABALLO, PETER E. KLOEDEN, AND BJ ¨
ORN SCHMALFUSS
o a su icien ly small ε > 0. Compa ing he coe icien s o (29) we ha e kψ( )k2≤
R2( ) i kψ(0)k2≤R2(0) whe e R2( ) is he solu ion o
dR2
d =2(a+
N
X
j=1
λjbj|z∗
j(θ ω)|+kT(θ ω)kkT−1(θ ω)kL +ε
2)R2
+1
εkT−1(θ ω)k2k (0)k2.
(30)
Unde he assump ions o he heo em, his equa ion has he unique exponen ially
s able s a iona y solu ion →R2(θ ω) de ined by he andom a iable
R2(ω) : = Z0
−∞
1
εkT−1(θ ω)k2k (0)k2×
×exp 
Z0
2(a+
N
X
j=1
λjbj|z∗
j(θτω)|+kT(θτω)kkT−1(θτω)kL +ε
2)dτ
d
(31)
o ω∈Ω, which in u n ollows easily by he a ia ion o cons an s o mula. This
andom a iable is empe ed (and ini e), see Lemma 4.6 below. In pa icula he
ball in Hgi en by G(ω) := B(0,2R(ω)) is mapped in o i sel :
ψ( , ω, G(ω)) ⊂G(θ ω) o ω∈Ω.
Indeed his ball is empe ed. In addi ion, his ball has he p ope y (11). To see
his we conside he di e en ial equa ion (30) wi h some andom empe ed ini ial
condi ion 2(ω). I we eplace ωby θ− ω o he solu ion o (30) a ime we ha e
by he a ia ion o cons an s o mula
2(θ− ω) exp 
Z0
2(a+
N
X
j=1
λjbj|z∗
j(θτω)|+kT(θτω)kkT−1(θτω)kL +ε
2)dτ

+Z0
1
εkT−1(θsω)k2k (0)k2×
×exp 
Z0
s
2(a+
N
X
j=1
λjbj|z∗
j(θτω)|+kT(θτω)kkT−1(θτω)kL +ε
2)dτ
ds.
This e m ends o R2(ω) as → −∞ which ollows by Bi kho ’s e godic heo em,
(23), (28), (25). In pa icula he i s e m ends o ze o. F om Theo em 2.4 we
jus ob ain he exis ence o a andom a ac o A={A(ω)}ω∈Ω⊂G.
Se
∆ψ( , ω, x1, x2) = ψ( , ω, x1)−ψ( , ω, x2).
Then we ha e
dk∆ψ( )k2
d ≤2(a+
N
X
j=1
λjbj|z∗
j(θ ω)|+kT(θ ω)kkT−1(θ ω)kL )k∆ψ( )k2.

17
We can conclude by he in a iance p ope y (9)
sup
y1,y2∈A(ω)
ky1−y2k2≤sup
x1,x2∈A(θ− ω)
kx1−x2k2×
×exp 
Z0
2(a+
N
X
j=1
λjbj|z∗
j(θτω)|+kT(θτω)kkT−1(θτω)kL )dτ

(32)
Since A ⊂ G, i is empe ed, and we ob ain om he p ope ies o z∗
jand T ha
he igh hand side ends o ze o o ω∈Ω, hence A(ω) is a andom ixed poin
deno ed by Y∗. To see ha Y∗is exponen ially a ac ing we no e ha
sup
x∈D(ω)
kψ( , ω, x)−Y∗(θ ω)k2= sup
x∈D(ω)
kψ( , ω, x)−ψ( , ω, Y ∗(ω))k2
≤sup
x∈D(ω)
kx−Y∗(ω)k2×
×exp 
Z
0
2(a+
N
X
j=1
λjbj|z∗
j(θτω)|+kT(θτω)kkT−1(θτω)kL )dτ
,
and he igh hand side ends o ze o exponen ially as o ω∈Ω. ¤
Since we now know ha A(ω) is a single poin we ha e simila ly o (32) and (10)
ha
Co olla y 4.5. The andom ixed poin Y∗(hence X∗) a ac s empe ed andom
se s in he pullback sense.
We now p o e he empe edness o R2.
Lemma 4.6. Fo ω∈Ω he mapping →R2(θ ω)has subexponen ial g ow h,
hence R2is empe ed.
P oo . We abb e ia e
α(ω) =a+
N
X
j=1
λjbj|z∗
j(ω)|+kT(ω)kkT−1(ω)kL +ε
2,Eα=: ¯α < 0,
β(ω) =1
εkT−1(ω)k2k (0)k2.
By he de ini ion o Ω we ha e
Z0
α(θτω)dτ ∼¯α| | o ω∈Ω, → −∞.
In addi ion →β(θ ω) has sub-exponen ial g ow h o → ±∞ such ha
R2(ω) = Z0
−∞
exp µZ0
α(θτω)¶β(θ ω)d < ∞.
18 TOM´
AS CARABALLO, PETER E. KLOEDEN, AND BJ ¨
ORN SCHMALFUSS
Fo an a bi a y c > 0, 0 < ε < min(−¯α, c
2)/4 and nega i e s < s0(ω, ε) we hen
ha e
ecs Z0
−∞
eR0
α(θτ+sω)dτ β(θs+ ω)d
≤ec
2sZ0
−∞
eR0
+s(α(θτω)−¯α)dτ−R0
s(α(θτω)−¯α)dτ−¯α +sc
2+log+β(θ +sω)d
≤ec
2sZ0
−∞
e−3ε( +s)−¯α +c
2sd ≤ec
2sZ0
−∞
eε d .
Bu he igh hand side ends o ze o o s→ −∞.
Simila ly, we ob ain he con e gence o →+∞, see also A nold [1] P oposi ion
4.1.3. ¤
Co olla y 4.7. Suppose ha and Bjcommu e, i.e. i holds
T(ω)−1 (T(ω)x) = (x), o ω∈Ω, x ∈H.
Then he conclusion o Theo em 4.4 also holds i ins ead o (28) we assume
a+L <0.
P oo . Since he expec a ion o z∗
jdepends on λjsuch ha
E|z∗
j| ≤ 1
pλj
,
we can choose λjsu icien ly small so ha PN
j=1 bjλjE|z∗
j|is also a bi a ily small.
¤
Rema k 4.8. Usually Bjand commu e i hey a e diagonal in some o hogonal
basis o H(see Kwiecinska [18] and ou example in Sec ion 6) .
The commu a i i y assump ions o Biensu e ha (20) can be ans o med in o a
andom di e en ial equa ion. Howe e , he e exis o he ans o ma ions o e y
special classes o s ochas ic e olu ion equa ions o be ans o med in o andom di -
e en ial equa ions, see e.g. Flandoli and Lisei [11] and Mohammed e al. [21]. Fo
he esul ing andom e olu ion equa ion ou heo y can be applied.
5. Non-linea pe u ba ions
We now wan o show ha i we change he nonlinea pa o (3) con inuously, hen
he ixed poin s also change con inuously. Fo his we s udy he amily o p oblems
indexed by n∈Z+gi en by
(33) dX =AXd + n(X)d +B(X)dW, X(0) = u0.
We suppose ha he cons an s µngi en by (13) co esponding o he unc ions
n, sa is y µn<0 uni o mly o n∈Z+. We will also deno e by Φn(·,0, u0), n =
0,1,2, ..., he solu ion o (33), and by X∗
n hei associa ed andom ixed poin s.
Then, ou objec i e is o p o e ha he X∗
na e close o X∗
0i he nis close o 0
in some sense.
19
Theo em 5.1. Conside he amily o s ochas ic e olu ion equa ions (33). Suppose
ha
µ:= sup
n∈Z+
(2a+ 2L n+LB)<0,
and, in addi ion, ha
lim
n→∞ n(u) = 0(u), o u∈H.
Then, o he co esponding ixed poin s we ha e
(L2) lim
n→∞ X∗
n=X∗
0.
P oo . One can ind he idea o he p oo in Zeidle [26] P oposi ion 1.2 . Deno ing
k · k2
L2=Ek·k2and aking in o accoun (12), we ha e
kX∗
n−X∗
0kL2=kΦn(1,−1, X∗
n(θ−1·)) −Φ0(1,−1, X∗
0(θ−1·))kL2
=kΦn(1,0, X∗
n)−Φ0(1,0, X∗
0)kL2
≤kΦn(1,0, X∗
n)−Φn(1,0, X∗
0)kL2+kΦn(1,0, X∗
0)−Φ0(1,0, X∗
0)kL2
≤eµkX∗
n−X∗
0kL2+kΦn(1,0, X∗
0)−Φ0(1,0, X∗
0)kL2,
so
(34) kX∗
n−X∗
0kL2≤1
1−eµkΦn(1,0, X∗
0)−Φ0(1,0, X∗
0)kL2.
Now i is no ha d o p o e ha he igh hand side ends o ze o. Indeed, we se
Xn( ) = Φn( , 0, X∗
0), n ∈Z+, co esponding o he solu ion o (3) wi h = n.
Then by he I o o mula we ob ain
d
d EkXn( )−X0( )k2≤µEkXn( )−X0( )k2+Ek n(X0( )− 0(X0( ))k2
EkXn(0) −X0(0)k2= 0.
The inequali y
k n(X0( ))k2≤2 sup
n∈Z+
k n(0)k2+ 2LkX0( )k2, L := sup
n∈Z+
L n<∞
allows us o ind an in eg able majo an o k n(X0( )) − 0(X0( ))k2such ha
poin wise con e gence o n(u) o 0(u) and he a ia ion o cons an s o mula
yield he con e gence o he igh hand side o (34). ¤
We now conside a amily o e olu ion equa ions (20) wi h = n. To ob ain a
amily o equa ions o he o m (26) we can apply he ans o ma ion Twhich is
independen o n.
Theo em 5.2. Suppose ha he Lipschi z cons an s o na e uni o mly bounded
by L, ha
a+
N
X
j=1
bjλjE|z∗
j|+LΠN
j=1E(kSBj(−z∗
j)kkSBj(z∗
j)k)<0,
and ha
lim
n→∞ n(x) = 0(x) o x∈H.
Le X∗
n, n ∈Z+,be he andom ixed poin s o (20) wi h nins ead o . Then
lim
n→∞ X∗
n(ω) = X∗
0(ω), o ω∈Ω.
20 TOM´
AS CARABALLO, PETER E. KLOEDEN, AND BJ ¨
ORN SCHMALFUSS
P oo . As in he p oo o Theo em 4.4 we in es iga e he andom dynamical sys ems
ψngene a ed by (26) wi h = nand ixed poin s Y∗
n(ω) a ac ing empe ed se s,
see Co olla y 4.5. We ha e supn∈Nk n(0)k<∞, so, as in he p oo o Theo em
4.4, he e exis s a empe ed se G(ω) con aining all ixed poin s Y∗
n(ω). This se is
gi en by a ball B(0,2R(ω)) whe e R2is a s a iona y solu ion o he one-dimensional
a ine di e en ial equa ion (30) whe e L has o be eplace by L= supn∈Z+L n
and k (0)kby supn∈Z+k n(0)k. By he assump ions R2(ω) exis s.
We ha e
kY∗
n(ω)−Y∗
0(ω)k=kψn( , θ− ω, Y ∗
n(θ− ω)) −ψ0( , θ− ω, Y ∗
0(θ− ω))k
≤ kψn( , θ− ω, Y ∗
n(θ− ω)) −ψn( , θ− ω, Y ∗
0(θ− ω))k
+kψn( , θ− ω, Y ∗
0(θ− ω)) −ψ0( , θ− ω, Y ∗
0(θ− ω))k.
Since he second ac o on he igh hand side o (32) is independen o ni we
eplace L by L, and Y∗
0, Y ∗
na e con ained in he empe ed ball G, we ha e o any
ε > 0, ω ∈Ω and ha he i s e m on he igh hand side is less han ε/2. In
addi ion, o his simila o he p oo o Theo em 5.1 he second e m on he igh
hand side is also less han ε/2 i nis la ge. We ob ain kY∗
n(ω)−Y∗
0(ω)k< ε o la ge
n. Since he ans o ma ion Tis a homeomo phism we ha e he conclusion. ¤
6. An example
Le us ake H=L2(O) and V=H1
0(O) whe e Ois a bounded domain in Rd
wi h a smoo h bounda y. Conside he Laplace ope a o A= ∆ wi h he Di ichle
bounda y condi ion. Theo em IX.31 in B ezis [2] ensu es he exis ence o a sequence
o eal numbe s {νn}n≥1such ha 0 < ν1< ν2<··· < νn<· · · , and νn→+∞
(namely, he eigen alues o he Laplacian), and a sequence {en}n≥1⊂V∩C∞(O) o
associa ed eigen ec o s (i.e. −∆en=νnenon O) which is a comple e o hono mal
basis in H.
On he o he hand, le us conside ope a o s Bi,i= 1,· · · , N such ha hey
a e diagonalizable in he same basis and a e bounded om abo e and below (see
Kwiecinska [18]), i.e., he e exis cons an s dk
i, k = 1, . . . , N such ha
(Bkei, ej) = dk
iδij, k = 1, . . . , N;i, j = 1,2, . . .
The assump ion ha he ope a o s Bka e bounded om abo e and below is equi -
alen o he condi ion ha he e exis posi i e cons an s mk, Mksuch ha
0< mk≤ |dk
i| ≤ Mk, k = 1, . . . , N;i= 1,2, . . .
Unde hese condi ions he ope a o s Bigene a e C0-g oups SBi( ) = eBi .
Finally, conside a Lipschi z con inuous unc ion om Hin o Hde ined as
(u)(x) = F(u(x)), x∈ O wi h F:R→RLipschi z con inuous wi h cons an
L . Recall ha hAu, ui≤−ν1kuk2 o all u∈V, and he e o e he semig oup
gene a ed by Asa is ies
kS( )k ≤ e−ν1 o all ≥0.
We now s udy he s ochas ic e olu ion equa ion
(35) dX = (AX + (X))d +
N
X
i=1
BiXdwi.
21
I ollows ha he cons an µin Theo em 3.3 sa is ies
µ=−2ν1+ 2L +LB≤ −2ν1+ 2L +
N
X
i=1
M2
i.
I µ < 0, Theo em 3.3 implies he exis ence o a unique s a iona y solu ion o ou
p oblem which is exponen ially s able in mean squa e. Also, hanks o Lemma 3.1
ii) and Theo em 3.5 he almos su e exponen ial s abili y o his s a iona y solu ion
holds.
Obse e ha µis nega i e only i he Lipschi z cons an s L and LBa e su icien ly
small. Howe e , by using he echnique o andom dynamical sys ems we can p o e
s abili y beha iou e en o la ge alues o he cons an LB.
Indeed, we will be able o apply Theo em 4.4. To his end, we conside ou equa ion
(35) in i s S a ono ich equi alen o m
dX ="(A−1
2
N
X
i=1
B2
i)X+ (X)#d +
N
X
i=1
BiX◦dwi.
Deno e C=A−1
2PN
i=1 B2
iwhich also gene a es a C0-semig oup SC( ) . I he
ope a o s A, B1, ..., BNcommu e mu ually, hen his semig oup SC( ) is gi en as
SC( ) = S( )e−
2PN
i=1 B2
i.
Now, by easy compu a ions (see [18]) we can deduce
e−
2PN
i=1 M2
i≤ ke−
2PN
i=1 B2
ik ≤ e−
2PN
i=1 m2
i
and assump ion (28) in Theo em 4.4 becomes
eµ=−ν1−1
2
N
X
i=1
m2
i+
N
X
i=1
MiλiE|z∗
i|+L
N
Y
i=1
E(kSBi(−z∗
i)kkSBi(z∗
i)k)<0.
Consequen ly, i he ope a o s Bia e such ha eµ < 0, we can apply Theo em 4.4
and ensu e ha he e exis s a unique exponen ially s able s a iona y solu ion gi en
by a andom ixed poin .
No ice ha his can p o ide be e s abili y esul s han he ones ob ained in Sec ion
3. Indeed, assume o ins ance ha Biu=miuwhe e mi∈R+ o i, ··· , N. This
means ha mi=Mi, and i is easy o check ha
eµ=−ν1−1
2
N
X
i=1
m2
i+
N
X
i=1
miλiE|z∗
i|+L ,
since now kSBi(−z∗
i)kkSBi(z∗
i)k= 1.Then, i
−ν1−1
2
N
X
i=1
m2
i+L <0
(wha happens i , o example, he noise in ensi ies mia e la ge enough), we can
choose s a iona y p ocess z∗
ico esponding o λisuch ha eµ < 0. Thus, some kind
o s abiliza ion has been ob ained o he non- i ial s a iona y solu ion.
Acknowledgemen s. This wo k was pa ially suppo ed by he DAAD (Ge -
many), and he Minis e io de Ciencia y Tecnolog´ıa (Spain) unde he p ojec s
HA2001-0075 and BFM2002-03068.

22 TOM´
AS CARABALLO, PETER E. KLOEDEN, AND BJ ¨
ORN SCHMALFUSS
T. Ca aballo would like o hank B. Schmal uss and his amily o he kind hospi-
ali y hey all o e ed him du ing his isi in No embe 2002.
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E-mail add ess, Tom´as Ca aballo: [email p o ec ed]
E-mail add ess, Pe e E. Kloeden: [email p o ec ed]
E-mail add ess, Bj¨o n Schmal uß: [email p o ec ed] me sebu g.de
(Tom´as Ca aballo) Dp o. 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)
(Pe e E. Kloeden) Fachbe eich Ma hema ik, Johann Wol gang Goe he-Uni e si ¨
a , D-
60054 F ank u am Main, Ge many
(Bj¨o n Schmal uß) Depa men o Applied Sciences, Uni e si y o Technology and Ap-
plied Sciences, Geusae S asse, D–06217 Me sebu g, Ge many,