ASYMPTOTIC EXPONENTIAL STABILITY FOR DIFFUSION PROCESSES
DRIVEN BY STOCHASTIC DIFFERENTIAL EQUATIONS
IN DUALS OF NUCLEAR SPACES
TOM´
AS CARABALLO
Dp o. Ecuaciones Di e enciales y An´alisis Num´e ico
Uni e sidad de Se illa. Apa ado de Co eos 1160
41080-Se illa, SPAIN
KAI LIU*
Depa men o P obabili y and S a is ics
The Uni e si y o She ield
The Hicks Building, Houns ield Road
She ield, S3 7RH, UK
ABSTRACT
The main objec i e o his pape is o in es iga e he asymp o ic s abili y
o di usion p ocesses d i en by a class o I ˆo s ochas ic di e en ial
equa ions in duals o nuclea spaces. A coe ci i y condi ion imposed on
his so o equa ion plays he ole o an exponen ial s abili y c i e ion.
An example is s udied o illus a e ou heo y.
AMS Classi ica ion: p ima y 93E03; seconda y 60H10.
Keywo ds: Almos su e exponen ial s abili y; L2-exponen ial s abili y; S ochas ic di u-
sion p ocesses in duals o nuclea spaces.
* Au ho o Co espondence.
1
1. In oduc ion
In he pape we shall s udy he exponen ial s abili y o s ochas ic di usion equa ions
in duals o nuclea spaces. These equa ions na u ally a ise in he esea ch o chemical
eac ion-di usion equa ions, neu ophysiology and u bulence, especially, in he ecen i e
pollu ion model esea ches (see [11], [15] and [16]). Roughly speaking, we shall conside
he ollowing s ochas ic di usion equa ion:
X =X0+Z
0
A(s, Xs)ds +Z
0
B(s, Xs)dWs(1.1)
whe e A:R+×Φ0→Φ0,B:R+×Φ0→ L(Φ0,Φ0) a e wo measu able mappings and W
is a Φ0- alued Wiene p ocess. He e Φ0is he dual space o a ce ain coun ably Hilbe ian
nuclea space and L(Φ0,Φ0) is he space o all bounded linea ope a o s om Φ0in o i sel .
Di usion equa ions o he ype (1.1) ha e been s udied by a numbe o au ho s, o
ins ance, G. Kallianpu and R.L. Wolpe [10], G. Kallianpu and J. Xiong [11], H. Tuckwell
[15] and J.B. Walsh [16] among o he s. The eade is e e ed o G. Kallianpu and J. Xiong
[11] o u he de ails conce ning ce ain p ope ies o he solu ions (1.1) and some ela ed
opics. In he pape , we a e pa icula ly in e es ed in he c i e ia o exponen ial s abili y
in he sense o mean squa e and pa hwise wi h p obabili y one o he s ong solu ions o
he equa ions (1.1).
I is a long his o y o he in es iga ion o he exponen ial s abili y o s ochas ic di -
e en ial equa ions in ini e dimensional spaces and, mo e ecen ly, o s ochas ic e olu ion
equa ions in Hilbe spaces. Fo in ini e dimensional case, we should men ion U.G. Hauss-
mann [6] (linea case) and A. Ichikawa [7] (semilinea case) o hei undamen al wo k
on his aspec . Ne e heless, o nuclea space- alued s ochas ic di e en ial equa ion si -
ua ions, o he bes o ou knowledge i seems ha nobody e e ca ied ou he s udy
o exponen ial s abili y ei he in he sense o mean squa e o pa hwise wi h p obabili y
one. This is he main ask in his pape o ill his gap. I is pa icula ly wo h poin ing
ou ha ou app oaches, which a e de o ed o he conside a ion o he s ochas ic di e en-
ial equa ions in duals o nuclea spaces (1.1), could e en be used o ex end he esul s o
[6][7] o co e gene al non-au onomous Hilbe space- alued s ochas ic di e en ial sys ems.
Fi s ly, we shall gi e su icien condi ions o he exponen ial s abili y in mean squa e o
he s ong solu ions o he equa ions (1.1). Nex , we ob ain exponen ial s abili y o pa hs
wi h p obabili y one. Ou a gumen is based on a coe ci i y condi ion which plays a key
ole o he exis ence and uniqueness o he equa ions (1.1). As a consequence, we will
obse e how a sui able coe ci i y condi ion may be ega ded as an exponen ial s abili y
c i e ion.
The exposi ion is as ollows. In Sec ion 2, we shall b ie ly collec some no ions and
no a ions which a e essen ial o ou s abili y analysis. Sec ion 3 is de o ed o he in es-
iga ion o exponen ially asymp o ic s abili y o s ong solu ions. Finally, in Sec ion 4 we
will illus a e he heo ems de i ed in he las sec ion by s udying an example.
2
2. P elimina ies
In his sec ion we a e going o s a e some basic no ions and no a ions in a sui able way.
In pa icula , he eade is s ongly e e ed o G. Kallianpu and J. Xiong [11] o a
sys ema ic and de ailed s a emen conce ning he ma e ial in his sec ion.
Le Φ be a sepa able F ´eche space which is a coun ably Hilbe ian space, ha is, i s
opology is gi en by an inc easing sepa able k·kn,n≥0, o compa ible Hilbe ian no ms.
In pa icula , h oughou his pape we suppose Φ is nuclea , p ecisely, o each n≥0 he e
exis s m > n such ha he canonical injec ion om Φmin o Φnis Hilbe -Schmid . He e
Φnis he comple ion o Φ wi h espec o k·kn. Le Φ0be he collec ion o all con inuous
linea maps om Φ o R, i.e., he dual space o Φ. We could show ha {Φn}n≥0is a
sequence o dec easing Hilbe ian spaces and Φ = ∩∞
n=0Φn. Iden i ying Φ0
0wi h Φ0by
Riesz’s ep esen a ion heo em, we deno e Φ0
nby Φ−nwi h no ms k·k−n,n≥0. Then
{Φ−n}n≥0is a sequence o inc easing Hilbe ian spaces, Φ0is sequen ially comple e and
Φ0=∪∞
n=0Φ−n.In he la e case, we shall deno e by {φp
j} ⊂ Φ a comple e o hono mal
sys em, o simply, CONS o Φpand {φ−p
j} he CONS o Φ−pconjuga e o {φp
j} o p≥0.
Le θpbe he isome y om Φ−p o Φpsuch ha θpφ−p
j=φp
j,∀j≥1.
A class o impo an examples o coun ably Hilbe ian spaces can be desc ibed app o-
p ia ely as ollows. Le Hbe a eal sepa able Hilbe space and A=−La closed densely
de ined sel -adjoin ope a o on Hsuch ha <−Lφ, φ >≤0 o φ∈Dom(L), he domain
o L. Le {T }be he semig oups on Hde e mined by A. Fu he assume ha some powe
o he esol en o Lis a Hilbe -Schmid ope a o , i.e.,
∃ 1such ha (λI +L)− 1is Hilbe -Schmid .(2.1)
This condi ion enables us o p o e ha he e exis 0 ≤λ1≤λ2≤ ··· and {φj} ⊂ H, a
CONS o H, such ha
Lφj=λjφj, o any j≥1.
De ine
Φ =nφ∈H:k(I+L) φk2
H<∞,∀ ∈Ro
=½φ∈H:
∞
X
j=1
(1 + λj)2 < φ, φj>2
H<∞,∀ ∈R¾,
and he inne p oduc <·,·> on Φ by
< φ, ψ > =
∞
X
j=1
(1 + λj)2 < φ, φj>H< ψ, φj>H
and
kφk2
=< φ, φ > .
Le Φ be he k·k -comple ion o Φ. We hen ha e
Φ =
Φ ,Φ0=[
Φ
3
and o ≤s,φ∈Φ, kφk ≤ kφksand u he mo e Φs⊂Φ wi h Φ0=H. Condi ion
(2.1) implies ha he injec ion om Φqin o Φpis Hilbe -Schmid o q≥p+ 1and
he e o e Φ is a coun ably Hilbe ian nuclea space, simply, CHNS. As usual, we also call
he compa ible amily (Φ, H, T ) o (Φ, H, L) a special compa ible amily.
We assume h oughou ha (Ω,F,{F } ≥0, P ) is a comple e p obabili y space wi h a
igh con inuous il a ion {F } ≥0. A map X: Ω →Φ0is a Φ0- alued andom a iable i
i is F/B(Φ0)-measu able, whe e B(Φ0) is he Bo el ield o he opological space Φ0(in he
sense o s ong opology). A amily {X ; ∈R+}o Φ0- alued andom a iables is called
a Φ0-p ocess.
In he es o his pape , we shall conce n wi h Φ0- alued ma ingales. In pa icula ,
we ha e he ollowing:
De ini ion 2.1. A Φ0- alued p ocess M={M } ≥0is a Φ0-ma ingale wi h espec
o {F } ≥0i o each φ∈Φ, M [φ] is a ma ingale wi h espec o {F }. I is called a
Φ0-squa e-in eg able-ma ingale i , in addi ion,
E³M [φ]2´<∞,∀φ∈Φ, ≥0.(2.2)
We le M(Φ0) ( esp. M2(Φ0)) deno e he collec ion o all Φ0-ma ingales ( esp. Φ0-squa e-
in eg able-ma ingales). We also le
M2,c(Φ0) = nM∈ M2(Φ0) : M [φ] has a con inuous e sion o each φ∈Φo.
De ini ion 2.2. A con inuous (in he sense o s ong opology) Φ0- alued s ochas ic
p ocess W= (W ) ≥0on (Ω,F, P) is called a cen e ed Φ0-Wiene p ocess wi h Q(·,·) i W
sa is ies he ollowing h ee condi ions:
a). W0= 0 a.s.;
b). Whas independen inc emen s, i.e., he andom a iables
W 1[φ1],(W 2−W 1)[φ2],···,(W n−W n−1)[φn] (2.3)
a e independen o any φ1,φ2,···,φn∈Φ, 0 ≤ 1≤ ··· ≤ n,n≥1;
c). Fo each ≥0 and φ∈Φ
E³eiW [φ]´=e− Q(φ,φ)/2(2.4)
whe e Qis a co a iance unc ional, i.e., a posi i e de ini e symme ic con inuous bilinea
o m on Φ ×Φ.
Clea ly, W∈ M2,c(Φ0), {W [φ] : φ∈Φ, ≥0}is a cen e ed Gaussian sys em and
E³W [ψ]Ws[φ]´= (s∧ )Q(ψ, φ), ψ, φ ∈Φ, s, ≥0.(2.5)
4
De ini ion 2.3. Le Hbe a sepa able Hilbe space wi h no m k·kH. A amily
{B (h) : ≥0, h ∈H}o eal- alued andom a iables is called a cyclind ical B ownian
mo ion (c.B.m) on Hwi h co a iance Σ i Σ is a con inuous sel -adjoin posi i e de ini e
ope a o on Hsuch ha he ollowing condi ions hold:
i). Fo each h∈Hsuch ha h6= 0, <Σh, h >−1/2
HB (h) is a one-dimensional
s anda d Wiene p ocess;
ii). Fo each ≥0, α1,α2∈Rand 1, 2∈H
B (α1 1+α2 2) = α1B ( 1) + α2B ( 2)a.s.;
iii). Fo each h∈H,{B (h)}is an FB
-ma ingale, whe e
FB
=σ{Bs(h) : s≤ , h ∈H}.
{B (h) : ≥0, h ∈H}is called a s anda d H-c.B.m o simply, H-c.B.m. i i is a H-c.B.m.
wi h co a iance Σ = I.
Fo each φ∈Φ, le ıφ := Q(φ, ·). Then ıis an injec i e linea ope a o om Φ
on o a linea subspace R(ı) o Φ0. In pa icula , o a bi a y 1, 2∈ R(ı), le HQ:=
Q(ı−1 1, ı−1 2). Then <·,·>HQis an inne p oduc on R(ı). Le k·kHQbe he no m on
R(ı) de e mined by he inne p oduc <·,·>HQand le HQbe he comple ion o R(ı)
wi h espec o k · kHQ. Then HQis a sepa able Hilbe space and HQ⊂Φ0. I could
also be shown ha he e exis s a one- o-one co espondence be ween a Φ0- alued Wiene
p ocess Wwi h co a iance Qand an HQ-c.B.m. B:
W =
∞
X
j=1
B ( j) j(2.6)
whe e { j}is a CONS o HQ;
B ( ) = lim
n→∞ W [ı−1 n],∀ ∈HQ(2.7)
whe e { n} ⊂ R(ı) con e ges o in HQ.
Conside he ollowing s ochas ic di usion equa ion (see [11] o u he de ails on
s ochas ic in eg al and ela ed p ope ies)
X =X0+Z
0
A(s, Xs)ds +Z
0
B(s, Xs)dWs(2.8)
whe e A:R+×Φ0→Φ0,B:R+×Φ0→ L(Φ0,Φ0) a e wo measu able mappings and
W is a Φ0- alued Wiene p ocess. He e L(Φ0,Φ0) deno es he collec ion o all con inuous
linea mappings om Φ0in o Φ0.
De ini ion 2.4. Le (Ω,F,{F } ≥0, P) be he s ochas ic basis and W a Φ0- alued
Wiene p ocess wi h co a iance unc ion Q. Suppose ha X0is a Φ−p- alued andom
5

a iable such ha EkX0k2
−p<∞. Then by a Φ−p- alued s ong solu ion on Ω o he SDE
(2.8) o ∈[0, T] we mean a p ocess X de ined on Ω such ha
(a). X is a Φ−p- alued F -measu able andom a iable;
(b). X ∈C([0, T],Φ−p), a.s.;
(c). The e exis s a sequence (σn) o bounded s opping imes on Ω inc easing o in ini y
such ha ∀n≥1
EZT∧σn
0kA(s, Xs)k−qds < ∞,(2.9)
and
EZT∧σn
0kB(s, Xs)k2
L(2)(HQ,Φ−p)ds < ∞.(2.10)
He e L(2)(HQ,Φ−p) deno es he class o all Hilbe -Schmid ope a o s om HQin o Φ−p
and qwill be in oduced in he ollowing assump ion (H1);
(d). The SDE (2.8) is sa is ied o all ∈[0, T] and almos all ω∈Ω.
I Tis eplaced by ∞, we call X a global s ong solu ion o (2.8).
As we a e mainly in e es ed in he s abili y analysis, one always assumes ha he
equa ion (2.8) has a unique global s ong solu ion. In pa icula , o his pu pose we shall
make he ollowing assump ion (H1) [11]:
The e exis s an index p0>0 such ha , ∀p≥p0,∃q≥pand a cons an K=K(p, q)>
0 such ha
(D1). (Con inui y) ∀ ∈R+, he maps ∈Φ−p→A( , )∈Φ−qand ∈Φ−p→
B( , )∈L(2)(HQ,Φ−p) a e con inuous;
(D2). (Coe ci i y) ∀ ∈R+and ∈Φ−p, we ha e
2A( , )[θp ] + kB( , )k2
L(2)(HQ,Φ−p)≤K(1 + k k2
−p); (2.11)
(D3). (G ow h) ∀ ∈R+and ∈Φ−p, we ha e
kA( , )k2
−q≤K(1 + k k2
−p); (2.12)
(D4). (Lipschi z) ∀ ∈R+, 1, 2∈Φ−p, we ha e
kA( , 1)−A( , 2)k−q≤Kk 1− 2k−p(2.13)
and
kB( , 1)−B( , 2)kL(2)(HQ,Φ−p)≤Kk 1− 2k−p.(2.14)
6
3. The Main Resul s
In his sec ion, we shall de o e ou sel es o he in es iga ion o exponen ial s abili y o
he equa ion (2.8). Fo simplici y, h oughou his sec ion we ake he special compa ible
amily (Φ, H, L) desc ibed in Sec ion 1 as ou basic CHNS. In pa icula , o ou end we
shall make he ollowing addi ional assump ion (H2):
∀ ∈R+, ∈Φ−p,p≥p0, he e exis posi i e cons an s ν > 0, µ > 0, p≤ ≤qand
posi i e unc ion γ( ), ∈R+, such ha
2A( , )[θq ] + kB( , )k2
L(2)(HQ,Φ− )≤ −νk k2
− +γ( )e−µ (3.1)
whe e p0,qa e in oduced as in he assump ion (H1) and γ( ) sa is ies ha o a bi a y
δ > 0, γ( ) = o(eδ ), as → ∞, i.e., lim →∞ γ( )/eδ = 0.
Be o e p oceeding o ou s abili y a gumen s, le us i s make he ollowing commen s
on he condi ion (H2):
Rema k 1. As is well known, he coe ci i y condi ion (2.11) plays an essen ial ole
in he es ablishmen o he exis ence and uniqueness o he equa ion (2.8). The u he
es ic i e coe ci i y condi ion (3.1) will play he ole o an exponen ial s abili y c i e ion
as desc ibed below.
Rema k 2. The exponen ial decay e m appea ing on he igh hand side o (3.1) is
o he essence o ou s abili y pu poses. In ac , o see his, le us simply conside he
ollowing one dimensional linea I ˆo equa ion:
Example 3.1. Assume X sa is ies he ollowing
dX =−pX d + (1 + )−qdW , ≥0
wi h ini ial da a X0= 0, whe e p,q > 0 a e wo posi i e cons an s and W is a one-
dimensional s anda d B ownian mo ion.
Clea ly, he le -hand side o he coe ci i y ype condi ion (3.1) now u ns ou o be
2<−p , > +h(1 + )−qi2=−2p 2+ (1 + )−2q.(3.2)
whe e <·,·>deno es he s anda d inne p oduc in R. Howe e , since he las e m
(1 + )−2qis no exponen ially dec easing, he solu ion is exponen ially uns able. Indeed,
i is easy o ob ain he explici solu ion
X =e−p Z
0
eps ·(1 + s)−qdWs=: e−p M , ≥0,
which immedia ely implies ha o a bi a ily gi en q > 0 Lyapuno exponen
lim
→∞
log E|X |2
= 0.
7
In he mean ime, no icing he law o he i e a ed loga i hm
lim sup
→∞
M
√2 log log = 1 a.s.
and
lim sup
→∞
log ³R
0e2ps(1 + s)−2qds´
= 2p,
we he e o e ge Lyapuno exponen
lim sup
→∞
1
log |X |= 0 a.s.
Tha is, in spi e o he ypical s abili y o an o dina y di e en ial equa ion
dX =−pX d ,
he polynomial ype decay o he noise e m is no su icien o ensu e he exponen ial
s abili y o i s s ochas ically pe u bed sys em.
Now we a e in a posi ion o ob ain ou main esul s in he pape .
Theo em 3.2. Suppose X is a solu ion o he equa ion (2.8) sa is ying (H1). Fu -
he mo e we assume he coe ci i y condi ion (3.1) holds. Then he e exis cons an s τ > 0,
C > 0such ha
EkX k2
− ≤C·e−τ ,∀ ≥0.(3.3)
Tha is, he s ong solu ion is exponen ially s able in mean squa e. In pa icula , cons an
τ > 0can be aken as ollows: τ < µ, i µ≤νand τ=ν, i µ > ν.
P oo . Fo a bi a y φ∈Φ, we ha e
X [φ] = Z
0
A(s, Xs)[φ]ds +X
jZ
0
< B(s, Xs)0φ, j>HQdWs[ı−1 j],(3.4)
whe e { j} ⊂ R(ı) is a CONS o HQand ıis de ined as in Sec ion 2. He e B(s, ·)0deno es
he dual ope a o o B(s, ·)∈ L(HQ,Φ− ), s≥0. I ollows om I ˆo’s o mula and
De ini ion 2.4 ha o a bi a y δ > 0 wi h µ−δ > 0, we ha e
e(µ−δ) ∧σnX ∧σn[φ]2−X0[φ]2
=(µ−δ)Z ∧σn
0
e(µ−δ)sXs[φ]2ds + 2 Z ∧σn
0
e(µ−δ)sXs[φ]A(s, Xs)[φ]ds
+ 2 X
jZ ∧σn
0
e(µ−δ)sXs[φ]HQdWs[ı−1 j]
+Z ∧σn
0
e(µ−δ)sQ(B(s, Xs)0φ, B(s, Xs)0φ)ds
8
whe e (σn) is he sequence o s opping imes de ined as in De ini ion 2.4. Now, since
R ∧σn
0e(µ−δ)sXs[φ]HQdW [ı−1 j], ∈R+, is a con inuous ma ingale, i ollows ha
E³Z ∧σn
0
e(µ−δ)sXs[φ]HQdWs[ı−1 j]´= 0, ∈R+.
The e o e, le ing φ=φ
k,n→ ∞,k∈Nand hen adding on index k∈N, we can deduce
by Fa ou’s lemma and he condi ion (3.1)
Ee(µ−δ) kX k2
−
≤EkX0k2
− + (µ−δ−ν)Z
0
e(µ−δ)sEkXsk2
− ds +Z
0
γ(s)e−δsds. (3.5)
I µ−ν≤0, we he e o e deduce
Ee(µ−δ) kX k2
− ≤EkX0k2
− +Z
0
γ(s)e−δsds,
ha is, le ing k(δ) = R∞
0γ(s)e−δsds, we ha e
EkX k2
− ≤³EkX0k2
− +k(δ)´e−(µ−δ) .
On he o he hand, i µ−ν > 0, i is always possible o choose a sui able δ > 0 such ha
µ−ν−δ > 0. Then, by i ue o G onwall’s lemma we easily de i e om (3.5) ha
Ee(µ−δ) kX k2
− ≤³EkX0k2
− +Z
0
γ(s)e−δsds´e(µ−δ−ν) .
Hence, le ing δ > 0 small enough immedia ely yields ha he e exis s a cons an k(δ)>0
such ha
EkX k2
− ≤³EkX0k2
− +k(δ)´e−ν .
Combining he a gumen s abo e, we hus ob ain ou conclusion.
Theo em 3.3. Assume he assump ions in Theo em 3.2 hold. Then he e exis
posi i e cons an s M,βand a subse Ω0⊂Ωwi h P(Ω0)=0such ha , o each ω6∈ Ω0,
he e exis s a posi i e andom numbe T(ω)such ha he ollowing holds:
kX k2
− ≤M·e−β ,∀ ≥T(ω).(3.6)
Tha is, he s ong solu ion is almos su ely s able.
P oo . Ou p oo s a e di ided in o he ollowing se e al s eps.
S ep 1. We i s ly claim ha he e exis s a cons an C > 0, independen o ∈R+, such
ha Z
s
EkB(u, Xu)k2
L(2)(HQ,Φ− )du ≤C < ∞,0≤s≤ . (3.7)
9