THE EXPONENTIAL BEHAVIOUR OF NONLINEAR
STOCHASTIC FUNCTIONAL EQUATIONS OF
SECOND ORDER IN TIME
Tom´
as Ca aballo, Ma ´
ıa J. Ga ido-A ienza and Jos´
e Real
Depa 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.
Abs ac
Su icien condi ions o exponen ial mean squa e s abili y o solu ions o delayed
s ochas ic pa ial di e en ial equa ions o second o de in ime a e es ablished. As a
consequence o hese esul s, some ones on he pa hwise exponen ial s abili y o he
sys em a e p o ed. The s abili y esul s de i ed a e applied also o pa ial di e en ial
equa ions wi hou he edi a y cha ac e is ics. The esul s a e illus a ed wi h se e al
examples.
1 In oduc ion
S ochas ic di e en ial delay equa ions and hei asymp o ic beha iou ha e been ecei ing
much a en ion in he las yea s (see [1], [2], [4], [8], [6], [7], [9], [11], [12], and he e e ences
he ein) since hese e a ded p oblems o en appea in Physics, Biology, Enginee ing, e c...
The delays can en e in he o mula ions in e y di e en ways, e.g., as a cons an o
a iable delay, as a dis ibu ed one, o e en some o hem can appea in he model a he
same ime. Also hese delay can be bounded ( ini e) o unbounded (in ini e). Howe e ,
he e is a possibili y o conside ing all o hem unde a uni ied o mula ion by using app o-
p ia e di e en ial unc ional equa ions. In his sense, we will ca y ou ou analysis in a
unc ional amewo k which will co e a wide a ie y o si ua ions con aining ini e delays
(see Sec ion 5).
1
On he o he hand, a e y in e es ing ques ion is o analyse he long- ime beha iou
o he solu ion o a s ochas ic unc ional equa ion. We ema k ha in some p oblems he
his o y o he phenomenon has a decisi e in luence on he u u e beha iou o he sys em
and, in some cases, no only a sho pe iod o he pas has o be aken in o accoun , bu a
la ge one. This ac mo i a es he p esen wo k.
The e exis s a wide li e a u e conce ning pa hwise exponen ial s abili y o pa abolic
s ochas ic e olu ion equa ions (wi h and wi hou delays). We men ion he e, amongs many
o he s, Ca aballo and Liu [3], Liu and Mao [10], Taniguchi [11], Taniguchi e al. [12] and
he e e ences he ein. Howe e , as a as we know, he e a e no pape s on he asymp o ic
s abili y o delay s ochas ic pa ial di e en ial equa ions o second o de in ime, which is
he main aim o his pape . In he case wi hou he edi a y cha ac e is ics his p oblem has
been conside ed by Cu ain [5], whe e one can ind su icien condi ions o he exponen ial
s abili y o he expec ed ene gy o he sys em, as well as o he exponen ial decay o he
sample pa hs, when he main ope a o gene a es a s ongly con inuous con ac ion semi-
g oup. In his pape we shall de elop he heo y in a a ia ional amewo k o non-linea
ope a o s in gene al and unde a unc ional o mula ion which co e s se e al kinds o delay
and, in pa icula , he non-delay case.
In o de o mo i a e ou heo y le us i s s udy he ollowing example.
The simples model o con inuum mechanics is gi en by he ib a ing s ing, subjec ed
o a cons an ension µ, which execu es small longi udinal ib a ions abou he posi ion o
s able equilib ium. Ou p oblem is o de e mine he la e al displacemen u(x, ) o a poin on
he s ing om i s equilib ium posi ion. Fo a cons an linea densi y %o he s ing, i we also
conside some ic ion p opo ional o eloci y, ϑ ( ), ha an ex e nal o ce ( , u( ), ( ))
ac s on he s ing, and ha u0and 0a e espec i ely he ini ial posi ion and eloci y, we
hen ob ain he p oblem
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∂2u
∂ 2−a2∂2u
∂x2+ϑ∂u
∂ + Φ µ , u( ),∂u
∂ ( )¶= 0,in [0,+∞)×[0,1],
u( , 0) = u( , 1) = 0, ∈[0,+∞),
u(0, x) = u0,∂u
∂ (0, x) = 0(x),in [0,1],
whe e Φ ¡ , u( ),∂u
∂ ( )¢co esponds o 1
% ¡ , u( ),∂u
∂ ( )¢,a=qµ
%,and ϑ > 0.
This model can be hough o be mo e ealis ic i we suppose ha Φ con ains some
andom ea u es, o example we can hink o Φ ¡ , u( ),∂u
∂ ( )¢=−σ∂u( )
∂x
dW ( )
d , ≥0,whe e
2
σ∈R,and W( ) is a one-dimensional Wiene p ocess. The e o e he equa ion becomes
∂2u
∂ 2−a2∂2u
∂x2+ϑ∂u
∂ =σ∂u
∂x( )dW( )
d , ≥0.
Cu ain p o es in [5] ha when σ2<4ϑπ2
4π2+ϑ(ϑ+√ϑ2+4π2)and a= 1, he null solu ion o his
p oblem is s able in mean squa e, i.e., i he andom e m is su icien ly small so ha his
ela ion is sa is ied. Bu , is i possible o deduce any exponen ial s abili y esul s o he
abo e sys em when σ2>4ϑπ2
4π2+ϑ(ϑ+√ϑ2+4π2)? As a consequence o he heo y we will de elop
in his pape , we will p o e ha i σ2<2ϑπ2
ϑ2+2π2 he sys em is exponen ially s able bo h in
mean squa e and pa hwise, so hese esul s imp o e he ones in [5].
On he o he hand, i we a e in e es ed in some p oblems conce ning he s abiliza ion
o con ollabili y o sys ems, i seems na u al o s udy equa ions in which some he edi a y
cha ac e is ics can appea . Fo example, in he abo e sys em we can conside he load Φ
does no depend jus on he p esen , bu on he his o y o he p ocess, i.e., i is modelled
o example by he exp ession Φ ¡ , u ,∂u
∂ ¢=−σ∂u( −τ( ))
∂x ˙
W( ), ≥0,whe e τ(·) is an
app op ia e delay unc ion (see Example 8, Sec ion 5).
The con en o he pape is as ollows. In Sec ion 2 we p esen he amewo k in which
ou analysis is ca ied ou , and in oduce some basic no a ions and assump ions. Sec ion 3
is de o ed o he main esul o his wo k, ha is, we es ablish a su icien condi ion ensu ing
mean squa e s abili y o delay s ochas ic pa ial di e en ial equa ions o second o de in
ime in a e y gene al si ua ion. We also indica e how his esul can be applied in some
pa icula cases. As a consequence o he mean squa e s abili y, in Sec ion 4 we ob ain
pa hwise exponen ial s abili y. Mo eo e , in his sec ion we will be conce ned wi h a mo e
gene al si ua ion in which ou heo y can be es ablished. In Sec ion 5 we include some
examples o illus a e hese esul s. Finally, some conclusions a e included in he las sec ion.
2 S a emen o he p oblem
Le Vand Hbe wo eal sepa able Hilbe spaces such ha V⊂H≡H∗⊂V∗whe e he
injec ions a e con inuous and dense.
We deno e by k·k,|·| and k·k∗ he no ms in V,Hand V∗ espec i ely; by (·,·) he inne
p oduc in H, and by h·,·i he duali y p oduc be ween V∗and V. Le us deno e by c > 0
a cons an such ha |x| ≤ ckxk,∀x∈V.
Assume {Ω,F, P}is a comple e p obabili y space wi h a no mal il a ion {F } ≥0, i.e.,
F0con ains he null se s in F, and F =∩s> Fs, o all ≥0.Deno e F =F0 o all ≤0.
3
Le us conside a eal alued {F }−Wiene p ocess {W( )} ≥0.
Gi en eal numbe s a < b and a sepa able Hilbe space H, we will deno e by I2(a, b;H)
he closed subspace o L2(Ω×(a, b),F⊗B([a, b]) ,dP⊗d ;H) o all s ochas ic p ocesses which
a e F -adap ed o almos e e y in (a, b) (in wha ollows, a.e. ),whe e B([a, b]) deno es
he Bo el σ-algeb a o subse s in [a, b]. I ϕ∈I2(a, b;H) we will w i e |ϕ|I2
H o deno e he
no m |ϕ|I2(a,b;H).
We deno e by L2(Ω; C(a, b;H)) he space o p ocesses X∈L2(Ω,F, dP;C(a, b;H)) such
ha X( ) is F -measu able o each in [a, b], whe e C(a, b;H) deno es he space o all
con inuous unc ions om [a, b] in o Hequipped wi h sup emum no m.
Le us ix h > 0 and conside T > 0.I we ha e a unc ion x∈C(−h, T;H), o
each ∈[0, T ] we deno e by x ∈C(−h, 0; H) he unc ion de ined by x (s) = x( +s),
−h≤s≤0.Mo eo e , i y∈L2(−h, T ;H) we also deno e by y ∈L2(−h, 0; H),a.e.
∈(0, T), he unc ion de ined by y (s) = y( +s),a.e. s∈(−h, 0).
Le A( ) : V→V∗, ≥0,be a amily o ope a o s sa is ying:
(A.1) A( ) is sel -adjoin o each ≥0.
(A.2) A( )∈ L(V, V ∗)∀ ≥0,and he e exis s cA>0 such ha kA( )uk∗≤cAkuk,
∀ ≥0,∀u∈V.
(A.3) ∃α > 0 such ha hA( )u, ui ≥ αkuk2,∀ ≥0,∀u∈V.
(A.4) hA(·)u, eui ∈ C1(0,+∞), ∀u, eu∈V, and hA0( )u, ui ≤ 0,∀ ≥0,∀u∈V, whe e
hA0( )u, euideno es d
d hA( )u, eui.
(A.5) he e exis s a Banach space Xsuch ha X⊂ {u∈V;A( )u∈H, ∀ ≥0}, he
injec ion o Xin Vis con inuous, and Xis dense in H.
Le B( , ·) : H→Hbe a amily o nonlinea ope a o s de ined a.e. ≥0 and sa is ying:
(B.1) ∀ ∈H, he map ∈(0,+∞)→B( , )∈His Lebesgue measu able.
(B.2) he map θ∈R→(B( , +θw), z)∈Ris con inuous ∀ , w, z ∈H, a.e. ≥0.
(B.3) he e exis s cB>0 such ha |B( , )| ≤ cB| |,∀ ∈H, a.e. ≥0.
(B.4) he e exis s β > 0 such ha (B( , )−B( , e ), −e )≥β| −e |2,∀ , e ∈H, a.e.
≥0.
4
Le F: [0,+∞)×C(−h, 0; V)×C(−h, 0; H)→Hand G: [0,+∞)×C(−h, 0; V)×
C(−h, 0; H)→Hbe wo amilies o nonlinea ope a o s de ined a.e. ≥0 such ha :
(F.1) ∀(ξ, η)∈C(−h, 0; V)×C(−h, 0; H) he map ∈(0,+∞)→F( , ξ, η)∈His Lebesgue
measu able,a.e. ≥0.
(F.2) F( , 0,0) = 0,a.e. ≥0.
(F.3) he e exis CF,H , CF,V >0 such ha ∀ξ, e
ξ∈C(−h, 0; V),∀η, eη∈C(−h, 0; H) and a.e.
≥0,
|F( , ξ, η)−F( , e
ξ, eη)|2≤CF,V ||ξ−e
ξ||2
C(−h,0;V)+CF,H |η−eη|2
C(−h,0;H).
(F.4) he e exis m0>0 and cons an s KF,H =KF,H (m0, h), KF,V =KF,V (m0, h)≥0 such
ha o all m∈[0, m0],∀x, ex∈C(−h, T;V),∀y, ey∈C(−h, T;H),and ∀ ≥0
Z
0
ems |F(s, xs, ys)−F(s, exs,eys)|2ds
≤KF,V Z
−h
ems kx(s)−ex(s)k2ds+KF,H Z
−h
ems |y(s)−ey(s)|2ds.
(G.1) ∀(ξ, η)∈C(−h, 0; V)×C(−h, 0; H) he map ∈(0,+∞)→G( , ξ, η)∈His Lebesgue
measu able,a.e. ≥0.
(G.2) G( , 0,0) = 0,a.e. ≥0.
(G.3) he e exis CG,H, CG,V >0 such ha ∀ξ, e
ξ∈C(−h, 0; V),∀η, eη∈C(−h, 0; H) and a.e.
≥0,
|G( , ξ, η)−G( , e
ξ, eη)|2≤CG,V ||ξ−e
ξ||2
C(−h,0;V)+CG,H |η−eη|2
C(−h,0;H).
(G.4) he e exis m0>0 and cons an s KG,H =KG,H(m0, h), KG,V =KG,V (m0, h)≥0
such ha o all m∈[0, m0],∀x, ex∈C(−h, T;V),∀y, ey∈C(−h, T;H),and ∀ ≥0
Z
0
ems |G(s, xs, ys)−G(s, exs,eys)|2ds
≤KG,V Z
−h
ems kx(s)−ex(s)k2ds+KG,H Z
−h
ems |y(s)−ey(s)|2ds.
Rema k 1 Assump ions (F.2),(G.2) a e mo i a ed by ou in e es in analyzing he s abili y
o he ze o solu ion o ou p oblem, bu hey a e no necessa y o p o e exis ence o solu ion.
5

We conside he ollowing p oblem,
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u∈I2(−h, T;V)∩L2(Ω; C(0, T;V)), o all T > 0,
∈I2(−h, T;H)∩L2(Ω; C(0, T;H)), o all T > 0,
u0( ) = ( ), ∈[0, T],
( ) + R
0A(s)u(s)ds+R
0B(s, (s))ds= 0+R
0F(s, us, s)ds
+R
0G(s, us, s)dW(s), ≥0,
u(0) = u0,
u( ) = ϕ1( ), ( ) = ϕ2( ),a.e. ∈(−h, 0),
(P)
whe e ϕ1∈I2(−h, 0; V), ϕ2∈I2(−h, 0; H), u0∈L2(Ω,F0, P;V) and 0∈L2(Ω,F0, P;H)
a e gi en.
A simila analysis o ha in Rema k 1 in [7] shows ha , unde ou assump ions (F.1) −
(F.4) and (G.1) −(G.4), all he in eg als appea ing in p oblem (P) a e well de ined, and
he e o e, he abo e p oblem makes sense.
On he o he hand, se e al esul s on he exis ence and uniqueness o solu ions o delay
s ochas ic e olu ion equa ions o second o de in ime can be seen in Ga ido-A ienza and
Real [7]. In pa icula , he ollowing one is a consequence o Theo em 4 in [7]:
Theo em 2 Assume ha hypo heses (A.1) −(A.5),(B.1) −(B.4),(F.1) −(F.4),(G.1) −
(G.4) hold. Then, i ϕ1∈I2(−h, 0; V),ϕ2∈I2(−h, 0; H),u0∈L2(Ω,F0, P;V)and
0∈L2(Ω,F0, P;H), he e exis s a unique solu ion (u, ) o p oblem (P), o all T > 0.
3 Exponen ial s abili y in mean squa e
In his sec ion we will es ablish a esul abou he mean squa e s abili y o he solu ion o
p oblem (P).
Theo em 3 Suppose ha assump ions (A.1) −(A.5),(B.1) −(B.4),(F.1) −(F.4),(G.1) −
(G.4) hold. In addi ion, assume ha he e exis some cons an s ε > 0and δ > 0such ha
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2α3/2Cδ> cKε,
and
Cδ³2β−(2K1/2
F,H +ε+KG,H )´>(4α2δ)−1(4αδ +c2(K1/2
F,H +cB)2)Kε,
(1)
6
whe e Cδ= 1 −δ−cK1/2
F,V α−1and Kε=KF,V ε−1+KG,V . Then, he ze o solu ion o
p oblem (P) is exponen ially s able in mean squa e, i.e., he e exis m∈(0, m0]and K1=
K1(m0, h)>0such ha , o all ≥0,
E³| ( )|2+ku( )k2´≤K1³E| 0|2+Eku0k2+|ϕ2|2
I2
H+kϕ1k2
I2
V´e−m ,(2)
o any solu ion (u, )o (P).
P oo . Le m∈(0, m0]. Applying I ˆo’s o mula o he p ocess
em | ( )|2+ em hA( )u( ), u( )i,
we ob ain o each ≥0 and P-a.s.
em | ( )|2+ em hA( )u( ), u( )i
=| 0|2+hA(0)u0, u0i+mZ
0
ems(| (s)|2+hA(s)u(s), u(s)i)ds
+Z
0
ems hA0(s)u(s), u(s)ids−2Z
0
ems(B(s, (s)), (s))ds
+ 2 Z
0
ems(F(s, us, s), (s))ds+Z
0
ems |G(s, us, s)|2ds
+ 2 Z
0
ems(G(s, us, s), (s))dW(s),
and hanks o (A.4) and (B.4),
em E| ( )|2+ em EhA( )u( ), u( )i(3)
≤E| 0|2+EhA(0)u0, u0i+ (m−2β)Z
0
emsE| (s)|2ds
+mZ
0
emsEhA(s)u(s), u(s)ids
+ 2 Z
0
emsE(F(s, us, s), (s))ds+Z
0
emsE|G(s, us, s)|2ds.
7
As em ≤1,∀ ∈[−h, 0], om (F.4) and (A.3) we ob ain
2Z
0
emsE(F(s, us, s), (s))ds
≤2µZ
0
emsE| (s)|2ds¶1/2µZ
0
emsE|F(s, us, s)|2ds¶1/2
≤2K1/2
F,H Z
−h
emsE| (s)|2ds+ 2 µKF,V Z
−h
emsEku(s)k2ds¶1/2µZ
0
emsE| (s)|2ds¶1/2
≤³2K1/2
F,H +ε´Z
0
emsE| (s)|2ds+KF,V (αε)−1Z
0
emsEhA(s)u(s), u(s)ids
+KF,V ε−1kϕ1k2
I2
V+ 2K1/2
F,H |ϕ2|2
I2
H,
o ε > 0,and by (G.4) and (A.3),
Z
0
emsE|G(s, us, s)|2ds≤KG,H |ϕ2|2
I2
H+KG,H Z
0
emsE| (s)|2ds
+KG,V kϕ1k2
I2
V+KG,V α−1Z
0
emsEhA(s)u(s), u(s)ids.
Thus, i we subs i u e hese inequali ies in o (3) we ha e
em E| ( )|2+ em EhA( )u( ), u( )i(4)
≤E| 0|2+EhA(0)u0, u0i+Kεkϕ1k2
I2
V+ (2K1/2
F,H +KG,H )|ϕ2|2
I2
H
+³m−2β+ 2K1/2
F,H +ε+KG,H ´Z
0
emsE| (s)|2ds
+¡m+Kεα−1¢Z
0
emsEhA(s)u(s), u(s)ids.
Now, we es ima e he las in eg al on he igh hand side o (4). Thanks o
d(em (u( ), ( ))) = mem (u( ), ( ))d + em | ( )|2d −em hA( )u( ), u( )id
−em (B( , ( )), u( ))d + em (F( , u , ), u( )) + em (G( , u , ), u( ))dW( ),
we deduce
em E(u( ), ( )) = E(u0, 0) + mZ
0
emsE(u(s), (s))ds
+Z
0
emsE| (s)|2ds−Z
0
emsEhA(s)u(s), u(s)ids
−Z
0
emsE(B(s, (s)), u(s))ds+Z
0
emsE(F(s, us, s), u(s))ds.
8
Consequen ly,
Z
0
emsEhA(s)u(s), u(s)ids
≤E(u0, 0) + Z
0
emsE| (s)|2ds
+c(m+cB)µZ
0
emsE| (s)|2ds¶1/2µα−1Z
0
emsEhA(s)u(s), u(s)ids¶1/2
+µZ
0
emsE|F(s, us, s)|2ds¶1/2µc2Z
0
emsEku(s)k2ds¶1/2
+³em E| ( )|2´1/2¡c2α−1em EhA( )u( ), u( )i¢1/2
≤E(u0, 0) + Z
0
emsE| (s)|2ds+cK1/2
F,V kϕ1k2
I2
V+c2(m+cB+K1/2
F,H )2(4αδ)−1|ϕ2|2
I2
H
+c2(m+cB+K1/2
F,H )2(4αδ)−1Z
0
emsE| (s)|2ds
+³δ+cK1/2
F,V α−1´Z
0
emsEhA(s)u(s), u(s)ids
+c(2α1/2)−1em E³| ( )|2+hA( )u( ), u( )i´,
and hus
CδZ
0
emsEhA(s)u(s), u(s)ids(5)
≤E(u0, 0) + cK1/2
F,V kϕ1k2
I2
V+c2(m+cB+K1/2
F,H )2(4αδ)−1|ϕ2|2
I2
H
+c(2α1/2)−1em E³| ( )|2+hA( )u( ), u( )i´
+ (4αδ)−1³4αδ +c2(m+cB+K1/2
F,H )2´Z
0
emsE| (s)|2ds.
Due o he i s condi ion in (1), he cons an Cδis posi i e. The e o e, we ha e
¡m+Kεα−1¢Z
0
emsEhA(s)u(s), u(s)ids
≤¡m+Kεα−1¢C−1
δhE(u0, 0) + c(2α1/2)−1em E³| ( )|2+hA( )u( ), u( )i´
+cK1/2
F,V kϕ1k2
I2
V+c2(m+cB+K1/2
F,H )2(4αδ)−1|ϕ2|2
I2
H
+(4αδ)−1³4αδ +c2(m+cB+K1/2
F,H )2´Z
0
emsE| (s)|2ds¸.
Thus, i we subs i u e his in o (4) i holds
9
h > 1, 2, 3≥0,a e gi en numbe s, : (− 1,0)→Ris a measu able bounded unc ion such
ha | (s)| ≤ L ,and g:R2→Ris a Lipschi z con inuous unc ion such ha g(0,0) = 0,
wi h
|g(x, y)−g(ex, ey)|2≤L2
g(|x−ex|2+|y−ey|2),∀(x, y),(ex, ey)∈R2.
The abo e p oblem can be se wi hin ou o mula ion by aking H=L2(0, π), V =
H1
0(0, π),
A( )u( ) = −∂2u
∂x2( ),∀u∈V, ∀ ≥0,
hB( , ), wi=γZπ
0
d
dx
dw
dxdx +Zπ
0e
k( , d
dx)dw
dxdx, ∀ , w ∈V, ∀ ≥0,
F( , ξ, η) = Z0
− 1
(s)η(s)ds, ∀( , ξ, η)∈R+×C(−h, 0; V)×C(−h, 0; H),
G( , ξ, η) = gµ∂ξ(− 2)
∂x , η(− 3)¶,∀( , ξ, η)∈R+×C(−h, 0; V)×C(−h, 0; H).
In his si ua ion, ope a o s A,B,Fand Gsa is y he hypo heses ensu ing exis ence and
uniqueness o a solu ion o he co esponding p oblem (see Ga ido-A ienza [6]). Conse-
quen ly, o ϕ1∈I2(−h, 0; V), ϕ2∈I2(−h, 0; H), u0∈L2(Ω,F0, P;V) and 0∈L2(Ω,F0, P;H)
gi en, he e exis s a unique solu ion u∈I2(−h, T;V)∩L2(Ω; C(0, T;V)),∂u
∂ ∈I2(−h, T;H)∩
L2(Ω; C(0, T;H)). The cons an s a e now
cB=ck, β =γ, α =c= 1, KF,H (m0) = 2
1L2
em0 1, KF,V (m0) = 0,
KG,V (m0) = L2
gem0 2, KG,H(m0) = L2
gem0 3.
Thanks o (15), he solu ion is exponen ially s able in mean squa e and almos su ely i we
impose
1L < γ, L2
g<min ½1,2γ−2L 1
1 + 2(1 + (ck+L 1)2)¾.
Example 10
Le us ake H=L2(O) and V=H1(O), whe e O ⊂ Rnis a bounded open se wi h smoo h
bounda y. Le us conside A( ) = −∆ o all ≥0; B( , ),whe e ∈L2(O), he unc ion o
L2(O) de ined, a.e. x∈ O,by B( , )(x) = k( , (x)),whe e k:R+×R→Ris a con inuous
map such ha he e exis ck,βk>0 such ha
|k( , a)| ≤ ck|a|,(k( , a)−k( , ea))(a−ea)≥βk|a−ea|2∀a, ea∈R,∀ ≥0.
Le us conside wo measu able unc ions :R+×R→Rand g:R+×R×Rn×R→R,
such ha ( , 0) = g( , 0,0,0) = 0,∀ ≥0, and we also suppose ha he e exis L , Lg>0
16

such ha
| ( , b)− ( ,eb)| ≤ L |b−eb|,
|g( , a, y, b)−g( , ea, ey,eb)|2≤L2
g(|a−ea|2+|y−ey|2+|b−eb|2),
∀ ≥0,∀a, ea, b,eb∈R,∀y, ey∈Rn.Conside also ou unc ions τi∈C1(R+), 1 ≤i≤4,
such ha 0 ≤τi( )≤h, ∀ ≥0,being τ∗
i= sup
0≤
τ0
i( )<1.As in Example 8, i we deno e
θi( ) = −τi( ), hen he e exis s ki>0 such ha θ−1
i( )≤ +ki,∀ ≥τi(0),1≤i≤4.
Fo ∈R+, ξ ∈C(−h, 0; V), η ∈C(−h, 0; H),deno e by F( , ξ, η) and G( , ξ, η) he
amilies o ope a o s de ined, a.e. x∈ O,by
F( , ξ, η)(x) = ( , η(−τ1( ))(x)),
G( , ξ, η)(x) = g( , ξ(−τ2( ))(x),∇ξ(−τ3( ))(x), η(−τ4( ))(x)).
Unde hese hypo heses, we can ensu e ha gi en ϕ1∈I2(−h, 0; H1(O)), ϕ2∈I2(−h, 0; L2(O)),
u0∈L2(Ω,F0, P;H1(O)) and 0∈L2(Ω,F0, P;L2(O)), he e exis s a unique solu ion
u∈I2(−h, T;H1(O)) ∩L2(Ω; C(0, T ;H1(O)), ∈I2(−h, T;L2(O)) ∩L2(Ω; C(0, T;L2(O)),
o he co esponden sys em (P), (see Ga ido-A ienza [6]).
This solu ion can be seen as a solu ion o he Neumann p oblem





























∂2u
∂ 2−∆u+kµ , ∂u
∂ ¶= ( , u( −τ1( )))
+gµ , u( −τ2( )),∇u( −τ3( )),∂u
∂ ( −τ4( ))¶dW( )
d ,in (0,+∞)×O,
∂u
∂ν = 0,on (0,+∞)×∂O,
u(0, x) = u0(x),∂u
∂ (0, x) = 0(x),in O,
u( ) = ϕ1( ),∂u( )
∂ =ϕ2( ), ∈(−h, 0),
whe e we deno e by ν he ou wa d uni no mal o ∂O.
In his si ua ion, i is no ha d o check ha
β=βk, cB=ck, α =c= 1,
KF,H (m0) = L2
1−τ∗
1
em0k1, KF,V (m0) = 0,
KG,H(m0) = L2
g
1−τ∗
4
em0k4, KG,V (m0) = L2
gmax ½em0k2
1−τ∗
2
,em0k3
1−τ∗
3¾.
17
So, using Rema k 4, he solu ion o ou p oblem is exponen ially s able in mean squa e and,
he e o e, almos su e exponen ially s able i we suppose
L2
g
1−τ∗
i
<1, i = 2,3,
L2
g
1−τ∗
4
<2βk−Ã2 + µck+L
(1 −τ∗
1)1/2¶2!L2
g
1−τ∗
i−2L
(1 −τ∗
1)1/2, i = 2,3.
6 Conclusions and inal ema ks
Some esul s on he exponen ial s abili y o unc ional s ochas ic pa ial di e en ial equa ions
o second o de in ime ha e been p o ed, which, in he pa icula case wi hou delay, also
imp o es a s abili y c i e ium in [5].
Howe e , ano he in e es ing ques ion is, in ou opinion, he analysis o he ac ual decay
a e o solu ions when we a e in a si ua ion in which he s abili y may no be exponen ial
(which uses o appea when one deals wi h nonlinea o non-au onomous p oblems). Only
a ew wo ks ha e been done conce ning he non-exponen ial s abili y o pa abolic s ochas ic
sys ems. I is wo h men ioning he pape by Liu [9] on he polynomial s abili y o semilin-
ea s ochas ic e olu ion equa ions which also co e s he delay si ua ion; on he o he hand,
Ca aballo e al. [2] p o e some esul s on he pa hwise s abili y wi h a gene al decay unc-
ion sa is ying sui able condi ions in bo h cases. I is ou in en ion o do an in es iga ion in
his di ec ion in a u u e pape .
Ano he poin is ha , al hough we ha e only conside ed he case o a eal Wiene p ocess,
he esul s can be ex ended o a Hilbe alued si ua ion. Howe e , we ha e p e e ed o
conside his amewo k o he sake o cla i y.
Acknowledgmen .
This wo k has been pa ially suppo ed by Jun a de Andaluc´ıa P ojec FQM314, and by
Minis e io de Ciencia y Tecnolog´ıa unde he p ojec s HA2001-0075 and BFM2002-03068.
Re e ences
[1] T. Ca aballo, Asymp o ic exponen ial s abili y o s ochas ic pa ial di e en ial equa-
ions wi h delay, S ochas ics S ochas ics Rep. 33 (1990), 27-47.
[2] T. Ca aballo, M.J. Ga ido-A ienza and J. Real, Asymp o ic s abili y o non-linea
s ochas ic e olu ion equa ions, S och. Anal. Appl. 21 (2003), no. 2 ( o appea ).
18
[3] T. Ca aballo and K. Liu, On exponen ial s abili y c i e ia o s ochas ic pa ial di e -
en ial equa ions, S ochas ic P ocesses and hei Applica ions 83 (1999), 289-301.
[4] T. Ca aballo, K. Liu and A. T uman, S ochas ic unc ional pa ial di e en ial equa ions:
exis ence, uniqueness and asymp o ic decay p ope y, P oc. R. Soc. Lond. A 456 (2000),
1775-1802.
[5] R.F. Cu ain, S abili y o s ochas ic pa ial di e en ial equa ion, J. Ma h. Anal. Appl.
79 (1981), 352-369.
[6] M.J. Ga ido-A ienza, “Algunos esul ados de exis encia, unicidad y es abilidad pa a
EDP uncionales es oc´as icas no lineales”, PhD. Thesis, Uni e sidad de Se illa, 2002.
[7] M.J. Ga ido-A ienza, J. Real, Exis ence and uniqueness o solu ions o delay s ochas ic
e olu ion equa ions o second o de in ime, Submi ed o S och. Dyn.
[8] H. Lisei, Conjuga ion o lows o s ochas ic and andom unc ional di e en ial equa-
ions, S och. Dyn. 1 (2001), no. 2, 283-298.
[9] K. Liu, Lyapuno unc ionals and asymp o ic s abili y o s ochas ic delay e olu ion
equa ions, S ochas ics S ochas ics Rep. 63 (1998) 1-26.
[10] K. Liu and X.R. Mao, Exponen ial s abili y o nonlinea s ochas ic e olu ion equa ions,
S ochas ic P ocesses and hei Applica ions 78 (1998), 173-193.
[11] T. Taniguchi, Asymp o ic s abili y heo ems o semilinea s ochas ic e olu ion equa-
ions in Hilbe spaces, S ochas ics S ochas ics Rep. 53 (1995), no. 1-2, 41–52.
[12] T. Taniguchi, K. Liu and A. T uman, Exis ence, uniqueness, and asymp o ic beha io
o mild solu ions o s ochas ic unc ional di e en ial equa ions in Hilbe spaces, J.
Di e en ial Equa ions 181 (2002), no. 1, 72–91.
19