EXISTENCE AND UNIQUENESS OF SOLUTIONS
FOR DELAY STOCHASTIC EVOLUTION
EQUATIONS
Tom´as CARABALLO, Ma ´ıa J. GARRIDO-ATIENZA and Jos´e REAL
Dp 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-mails: [email p o ec ed] ; mga ido@nume .us.es; eal@nume .us.es
Abs ac
Some esul s on he exis ence and uniqueness o solu ions o s ochas ic e olu ion equa-
ions con aining some he edi a y cha ac e is ics a e p o ed. In ac , ou heo y is de eloped
om a a ia ional poin o iew and in a gene al unc ional se ing which pe mi us o deal
wi h se e al kinds o delay e ms in a uni ied o mula ion.
1. In oduc ion and s a emen o he p oblem
When one wan s o model some e olu ion phenomena a ising in physics, biology, enginee -
ing, e c., some he edi a y cha ac e is ics such as a e -e ec , ime-lag, ime-delay can appea
in he a iables. Typical examples can be ound in he esea ches o ma e ials wi h e mal
memo y, biochemical eac ions, popula ion models, e c. (see, o ins ance, Ruess [10], Wu
[11] and e e ences ci ed he ein). This enables us o hink ha he p oblem could be be e
modeled by conside ing a unc ional di e en ial equa ion which akes in o accoun he his o y
o he sys em. Howe e , in mos cases, some kind o andomness can appea in he p oblem, so
ha he sys em should be modeled by a s ochas ic o m o he unc ional equa ion. Mo i a ed
by hese ac s, ou main pu pose in his pape is o analyse he exis ence and uniqueness o
solu ions o a class o nonlinea s ochas ic PDEs wi h ime delays in a a ia ional con ex ,
which, in pa icula , ex end and comple e he esul s in Ca aballo [2] and Ca aballo e al. [4].
Fi o all, we would like o men ion ha , in he de e minis ic amewo k, he e exis s a
wide li e a u e on he exis ence o di e en kind o solu ions (s ong, mild, in eg al, e c.) o
unc ional di e en ial equa ions e en in he mo e gene al con ex o di e en ial inclusions. I
is well wo h eading he wo k by Ruess [10] whe e one can ind a desc ip ion o he di e en
echniques used o handle his ques ion, in addi ion o a la ge lis o e e ences conce ning hese
me hods (e.g. me hod o lines, Gale kin app oxima ions, Ka o app oximan s, e c.). Howe e ,
om a a ia ional poin o iew, only a ew wo ks ha e been published (A ola [1] o linea and
semilinea e a ded equa ions, Ca aballo [3] o a mo e gene al nonlinea mono one si ua ion
in he unc ional amewo k, among o he s).
As o he s ochas ic p oblem in he a ia ional se ing, e en much less has been done.
As a as we know, only he wo ks by Real [8,9] (in he linea case), Ca aballo [2] (nonlinea
p oblem wi h a iable delay) and Ca aballo e al. [4] ha e appea ed up o da e. Bu , as
he assump ions in hese wo ks a e a he es ic i e so ha he ope a o s in ol ed in he
equa ions canno be gene al enough, we a e now in e es ed in de eloping a heo y which, in
pa icula , con ains he p e ious wo ks and which pe mi s us o p o e exis ence and uniqueness
o solu ion o a wide class o sys ems.
1
To s a o , le us s a e he abs ac amewo k in which ou analysis will be ca ied ou .
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, H and V∗ espec i ely; by ((·,·)) and (·,·)
he scala p oduc s in Vand H espec i ely; and by h·,·i he duali y p oduc be ween V∗and
V.
Assume ha {Ω,F, P}is a comple e p obabili y space, equipped wi h a no mal il a ion
{F } ≥0,i.e., F0con ains all A∈ F such ha P(A) = 0 and F =T
s>
Fs,∀ ≥0.Deno e
F =F0 o all ≤0.
We suppose also gi en {W( )} ≥0, a eal alued {F } −Wiene p ocess.
Gi en eal numbe s a<b, and a sepa able Hilbe space Hwe will deno e by I2(a, b;H)
he space o all p ocesses X∈L2(Ω ×(a, b),F ⊗ B((a, b)), dP ⊗d ;H) (whe e B((a, b)) deno es
he Bo el σ−algeb a on (a, b)) such ha X( ) is F −measu able a.e. ∈(a, b). The space
I2(a, b;H) is a closed subspace o L2(Ω ×(a, b),F ⊗ B((a, b)), dP ⊗d ;H).
We will deno e by C(a, b;H) he Banach space o all con inuous unc ions om [a, b] in o
Hequipped wi h sup no m. We will w i e L2(Ω; C(a, b;H)) ins ead o L2(Ω,F, dP;C(a, b;H)).
Le us also conside wo ixed eal numbe s T > 0 and h > 0. I we conside a unc ion
x∈C(−h, T;H), o each ∈[0, T] we will deno e by x ∈C(−h, 0; H) he unc ion de ined
by x (s) = x( +s)∀s∈[−h, 0]. Mo eo e , i y∈L2(−h, T;H), we will also deno e by
y ∈L2(−h, 0; H), o a.e. ∈(0, T ), he unc ion de ined by y (s) = y( +s) a.e. s∈(−h, 0).
Le A( , ·) : V→V∗be a amily o nonlinea ope a o s de ined a.e. ∈(0, T ) and
sa is ying:
(A.1) (Measu abili y) ∀ ∈V, he map ∈(0, T )→A( , )∈V∗is Lebesgue measu able.
(A.2) (Hemicon inui y) he map
θ∈IR → hA( , u +θ ), wi ∈ IR
is con inuous ∀u, , w ∈V, and a.e. ∈(0, T).
(A.3) (Boundedness) he e exis s c > 0 such ha kA( , )k∗≤ck k ∀ ∈V, a.e. ∈
(0, T).
(A.4) (Mono onici y and Coe ci i y): he e exis α > 0 and λ∈IR such ha
−2hA( , u)−A( , ), u − i+λ|u− |2≥αku− k2,∀u, ∈V, a.e. ∈(0, T).
Le F1: (0, T)×C(−h, 0; H)→V∗and F2: (0, T )×C(−h, 0; V)→V∗be wo amilies o
nonlinea ope a o s de ined a.e. ∈(0, T) such ha :
(F1.1) ∀ξ∈C(−h, 0; H), he map ∈(0, T)7−→ F1( , ξ)∈V∗is Lebesgue measu able,
(F1.2) F1( , 0) = 0,a.e. ∈(0, T),
(F1.3) he e exis s CF1>0 such ha
2
kF1( , ξ)−F1( , η)k2
∗≤CF1|ξ−η|2
C(−h,0;H),∀ξ, η ∈C(−h, 0; H),a.e. ∈(0, T),
(F2.1) ∀ξ∈C(−h, 0; V), he map ∈(0, T)7−→ F2( , ξ)∈V∗is Lebesgue measu able,
(F2.2) F2( , 0) = 0,a.e. ∈(0, T),
(F2.3) he e exis s CF2>0 such ha
kF2( , ξ)−F2( , η)k2
∗≤CF2kξ−ηk2
C(−h,0;V),∀ξ, η ∈C(−h, 0; V),a.e. ∈(0, T),
(F2.4) he e exis s KF2>0 such ha ∀x, y ∈C(−h, T;V),and ∀ ∈[0, T],
Z
0
kF2(s, xs)−F2(s, ys)k2
∗ds ≤KF2Z
−h
kx(s)−y(s)k2ds.
Le also G0: (0, T)×C(−h, 0; H)→Hand G1: (0, T)×C(−h, 0; V)→Hbe ano he
wo amilies o nonlinea ope a o s de ined a.e. ∈(0, T ) such ha :
(G0.1) ∀ξ∈C(−h, 0; H), he map ∈(0, T)7−→ G0( , ξ)∈His Lebesgue measu able,
(G0.2) G0( , 0) = 0,a.e. ∈(0, T),
(G0.3) he e exis s CG0>0 such ha
|G0( , ξ)−G0( , η)|2≤CG0|ξ−η|2
C(−h,0;H),∀ξ, η ∈C(−h, 0; H),a.e. ∈(0, T),
(G1.1) ∀ξ∈C(−h, 0; V), he map ∈(0, T)7−→ G1( , ξ)∈His Lebesgue measu able,
(G1.2) G1( , 0) = 0,a.e. ∈(0, T),
(G1.3) he e exis s CG1>0 such ha
|G1( , ξ)−G1( , η)|2≤CG1kξ−ηk2
C(−h,0;V),∀ξ, η ∈C(−h, 0; V),a.e. ∈(0, T),
(G1.4) he e exis s KG1>0 such ha ∀x, y ∈C(−h, T;V),and ∀ ∈[0, T],
Z
0
|G1(s, xs)−G1(s, ys)|2ds ≤KG1Z
−h
kx(s)−y(s)k2ds.
We conside he p oblem
3
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u∈I2(−h, T;V)∩L2(Ω; C(−h, T ;H)),
u( ) = ψ(0) + Z
0
A(s, u(s)) ds +Z
0
(F1(s, us) + F2(s, us) + (s)) ds
+Z
0
(G0(s, us) + G1(s, us) + g(s)) dW(s), ∈[0, T],
u( ) = ψ( ), ∈[−h, 0],
(P)
whe e ∈I2(0, T;V∗), g∈I2(0, T ;H) and ψ∈I2(−h, 0; V)∩L2(Ω; C(−h, 0; H)) a e gi en.
Rema k 1.1. I is no di icul o deduce om (F1.1)-(F1.3) ha i u∈I2(−h, T ;V)∩
L2(Ω; C(−h, T;H)), he p ocess F1( , u ) belongs o I2(0, T;V∗). Also, by means o (G0.1)-
(G0.3), he p ocess G0( , u ) belongs o I2(0, T ;H).
Rema k 1.2. Obse e ha by (F2.1)-(F2.3), o a gi en x∈C(−h, T ;V), he unc ion
Fx
2: (0, T)→V∗de ined by Fx
2( ) = F2( , x ) a.e. ∈(0, T), belongs o L2(0, T ;V∗). Then,
hanks o (F2.4), he mapping
Ξ : x∈C(−h, T;V)7→ Fx
2∈L2(0, T;V∗)
has a unique ex ension o a mapping e
Ξ which is uni o mly con inuous om L2(−h, T;V) in o
L2(0, T;V∗). F om now on, we will also w i e F2( , x ) = e
Ξ(x)( ) o each x∈L2(−h, T;V),
and o e e y x, y ∈L2(−h, T;V) i holds
Z
0
kF2(s, xs)−F2(s, ys)k2
∗ds ≤KF2Z
−h
kx(s)−y(s)k2ds ∀ ∈[0, T].(1.1)
By a simila a gumen , we can de ine G1( , x )∈L2(0, T ;H) o each x∈L2(−h, T;V), and
∀x, y ∈L2(−h, T;V) i ollows
Z
0
|G1(s, xs)−G1(s, ys)|2ds ≤KG1Z
−h
kx(s)−y(s)k2ds ∀ ∈[0, T].(1.2)
Thus, i u∈I2(−h, T ;V) is gi en, he p ocess F2( , u ) belongs o I2(0, T;V∗), he p ocess
G1( , u ) belongs o I2(0, T;H), and, consequen ly, ∀u, ∈I2(−h, T;V) we ob ain
Z
0
kF2(s, us)−F2(s, s)k2
∗ds ≤KF2Z
−h
ku(s)− (s)k2ds ∀ ∈[0, T], P −a.s., (1.3)
and
Z
0
|G1(s, us)−G1(s, s)|2ds ≤KG1Z
−h
ku(s)− (s)k2ds ∀ ∈[0, T], P −a.s.. (1.4)
As a consequence o he p eceding ema ks, he e ms appea ing in p oblem (P) make
sense. Now, we a e in e es ed in es ablishing some esul s on he exis ence and uniqueness o
solu ion o (P) unde some addi ional assump ions. To his espec , i is wo h men ioning
ha in he absence o he edi a y cha ac e is ics (i.e. when h= 0), ou p oblem has been sol ed
by Pa doux [6] (see also Da P a o and Zabczyk [5] o a di e en app oach); in he linea case
4
con aining a iable delays, i has also been ea ed by Real [9]; Ca aballo [2] conside ed he
nonlinea mono one si ua ion wi h a iable delay bu o bounded ope a o s Fiand Gi, and
inally, Ca aballo e al. [4] p o ided an answe o ou p oblem in he pa icula si ua ions in
which F2≡0, F1( , ·) is a amily o ope a o s om Vin o Hand, wha is mo e impo an ,
unde s onge assump ions on he amily o ope a o s which do no allow us o co e a wide
class o applica ions (e.g. in he case o unbounded ope a o s, essen ially he ones con aining
dis ibu ed delays sa is y he assump ions in [4]). Thus, on he one hand, he esul s we shall
ob ain can be conside ed as ex ensions o he nonlinea case o hose ob ained in Real [9]. On
he o he hand, he p esence o he e m F1and he hypo heses ha we shall impose on F2
and G1, pe mi us, as we ha e al eady men ioned, o ea examples which canno be handled
wi h he esul s in Ca aballo e al. [4].
The pape is o ganized as ollows. In Sec ion 2, we p o e a i s esul on he exis ence
and uniqueness o solu ion o he p oblem (P) in he pa icula case F2≡G1≡0. Then, in
Sec ion 3, we es ablish an exis ence and uniqueness esul o he comple e p oblem. Finally,
an example is conside ed in he las Sec ion o illus a e ou esul s.
2. A i s exis ence and uniqueness esul
In his sec ion, we shall conside he p oblem
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u∈I2(−h, T;V)∩L2(Ω; C(−h, T ;H)),
u( ) = ψ(0) + Z
0
A(s, u(s)) ds +Z
0
(F1(s, us) + (s)) ds
+Z
0
(G0(s, us) + g(s)) dW(s), ∈[0, T],
u( ) = ψ( ), ∈[−h, 0].
(P0)
We can now p o e he ollowing esul :
Theo em 2.1 Assume ha hypo heses (A.1)-(A.4),(F1.1)-(F1.3) and (G0.1)-(G0.3) hold.
Then, o e e y ψ∈I2(−h, 0; V)∩L2(Ω; C(−h, 0; H)), ∈I2(0, T;V∗)and g∈I2(0, T;H),
he e exis s a unique solu ion u o he p oblem (P0).
P oo .
Uniqueness o solu ions. Assume ha u, ∈I2(−h, T;V)∩L2(Ω; C(−h, T;H)) a e
wo solu ions o (P0). Then, I ˆo’s o mula and condi ion (A.4) imply ha o all ∈[0, T]
|u( )− ( )|2+αZ
0
ku(s)− (s)k2ds
≤λZ
0
|u(s)− (s)|2ds
+ 2 Z
0
hF1(s, us)−F1(s, s), u(s)− (s)ids
+ 2 Z
0
(G0(s, us)−G0(s, s), u(s)− (s)) dW(s)
+Z
0
|G0(s, us)−G0(s, s)|2ds.
5

The e o e,
E·sup
0≤s≤
|u(s)− (s)|2¸+αE Z
0
ku(s)− (s)k2ds
≤2|λ|EZ
0
|u(s)− (s)|2ds + 2EZ
0
|G0(s, us)−G0(s, s)|2ds
+ 4EZ
0
kF1(s, us)−F1(s, s)k∗ku(s)− (s)kds
+ 4E·sup
0≤s≤ Zs
0
(G0(θ, uθ)−G0(θ, θ), u(θ)− (θ)) dW(θ)¸(2.1)
o all ∈[0, T]. Now, we can es ima e he e ms on he igh -hand side o (2.1).
On he one hand,
4EZ
0
kF1(s, us)−F1(s, s)k∗ku(s)− (s)kds
≤EZ
0·8
αkF1(s, us)−F1(s, s)k2
∗+α
2ku(s)− (s)k2¸ds
≤8
αCF1EZ
0
|us− s|2
C(−h,0;H)ds +α
2EZ
0
ku(s)− (s)k2ds
≤8
αCF1EZ
0
sup
0≤ ≤s
|u( )− ( )|2ds +α
2EZ
0
ku(s)− (s)k2ds. (2.2)
On he o he hand, Bu kholde -Da is-Gundy’s inequali y yields ha
4E·sup
0≤s≤ Zs
0
(G0(θ, uθ)−G0(θ, θ), u(θ)− (θ)) dW(θ)¸
≤12E(sup
0≤s≤
|u(s)− (s)|·Z
0
|G0(θ, uθ)−G0(θ, θ)|2dθ¸1
2)
≤1
2Eµsup
0≤s≤
|u(s)− (s)|2¶+ 72EZ
0
|G0(θ, uθ)−G0(θ, θ)|2dθ
≤1
2Eµsup
0≤s≤
|u(s)− (s)|2¶+ 72CG0EZ
0
|uθ− θ|2
C(−h,0;H)dθ
≤1
2Eµsup
0≤s≤
|u(s)− (s)|2¶+ 72CG0EZ
0
sup
0≤ ≤θ
|u( )− ( )|2dθ. (2.3)
Thus, (2.1)-(2.3) imply ha o all ∈[0, T]
1
2E·sup
0≤s≤
|u(s)− (s)|2¸+α
2EZ
0
ku(s)− (s)k2ds
≤·2|λ|+8
αCF1+ 74CG0¸EZ
0
sup
0≤ ≤θ
|u( )− ( )|2dθ.
Now, uniqueness ollows immedia ely om G onwall’s lemma.
6
Exis ence o solu ions: We deno e u0≡0, and de ine by ecu ence a sequence {un}n≥1
o p ocesses as solu ions o he p oblem
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un∈I2(−h, T;V)∩L2(Ω; C(−h, T ;H)),
un( ) = ψ(0) + Z
0
(A(s, un(s)) −λ
2un(s)) ds +λ
2Z
0
un−1(s)ds
+Z
0
(F1(s, un−1
s) + (s)) ds +Z
0
(G0(s, un−1
s) + g(s)) dW(s), ∈[0, T],
un( ) = ψ( ), ∈[−h, 0].
(P0
n)
Obse e ha u0≡0∈I2(−h, T;V)∩L2(Ω; C(−h, T;H)), and by Rema k 1.1, i un−1∈
I2(−h, T;V)∩L2(Ω; C(−h, T ;H)),i ollows ha F1( , un−1
)∈I2(0, T;V∗),and G0( , un−1
)∈
I2(0, T;H),and consequen ly, om he esul s in Pa doux [6], he e exis s a unique un∈
I2(−h, T;V)∩L2(Ω; C(−h, T ;H)) which is a solu ion o (P0
n).
Now, we wan o p o e ha {un}n≥1con e ges in I2(−h, T;V)∩L2(Ω; C(−h, T;H)) o
a p ocess uwhich will be he solu ion o p oblem (P0).
Applying I ˆo’s o mula o he p ocess un+1( )−un( ), n≥1, and using condi ion (A.4),
we ob ain
¯¯un+1( )−un( )¯¯2+αZ
0°
°un+1(s)−un(s)°
°2ds
≤λZ
0
(un+1(s)−un(s), un(s)−un−1(s)) ds
+ 2 Z
0F1(s, un
s)−F1(s, un−1
s), un+1(s)−un(s)®ds
+ 2 Z
0
(G0(s, un
s)−G0(s, un−1
s), un+1(s)−un(s)) dW(s)
+Z
0¯¯G0(s, un
s)−G0(s, un−1
s)¯¯2ds (2.4)
o all ∈[0, T].
Consequen ly, (2.4) yields
E·sup
0≤s≤ ¯¯un+1(s)−un(s)¯¯2¸+αE Z
0°
°un+1(s)−un(s)°
°2ds
≤2|λ|EZ
0¯¯un+1(s)−un(s)¯¯¯¯un(s)−un−1(s)¯¯ds
+ 4EZ
0¯¯F1(s, un
s)−F1(s, un−1
s), un+1(s)−un(s)®¯¯ds
+ 4E·sup
0≤s≤ Zs
0
(G0(θ, un
θ)−G0(θ, un−1
θ), un+1(θ)−un(θ)) dW(θ)¸
+ 2EZ
0¯¯G0(s, un
s)−G0(s, un−1
s)¯¯2ds. (2.5)
Now, obse e ha
7
2|λ|EZ
0¯¯un+1(s)−un(s)¯¯¯¯un(s)−un−1(s)¯¯ds
≤2β|λ|EZ
0°
°un+1(s)−un(s)°
°¯¯un(s)−un−1(s)¯¯ds
≤α
3EZ
0°
°un+1(s)−un(s)°
°2ds +3λ2β2
αEZ
0
sup
0≤θ≤s¯¯un(θ)−un−1(θ)¯¯2ds, (2.6)
whe e β > 0 is a cons an such ha | | ≤ βk k,∀ ∈V.
On he o he hand, hanks o condi ion (F1.3), we can ob ain
4EZ
0¯¯F1(s, un
s)−F1(s, un−1
s), un+1(s)−un(s)®¯¯ds
≤4EZ
0°
°F1(s, un
s)−F1(s, un−1
s)°
°∗°
°un+1(s)−un(s)°
°ds
≤EZ
0·12
α°
°F1(s, un
s)−F1(s, un−1
s)°
°2
∗+α
3°
°un+1(s)−un(s)°
°2¸ds
≤12
αCF1EZ
0¯¯un
s−un−1
s¯¯2
C(−h,0;H)ds +α
3EZ
0°
°un+1(s)−un(s)°
°2ds
≤12
αCF1EZ
0
sup
0≤ ≤s¯¯un( )−un−1( )¯¯2ds +α
3EZ
0°
°un+1(s)−un(s)°
°2ds. (2.7)
In a simila manne as o uniqueness, we can ob ain om Bu kholde -Da is-Gundy’s inequa-
li y ha
4E·sup
0≤s≤ Zs
0
(G0(θ, un
θ)−G0(θ, un−1
θ), un+1(θ)−un(θ)) dW(θ)¸
≤1
2Eµsup
0≤s≤ ¯¯un+1(s)−un(s)¯¯2¶+ 72CG0EZ
0
sup
0≤ ≤θ¯¯un( )−un−1( )¯¯2dθ. (2.8)
Then, we can ge om (2.5)-(2.8) and (G0.3), ha he e exis s a posi i e cons an ksuch ha
o all n≥1 and all ∈[0, T]
1
2E·sup
0≤s≤ ¯¯un+1(s)−un(s)¯¯2¸+α
3EZ
0°
°un+1(s)−un(s))°
°2ds
≤k
2EZ
0
sup
0≤ ≤θ¯¯un( )−un−1( )¯¯2dθ. (2.9)
Now, we de ine
ρn( ) = 1
2E·sup
0≤s≤ ¯¯un+1(s)−un(s)¯¯2¸+α
3EZ
0°
°un+1(s)−un(s))°
°2ds, ∀n≥1,∀ ∈[0, T].
Then, (2.9) immedia ely implies ha
ρn( )≤kZ
0
ρn−1(s)ds, ∀n≥1,∀ ∈[0, T],
8
and, consequen ly, by i e a ing he p eceding inequali y, we ob ain
ρn( )≤kn−1Tn−1
(n−1)! ρ1(T),∀n≥1,∀ ∈[0, T].(2.10)
Since un+1( ) = un( ),∀ ∈[−h, 0], (2.10) implies ha {un}n≥1is a Cauchy sequence in
I2(−h, T;V)∩L2(Ω; C(−h, T ;H)).Thus, he e exis s u∈I2(−h, T;V)∩L2(Ω; C(−h, T;H))
such ha
un→uin I2(−h, T;V)∩L2(Ω; C(−h, T ;H)).
Thanks o condi ions (F1.3) and (G0.3), we ha e in pa icula ha
F1( , un
)→F1( , u ) in I2(0, T;V∗),
and
G0( , un
)→G0( , u ) in I2(0, T;H).
Mo eo e , by (A.3), he sequence {A( , un( ))}n≥1is bounded in I2(0, T;V∗). Thus, he e
exis a subsequence {A( , unk( ))}nk≥1⊂ {A( , un( ))}n≥1and ξ∈I2(0, T;V∗),such ha
A( , unk( )) * ξ in I2(0, T ;V∗),
whe e *deno es weak con e gence. Thus, we can ake limi s in (P0
nk), and ob ain ha uis
solu ion o 
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u∈I2(−h, T;V)∩L2(Ω; C(−h, T ;H)),
u( ) = ψ(0) + Z
0
ξ(s)ds +Z
0
(F1(s, us) + (s)) ds
+Z
0
(G0(s, us) + g(s)) dW(s), ∈[0, T],
u( ) = ψ( ), ∈[−h, 0].
(P00)
To simpli y he no a ion, obse e ha ξis uniquely de e mined by u, and hus, he whole
sequence {A( , un( ))}n≥1con e ges weakly o ξin I2(0, T;V∗).
In o de o p o e ha uis in ac a solu ion o p oblem (P0), we only need o p o e ha
ξ( ) = A( , u( )) in (0, T).
Fi s o all, applying I ˆo’s o mula o |un( )|2and o |u( )|2on he in e al [0, T], we
ob ain
E|un(T)|2=E|ψ(0)|2+ 2EZT
0
hA(s, un(s)), un(s)ids
+λE ZT
0
(un(s), un−1(s)) ds
−λE ZT
0
|un(s)|2ds + 2EZT
0F1(s, un−1
s) + (s), un(s)®ds
+EZT
0
|G0(s, un−1
s) + g(s)|2ds (2.12)
and
E|u(T)|2=E|ψ(0)|2+ 2EZT
0
hξ(s), u(s)ids + 2EZT
0
hF1(s, us) + (s), u(s)ids
+EZT
0
|G0(s, us) + g(s)|2ds. (2.13)
9
and consequen ly,
lim
k→∞(xmk−ymk) = 2EZT
0
h−A( , X( )) −F2( , X ), u( )id
+ 2EZT
0
hη( )−A( , X( )) + σ( )−F2( , X ),−X( )id
+EZT
0
|G1( , X )|2d −2EZT
0
(ζ( ), G1( , X )) d
−2αE ZT
0
((u( ), X( ))) d +αE ZT
0
kX( )k2d . (3.12)
Applying I ˆo’s o mula o |umk( )|2on he in e al [0, T],
E|umk(T)|2≤E|ψ(0)|2+EZT
0
|G1( , umk
) + g( )|2d
+ 2EZT
0
hA( , umk( )) + F2( , umk
) + ( ), umk( )id ,
and, hus
ymk≥E|umk(T)|2−E|ψ(0)|2−EZT
0
|g( )|2d +αE ZT
0
kumk( )k2d
−2EZT
0
(G1( , umk
), g( )) d −2EZT
0
h ( ), umk( )id .
Le ing k→ ∞,
lim in
k→∞ ymk≥E|u(T)|2−E|ψ(0)|2−EZT
0
|g( )|2d +αE ZT
0
ku( )k2d
−2EZT
0
(ζ( ), g( )) d −2EZT
0
h ( ), u( )id . (3.13)
Applying once again I ˆo’s o mula o |u( )|2on [0, T],
E|u(T)|2=E|ψ(0)|2+EZT
0
|ζ( ) + g( )|2d + 2EZT
0
hη( ) + σ( ) + ( ), u( )id ,
and so, om (3.13)
lim in
k→∞ ymk≥2EZT
0
hη( ) + σ( ), u( )id +EZT
0
|ζ( )|2d +αE ZT
0
ku( )k2d . (3.14)
F om (3.12) and (3.14) we ha e
0≥lim in
k→∞ xmk≥2EZT
0
hη( )−A( , X( )) + σ( )−F2( , X ), u( )−X( )id
+EZT
0
|ζ( )−G1( , X )|2d +αE ZT
0
ku( )−X( )k2d . (3.15)
16

I we ake X( ) = u( ) in (3.15), i ollows ha ζ( ) = G1( , u ), ∈[0, T ].Now, we will se
X( ) = u( )−δZ( ),whe e δ > 0 and Z∈I2(−h, T;V) is such ha Z= 0 in (−h, 0). Then,
by (3.15),
0≥2EZT
0
hη( )−A( , u( )−δZ( )) + σ( )−F2( , u −δZ ), δZ( )id
+EZT
0
|G1( , u )−G1( , u −δZ )|2d . (3.16)
Di iding by δin (3.16), and le ing δ→0,we ge by (A.2), (F2.2) and (F2.4),
2EZT
0
hη( )−A( , u( )) + σ( )−F2( , u ), Z( )id ≤0,
and since Z∈I2(0, T;V) is a bi a y, clea ly η( ) + σ( ) = A( , u( )) + F2( , u ) in [0, T].
S ep 2. Now, we conside p oblem (P) unde he condi ions in he heo em. We deno e
u0≡0, and de ine by ecu ence a sequence {un}n≥1o p ocesses by

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
un∈I2(−h, T;V)∩L2(Ω; C(−h, T ;H)),
un( ) = ψ(0) + Z
0
(A(s, un(s)) −λ
2un(s)) ds +λ
2Z
0
un−1(s)ds
+Z
0
(F1(s, un−1
s) + F2(s, un
s) + (s)) ds
+Z
0
(G0(s, un−1
s) + G1(s, un
s) + g(s)) dW(s), ∈[0, T],
un( ) = ψ( ), ∈[−h, 0].
(Pn)
Obse e ha i un−1∈I2(−h, T;V)∩L2(Ω; C(−h, T;H)), hen F1( , un−1
)∈I2(0, T;V∗),
and G0( , un−1
)∈I2(0, T;H).Mo eo e , he amily o ope a o s de ined by e
A( , ) = A( , )−
λ
2 ∀ ∈V, a.e. ∈(0, T), sa is ies condi ions (A.1) −(A.5) wi h λ= 0.Consequen ly, we can
use S ep 1 o ensu e ha p oblem (Pn) has a unique solu ion.
Now, a guing as in he p oo o Theo em 2.1, we can p o e ha {un}n≥1is a Cauchy
sequence in I2(−h, T;V)∩L2(Ω; C(−h, T;H)), and hus, i con e ges o a p ocess u∈
I2(−h, T;V)∩L2(Ω; C(−h, T ;H)), which will be he solu ion o (P).
In o de o ob ain ou objec i e, we i s apply I ˆo’s o mula o he p ocess un+1( )−un( ),
≥0, n≥1, and using (A.5) we ha e
¯¯un+1( )−un( )¯¯2+αZ
0°
°un+1(s)−un(s)°
°2ds
≤λZ
0
(un+1(s)−un(s), un(s)−un−1(s)) ds
+ 2 Z
0F1(s, un
s)−F1(s, un−1
s), un+1(s)−un(s)®ds
+ 2 Z
0
(G0(s, un
s)−G0(s, un−1
s), G1(s, un+1
s)−G1(s, un
s)ds
17
+Z
0¯¯G0(s, un
s)−G0(s, un−1
s)¯¯2ds
+ 2 Z
0
(G0(s, un
s)−G0(s, un−1
s), un+1(s)−un(s)) dW(s)
+ 2 Z
0
(G1(s, un+1
s)−G1(s, un
s), un+1(s)−un(s)) dW(s),(3.17)
which, oge he wi h condi ions (F1.4), (G0.4) and (G1.4) imply
E¯¯un+1( )−un( )¯¯2+αE Z
0°
°un+1(s)−un(s)°
°2ds
≤3α
4EZ
0°
°un+1(s)−un(s)°
°2ds +λ2β2
αZ
0
sup
0≤θ≤s
E¯¯un(θ)−un−1(θ)¯¯2ds
+4
αCF1Z
0
sup
0≤θ≤s
E¯¯un(θ)−un−1(θ)¯¯2ds
+µ3KG1
α+ 1¶CG0Z
0
sup
0≤θ≤s
E¯¯un(θ)−un−1(θ)¯¯2ds, (3.18)
whe e β > 0 is he cons an such ha | | ≤ βk k ∀ ∈V. Consequen ly, (3.18) yields
sup
0≤s≤
E¯¯un+1(s)−un(s)¯¯2+α
4EZ
0°
°un+1(s)−un(s)°
°2ds
≤kZ
0
sup
0≤θ≤s
E¯¯un(θ)−un−1(θ)¯¯2ds, (3.19)
o all ∈[0, T] and all n≥1, whe e k=2λ2β2
α+8
αCF1+ 2 µ3KG1
α+ 1¶CG0. Now, i we
deno e
ρn( ) = sup
0≤s≤
E¯¯un+1(s)−un(s)¯¯2+α
4EZ
0°
°un+1(s)−un(s)°
°2ds, ∀n≥1,∀ ∈[0, T],
we can deduce om (3.19) ha
ρn( )≤(kT)n−1
(n−1)! ρ1(T),∀n≥1,∀ ∈[0, T],
and hus, ∀n≥1,
sup
0≤s≤T
E¯¯un+1(s)−un(s)¯¯2+α
4EZT
0°
°un+1(s)−un(s)°
°2ds ≤(kT)n−1
(n−1)! ρ1(T),(3.20)
and, in pa icula , {un}n≥1is a Cauchy sequence in I2(−h, T;V).
Now, in o de o p o e ha {un}n≥1is a Cauchy sequence in L2(Ω; C(−h, T ;H)), we
conside again (3.17), ake sup
0≤s≤T
and, inally, expec a ion, so ha we ob ain
Eµsup
0≤s≤T¯¯un+1(s)−un(s)¯¯2¶
18
≤ |λ|EZT
0¯¯(un+1(s)−un(s), un(s)−un−1(s))¯¯ds
+ 2EZT
0¯¯F1(s, un
s)−F1(s, un−1
s), un+1(s)−un(s)®¯¯ds
+ 2EZT
0¯¯(G0(s, un
s)−G0(s, un−1
s), G1(s, un+1
s)−G1(s, un
s)¯¯ds
+ 2Eµsup
0≤s≤TZs
0
(G0(θ, un
θ)−G0(θ, un−1
θ), un+1(θ)−un(θ)) dW(θ)¶
+ 2Eµsup
0≤s≤TZs
0
(G1(θ, un+1
θ)−G1(θ, un
θ), un+1(θ)−un(θ)) dW(θ)¶
+EZT
0¯¯G0(s, un
s)−G0(s, un−1
s)¯¯2ds (3.21)
o all ∈[0, T] and all n≥1.
Using (F1.4),we ob ain
2EZT
0¯¯F1(s, un
s)−F1(s, un−1
s), un+1(s)−un(s)®¯¯ds
≤CF1ZT
0
sup
0≤θ≤s
E¯¯un(θ)−un−1(θ)¯¯2ds +EZT
0°
°un+1(s)−un(s)°
°2ds. (3.22)
F om Bu kholde -Da is-Gundy’s inequali y, (G0.4) and (G1.4), we ha e
2Eµsup
0≤s≤TZs
0
(G0(θ, un
θ)−G0(θ, un−1
θ), un+1(θ)−un(θ)) dW(θ)¶
≤1
3Eµsup
0≤s≤T¯¯un+1(s)−un(s)¯¯2¶+ 27CG0ZT
0
sup
0≤θ≤s
E¯¯un(θ)−un−1(θ)¯¯2ds, (3.23)
EZT
0¯¯G0(s, un
s)−G0(s, un−1
s)¯¯2ds ≤CG0ZT
0
sup
0≤θ≤s
E¯¯un(θ)−un−1(θ)¯¯2ds, (3.24)
2EZT
0¯¯(G0(s, un
s)−G0(s, un−1
s), G1(s, un+1
s)−G1(s, un
s)¯¯ds
≤CG0ZT
0
sup
0≤θ≤s
E¯¯un(θ)−un−1(θ)¯¯2ds +KG1EZT
0°
°un+1(s)−un(s)°
°2ds, (3.25)
and
2Eµsup
0≤s≤TZs
0
(G1(θ, un+1
θ)−G1(θ, un
θ), un+1(θ)−un(θ)) dW(θ)¶
≤1
3Eµsup
0≤s≤T¯¯un+1(s)−un(s)¯¯2¶+ 27KG1EZT
0°
°un+1(s)−un(s)°
°2ds. (3.26)
Also,
|λ|EZT
0
(un+1(s)−un(s), un(s)−un−1(s)) ds
≤1
2EZT
0°
°un+1(s)−un(s)°
°2ds +λ2β2
2ZT
0
sup
0≤θ≤s
E¯¯un(θ)−un−1(θ)¯¯2ds. (3.27)
19
F om (3.20)-(3.27), we deduce ha {un}n≥1is a Cauchy sequence in L2(Ω; C(−h, T;H)).
Thus, he e exis s usuch ha un→uin I2(−h, T;V)∩L2(Ω; C(−h, T;H)). Now, by a
simila a gumen o he one in he p oo o heo em 2.1, we can deduce ha uis he solu ion
o p oblem (P).
Rema k 3.1. The hypo hesis conce ning he compac ness o he injec ion V⊂Hcan
be omi ed i , o example, ψ≡0.
Rema k 3.2. Theo ems 2.1. and 3.1. can be ex ended o he case in which W( ) is an
IRn- alued (o Hilbe alued) Wiene p ocess.
4. An example
To illus a e ou heo y, mainly Theo em 3.1, we shall conside he ollowing si ua ion,
which canno be handled wi h he esul s in Ca aballo [2] o Ca aballo e al. [4].
Assume O ⊂ IRnis a bounded open se . Le us se H=L2(O), V =H1
0(O) and
V∗=H−1(O).
Le φ: [0, T]×IRn→IRnbe a con inuous map such ha he e exis s cφ>0 such ha
|φ( , x)|IRn≤cφ|x|IRn o all ( , x)∈[0, T]×IRn, and suppose ha
(φ( , x)−φ( , y)) ·(x−y)≤0∀ ∈[0, T ],∀x, y ∈IRn,(4.1)
whe e we deno e by · he escala p oduc in IRn. I is easy o see ha he amily o ope a o s
A( , ·) de ined by
hA( , u), i=−ZO
∇u(x)· ∇ (x)dx −ZO
φ( , ∇u(x)) · ∇ (x)dx ∀ ∈[0, T],∀u, ∈V, (4.2)
sa is ies hypo heses (A.1)-(A.4), wi h λ= 0 and α≤2.
Le us conside now a measu able map k1: [0, T]×IR →IRnand a measu able unc ion
ω1: [0, T]→IR such ha 0 ≤ω1( )≤h o all ∈[0, T ].Suppose ha k1( , 0) = 0,∀ ∈[0, T ],
and ha he e exis s Lk1>0 such ha
|k1( , a)−k1( , b)|IRn≤Lk1|a−b|,∀ ∈[0, T],∀a, b ∈IR.(4.3)
Deno e by F1( , ·) he amily o ope a o s de ined by
hF1( , ξ), i=−ZO
k1( , ξ(−ω1( ))(x)) · ∇ (x)dx, ∀ξ∈C(−h, 0; H),∀ ∈V, (4.4)
o each ∈[0, T].
Then, he amily F1( , ·) sa is ies assump ions (F1.1)-(F1.4), wi h CF1=L2
k1and CF1=
L2
k1.
Conside also k2: [0, T]×IRn→IRn, measu able, and ω2∈C1([0, T]) such ha 0 ≤
ω2( )≤h o all ∈[0, T], and ω∗
2= max ∈[0,T ]ω0
2( )<1. Suppose ha k2( , 0) = 0,∀ ∈
[0, T], and ha he e exis s Lk2>0 such ha
|k2( , x)−k2( , y)|IRn≤Lk2|x−y|IRn,∀ ∈[0, T ],∀x, y ∈IRn.(4.5)
20
Deno e by F2( , ·) he amily o ope a o s de ined by
hF2( , ξ), i=−ZO
k2( , ∇ξ(−ω2( ))(x)) · ∇ (x)dx, ∀ξ∈C(−h, 0; V),∀ ∈V, (4.6)
o each ∈[0, T].Then, he amily F2( , ·) sa is ies hypo heses (F2.1)-(F2.4), wi h CF2=L2
k2
and KF2=L2
k2
1−ω∗
2
.
Finally, le l0: [0, T ]×IR →IR and l1: [0, T]×IRn→IR be wo measu able unc ions,
such ha l0( , 0) = l1( , 0) = 0 o all ∈[0, T], and he e exis Ll0>0 and Ll1>0 such ha
|l0( , a)−l0( , b)| ≤ Ll0|a−b|,∀ ∈[0, T],∀a, b ∈IR,(4.6)
and
|l1( , x)−l1( , y)| ≤ Ll1|x−y|IRn,∀ ∈[0, T ],∀x, y ∈IRn.(4.7)
Le us also ix ρi: [0, T]→IR, i= 0,1, wo measu able unc ions such ha 0 ≤ρi( )≤h o
all ∈[0, T] and i= 0,1, ρ1∈C1([0, T]), and ρ∗
1= max ∈[0,T ]ρ0
1( )<1.
Then, i we de ine
G0( , ξ)(x) = l0( , ξ(−ρ0( ))(x)),∀ ∈[0, T],∀ξ∈C(−h, 0; H),a.e. x ∈ O,(4.8)
and
G1( , ξ)(x) = l1( , ∇ξ(−ρ1( ))(x)),∀ ∈[0, T],∀ξ∈C(−h, 0; V),a.e. x ∈ O,(4.9)
i is easy o check ha G0sa is ies (G0.1)-(G0.4), and ha G1sa is ies hypo heses (G1.1)-
(G1.4), wi h CG0=CG0=L2
l0,CG1=L2
l1, and KG1=L2
l1
1−ρ∗
1
.
As o hypo hesis (A.5), i is ul illed wi h b
λla ge enough p o ided
2Lk2
p1−ω∗
2
+L2
l1
1−ρ∗
1
<2.(4.10)
Consequen ly, unde all he hypo heses abo e, we can ensu e ha gi en ψ∈I2(−h, 0; H1
0(O))∩
L2(Ω; C(−h, 0; L2(O))), ∈I2(0, T;H−1(O)), and g∈I2(0, T;L2(O)), he e exis s a unique
solu ion u∈I2(−h, T;H1
0(O)) ∩L2(Ω; C(−h, T;L2(O))) o he co esponding p oblem (P).
Such a solu ion, sa is ies, in a gene alized sense, he p oblem
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∂u( )
∂ = ∆u( ) + ∇ · (φ( , ∇u( ))) + ∇ · (k2( , ∇u( −ω2( )))) + ∇ · (k1( , u( −ω1( ))))
+ ( ) + (l1( , ∇u( −ρ1( ))) + l0( , u( −ρ0( ))) + g( )) ∂W ( )
∂ in O × (0, T ),
u(0) = 0 on ∂O × (0, T),
u( ) = ψ( ) in O × [−h, 0],
whe e, o a ec o unc ion ~ = ( 1, ..., n) de ined on O, we deno e by ∇ ·~ he di e gence o
~ de ined by ∇ · ~ =
n
X
i=1
∂ i
∂xi
.
21

Acknowledgemen . This wo k has been pa ly suppo ed by Jun a de Andalucia P ojec
FQM314.
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