Exis ence and uniqueness o solu ions o
Non–Linea S ochas ic Pa ial Di e en ial Equa ions
TOM´
AS CARABALLO GARRIDO
Depa amen o de An´alisis Ma em´a ico, Uni e sidad de Se illa,
Apa ado de Co eos 1.160.
41080-SEVILLA, Spain.
ABSTRACT:
We s a e some esul s on exis ence and uniqueness o he solu ion o non linea s ochas ic PDEs
wi h de ia ing a gumen s. In ac , we conside he equa ion
dx( )+(A( , x( )) + B( , x(τ( ))) + ( )) d = (C( , x(ρ( ))) + g( )) dw ,
whe e A( , .), B( , .)and C( , .)a e sui able amilies o non linea ope a o s in Hilbe spaces,
w is a Hilbe alued Wiene p ocess, and τ , ρ a e unc ions o delay. I Asa is ies a coe ci i y
condi ion and a mono onici y hypo hesis, and i B , C a e Lipschi z con inuous, we p o e ha
he e exis s a unique solu ion o an ini ial alue p oblem o he p eceden equa ion. Some examples
o in e es o he applica ions a e gi en o illus a e he esul s.
1. In oduc ion
The s udy o s ochas ic PDE’s has g ea ly de eloped o e he las yea s. S ochas ic PDE’s
a e used in modelling physical phenomena [5], popula ion biology [7], il e ing [13], e c.
The main aim o his pape is o s udy his ype o equa ion wi h delay e ms. In ac , we
p o e exis ence and uniqueness o solu ion (in I ˆo’s sense) o a a he gene al ype o s ochas ic
PDEs wi h non linea mono one ope a o s and wi h delays. We deal wi h he ollowing s ochas ic
pa abolic equa ion:
(1) ½dx( )+[A( , x( )) + B( , x(τ( ))) + ( )] d = [C( , x( ho( ))) + g( )] dw , > 0
x(0) = x0,
whe e A( , .), B( , .), C( , .) a e amilies o ope a o s in Hilbe spaces, non linea e en ually, and
sa is ying a mono onici y condi ion; w is a Hilbe alued Wiene p ocess, and τ , ρ a e delay
unc ions.
1
When he e a e no delays ( τ( ) = ρ( ) = ), he equa ion (1) has been s udied: in he case
B=C= 0, o Anon linea , in Bensoussan [2] and Cu ain [6], and o some ype o non linea
ope a o s A, in Bensoussan–Temam [3,4] and Ma cus [10]; in he case C6= 0 , B = 0, o linea
Aand C, in Balak ishann [1], o linea Aand non linea Cin Dawson [6], and o non linea
mono one Aand Lipschi z con inuous Cin Pa doux [12].
In he case wi h de ia ing a gumen s, Real [14,15] s udies a a he gene al case when all o
he ope a o s a e linea and he e exis s a e m which is a non con inuous ma ingale. Howe e ,
we ha e no ound in he li e a u e he case we a e going o analyze he e.
We will adap o ou p oblem one o he mos impo an me hod o sol ing non linea PDEs
(see Lions [9]): he mono onici y me hod. Pa doux [12] also used an adap a ion o ha me hod
o ano he ype o non linea mono one equa ions: when B= 0 and wi hou delays.
In Sec ion 2 we shall s a e he p oblem and he no a ion we a e going o use. Uniqueness
o solu ion will be p o ed, in Sec ion 3, using I ˆo’s o mula. In Sec ion 4, we s a e he exis ence
esul s. Some ex ensions o he esul s a e gi en in Sec ion 5. Finally, we illus a e ou heo y wi h
se e al impo an examples appea ing in he applica ions.
2. S a emen o he p oblem
The heo y o s ochas ic in eg als in Hilbe spaces is well de eloped (see [8], [12], o example).
The a ia ional me hod we a e going o use, o ces us o wo k wi h he classical pai o eal
sepa able Hilbe spaces V , H sa is ying V ,→H(injec ion con inuous and dense). We iden i y
Hwi h i s dual space, and deno e by V0 he dual o V. Then, we ha e
V ,→H≡H0,→V0.
We will deno e by k.k,|.|and k.k∗ he no ms in V,Hand V0 espec i ely; by h., .i he
duali y p oduc be ween V0, V , and by (.,.) he scala p oduc in H.
Le us ix T > 0 and, le w be a Wiene p ocess de ined on he comple e p obabili y space
(Ω,F, P) and aking alues in he sepa able Hilbe space K, wi h inc emen al co a iance ope a o
W. Le (F ) ≥0be he σ-algeb a gene a ed by {ws,0≤s≤ }, hen w is a ma ingale ela i e o
(F ) ≥0and we ha e he ollowing ep esen a ion o w :
w =
∞
X
i=1
βi
ei,
whe e (ei) is an o hono mal se o eigen ec o s o W,βi
a e mu ually independen eal Wiene
p ocesses wi h inc emen al co a iance λi>0, W ei=λieiand W=P∞
i=1 λi( deno es he
ace o an ope a o , see [8], [12], [13]).
2
As an abuse o no a ion, we also use |.| o he no m in he linea con inuous ope a o space
L(K, H).
We deno e by Ip(0, T;V), o p > 1, he space o V– alued p ocesses (x( )) ∈[0,T ](we will
w i e x( ) o sho ) measu able ( om [0, T ]×Ω in V), and sa is ying:
i) x( ) is F −measu able a.e. in (in he sequel, we will w i e a.e. .)
ii) ERT
0|x |pd < +∞.
I is easy o check ha Ip(0, T;V) is a closed subspace o Lp(Ω×[0, T],F ⊗B([0, T]), dP ⊗d ;V),
whe e by B([0, T]) we deno e he Bo el σ–algeb a.
Fo sho , we shall w i e L2(Ω; C(−h, T;H)) ins ead o L2(Ω,F, dP ;C(−h, T;H)) ,whe e
C(−h, T;H) deno es he space o con inuous unc ions om [−h, T] o H.
Le A( , .) : V→V0be a amily o non linea ope a o s de ined a.e. ., and le p > 1.We
make he ollowing hypo heses:
(a.1) Coe ci i y: ∃α > 0, λ ∈Rsuch ha :
2hA( , x), xi+λ|x|2≥αkxkp,∀x∈V , a.e. .
(a.2) Mono onici y: 2hA( , x)−A( , y), x −yi+λ|x−y|2≥0,∀x, y ∈V , a.e. .
(a.3) Boundedness: ∃β > 0 : kA( , x)k∗≤βkxkp−1,∀x∈V , a.e. .
(a.4) Hemicon inui y: θ∈R→ hA( , x +θy), zi ∈ Ris con inuous ∀x, y, z ∈V , a.e. .
(a.5) Measu abili y:
∈(0, T)→A( , x)∈V0is Lebesgue −measu able ∀x∈V , a.e. .
Le B( , .) : H→Hbe a amily o ope a o s de ined a.e. ., and sa is ying:
(b.1) B( , 0) = 0
(b.2) Lipschi z condi ion: ∃k1such ha
|B( , x)−B( , y)| ≤ k1|x−y|,∀x, y ∈H , a.e. .
(b.3) Measu abili y: ∈(0, T)→B( , x)∈His Lebesgue–measu able, ∀x∈V .
And le C( , .) : H→ L(K, H) be ano he amily de ined a.e. . and e i ying:
(c.1) C( , 0) = 0
(c.2) Lipschi z condi ion: ∃k2such ha
|C( , x)−C( , y)| ≤ k2|x−y|,∀x, y ∈H , a.e. .
(c.3) Measu abili y: ∈(0, T)→C( , x)∈ L(K, H) is Lebesgue–measu able ∀x∈H .
We also conside wo measu able unc ions (o delay) ρ, τ : [0, T]→[0, T] , such ha
(ρ.τ) 0 ≤ρ( ), τ( )≤ , ∀ ∈[0, T].
Fo , g we suppose ha
3
( .g) ∈I2(0, T;H), g ∈I2(0, T;L(K, H)).
And inally, we a e gi en an ini ial alue x0∈L2(Ω,F0, P;H).
Now, we s a e he ollowing p oblem:
(PC)


To ind a p ocess x∈Ip(0, T ;V)∩L2(Ω; C(0, T;H)) such ha :
x( ) + R
0[A(s, x(s)) + B(s, x(τ(s))) + (s)] ds
=x0+R
0[C(s, x(ρ(s))) + g(s)] dws, P −a.s., ∀ ∈[0, T ].
Rema k 2.1. We obse e ha , i x∈L2(0, T ;H) hen, by (b.1)–(b.3), B(x)∈L2(0, T;H) ,
whe e B(x)( ) = B( , x( )) .Mo eo e , x∈L2(0, T;H)→B(x)∈L2(0, T ;H) is con inuous,
and so, measu able. Since x∈H→B( , x)∈His con inuous a.e. ., i ollows ha , i x( )
is an H– alued s ochas ic p ocess and F –adap ed, hen B( , x( )) also is. In addi ion, i x∈
L2(Ω ×(0, T); H), hen B(x)∈L2(Ω ×(0, T ); H) oo. Finally, i xnis a bounded sequence in
L2(Ω ×(0, T); H), B(xn) also is bounded.
Simila obse a ions a e deduced om (c.1)–(c.3) o C:L2(0, T ;H)→L2(0, T;L(K, H))
de ined by C(x)( ) = C( , x( )) .
These ema ks and he measu abili y o ρand τimply ha he in eg als appea ing in (P C)
a e well de ined.
3. Uniqueness o solu ion
In his Sec ion we will p o e ha he e exis s a mos one solu ion o (PC). We will ob ain
his esul om (a.2) and I ˆo’s o mula (see [8], [13] o ha o mula).
Theo em 3.1
Assume he hypo heses in Sec ion 2. Then, he e exis s a mos one solu ion o (P C)in
Ip(0, T;V)∩L2(Ω; C(0, T ;H)) .
P oo . Suppose ha x, y ∈Ip(0, T;V)∩L2(Ω; C(0, T ;H)) a e solu ions o (PC). Then, applying
I ˆo’s o mula, we ob ain
E|x( )−y( )|2
(3.1)
=−2EZ
0
hA(s, x(s)) −A(s, y(s)), x(s)−y(s)ids
−2EZ
0
(B(s, x(τ(s))) −B(s, y(τ(s))), x(s)−y(s)) ds
+EZ
0
£(C(s, x(ρ(s))) −C(s, y(ρ(s)))) W(C(s, x(ρ(s))) −C(s, y(ρ(s))))∗¤ds.
4
Now, by pu ing z( ) = x( )−y( ) and using condi ions (a.2), (b.2), (c.2), i ollows
E|z( )|2≤λE Z
0
|z(s)|2ds + 2k1EZ
0
|z(τ(s))||z(s)|ds(3.2)
+k2
2 (W)EZ
0
|z(ρ(s))|2ds .
We a e going o es ima e he e ms on he igh -hand side o (3.2). Fi s ,
λE Z
0
|z(s)|2ds ≤ |λ|Z
0
sup
∈[0,s]
E|z( )|2ds.(3.3)
Using (ρ.τ) we ge
2EZ
0
|z(τ(s))||z(s)|ds ≤EZ
0
|z(τ(s))|2ds +EZ
0
|z(s)|2ds(3.4)
≤2Z
0
sup
∈[0,s]
E|z( )|2ds ,
EZ
0
|z(ρ(s))|2ds ≤Z
0
sup
∈[0,s]
E|z( )|2ds .(3.5)
Consequen ly, (3.2) −(3.5) yield
(3.6) sup
∈[0, ]
E|z( )|2≤£|λ|+ 2k1+ 2k2
2 (W)¤Z
0
sup
∈[0,s]
E|z( )|2ds .
Finally, G onwall’s Lemma implies
(3.7) sup
∈[0, ]
E|z( )|2= 0 ,∀ ∈[0, T].
Ob iously, uniqueness ollows om (3.7).
4. Exis ence o solu ion
Fi s , we s a e a heo em on exis ence and uniqueness o solu ion o a s ochas ic e olu ion
equa ion, and an ene gy equali y. Nex , we will p o e he exis ence o solu ion o (PC) using his
esul .
Theo em 4.1
Assume he hypo heses in Sec ion 2, wi h λ= 0 . Then, he e exis s a unique p ocess
x∈Ip(0, T;V)∩L2(Ω; C(0, T ;H)) such ha
x( ) + Z
0
[A(s, x(s)) + (s)] ds =x0+M , P −a.s. , ∀ ∈[0, T],
5

whe e M is a H– alued con inuous, squa e in eg able F –ma ingale. The solu ion also e i ies
he ollowing ene gy equali y:
|x( )|2+ 2 Z
0
hA(s, x(s)), x(s)ids + 2 Z
0
( (s), x(s)) ds(4.1)
=|x0|2+ 2 Z
0
(x(s), dMs) + hhMii , P −a.s. , ∀ ∈[0, T],
whe e hhMii deno es he quad a ic a ia ion o M (see M´e i ie and Pellaumail [11]).
P oo . See [11], [15] and he e e ences gi en he e.
Rema k 4.1. We obse e ha , in ou si ua ion (see he p oo o heo em 4.2), he ma ingale M
will be R
0g(s)dwsand hence, he ene gy equali y yields
E|x( )|2+ 2EZ
0
hA(s, x(s)), x(s)ids + 2EZ
0
( (s), x(s)) ds(4.2)
=E|x0|2+EZ
0
(g(s)Wg(s)∗)ds , P −a.s. , ∀ ∈[0, T].
In Pa doux [12,13] and Ichikawa [8] we can ind a a he gene al I ˆo’s o mula.
Now, using a Pica d’s scheme, we can p o e he exis ence o solu ion o he p oblem (PC).
Theo em 4.2
Assume he condi ions in Sec ion 2. Then, he e exis s a unique solu ion o (PC)in
Ip(0, T;V)∩L2(Ω; C(0, T ;H)) .
P oo . Uniqueness holds om heo em 3.1. Fo he exis ence, we conside he equa ions
x1( ) + Z
0·A(s, x1(s)) + λ
2x1(s)¸ds +Z
0
(s)ds(4.3)
=x0+Z
0
g(s)dws
xn+1( ) + Z
0·A(s, xn+1(s)) + λ
2xn+1(s)¸ds(4.4)
+Z
0
B(s, xn(τ(s))) ds +Z
0
(s)ds
=x0+Z
0
λ
2xn(s)ds +Z
0
C(s, xn(ρ(s))) dws
+Z
0
g(s)dws,∀n= 1,2,3, ...
By (a.1)–(a.5), he amily A1( , .) : V→V0de ined by A1( , x) = A( , x) + (λ/2)x , sa is ies
he assump ions in heo em 4.1. Consequen ly, (4.3) has a unique solu ion x1∈Ip(0, T;V)∩
L2(Ω; C(0, T;H)) .
We no e ha , om (b.2),(c.2) and he measu abili y o he unc ions ρ , τ , i ollows:
6
i) The map ( , ω)∈(0, T )×Ω→B( , x1(τ( )) ∈Hbelongs o I2(0, T;H).
ii) The map ( , ω)∈(0, T)×Ω→C( , x1(ρ( )) ∈Hbelongs o I2(0, T;L(K, H)) ,and so,
R·
0C(s, x1(ρ(s))) dwsis a con inuous and squa e in eg able F –ma ingale.
Again, by hese ema ks, we can apply heo em 4.1 and we ge ha he e exis s a unique
p ocess x2∈Ip(0, T;V)∩L2(Ω; C(0, T;H)) , which is solu ion o (4.4) o n= 1 . By ecu ence,
we ob ain a sequence o solu ions o (4.3) −(4.4) , {xn}n≥1⊂Ip(0, T;V)∩L2(Ω; C(0, T;H)) .
In he sequel, we shall p o e ha he sequence {xn}is con e gen o a p ocess in Ip(0, T;V)∩
L2(Ω; C(0, T;H)) , and his p ocess is he solu ion o (PC). We shall spli his p oo in ou s eps.
STEP 1.– {xn}is a Cauchy sequence in L2(Ω; C(0, T;H)).
Indeed, o n > 1, i ollows om I ˆo’s o mula o he p ocess xn+1( )−xn( ),
|xn+1( )−xn( )|2+ 2 Z
0
hA(xn+1)−A(xn), xn+1 −xnids(4.5)
+λZ
0
|xn+1 −xn|2ds + 2 Z
0¡B(xn
τ)−B(xn−1
τ), xn+1 −xn¢ds
=λZ
0¡xn+1 −xn, xn−xn−1¢ds + 2 Z
0¡xn+1 −xn,(C(xn
ρ)−C(xn−1
ρ)) dws¢
+Z
0
h¡C(xn
ρ)−C(xn−1
ρ)¢W¡C(xn
ρ)−C(xn−1
ρ)¢∗ids ,
whe e, by de ini ion: xn:= xn(s), A(xn) := A(s, xn(s)) , B(xn
τ) := B(s, xn(τ(s))) and
C(xn
ρ) := C(s, xn(ρ(s))) .
¿F om (a.2),
|xn+1( )−xn( )|2≤|λ|Z
0
|xn+1 −xn||xn−xn−1|ds(4.6)
+ 2 ¯¯¯¯Z
0¡xn+1 −xn,(C(xn
ρ)−C(xn−1
ρ)) dws¢¯¯¯¯
+Z
0¯¯¯ h¡C(xn
ρ)−C(xn−1
ρ)¢W¡C(xn
ρ)−C(xn−1
ρ)¢∗i¯¯¯ds
+ 2 Z
0
|B(xn
τ)−B(xn−1
τ)||xn+1 −xn|ds .
Consequen ly, (4.6) yields
E·sup
0≤θ≤
|xn+1(θ)−xn(θ)|2¸≤ |λ|EZ
0
|xn+1 −xn||xn−xn−1|ds(4.7)
+ (W)EZ
0
|C(xn
ρ)−C(xn−1
ρ)|2ds
+ 2EZ
0
|B(xn
τ)−B(xn−1
τ)||xn+1 −xn|ds
+ 2E·sup
0≤θ≤ ¯¯¯¯Z
0¡xn+1 −xn,(C(xn
ρ)−C(xn−1
ρ)) dws¢¯¯¯¯¸.
Now, we es ima e he e ms on he igh -hand side o (4.7), and we apply he inequali y
2ab ≤a2
l2+l2b2, a, b ∈R, l > 0,
7
o sui able l.
|λ|EZ
0
|xn+1 −xn||xn−xn−1|ds(4.8)
≤1
4E·sup
0≤θ≤
|xn+1(θ)−xn(θ)|2¸+λ2TZ
0
E·sup
0≤θ≤s
|xn(θ)−xn−1(θ)|2¸ds .
(W)EZ
0
|C(xn
ρ)−C(xn−1
ρ)|2ds(4.9)
≤ (W)k2
2Z
0
E·sup
0≤θ≤s
|xn(θ)−xn−1(θ)|2¸ds .
2EZ
0
|B(xn
τ)−B(xn−1
τ)||xn+1 −xn|ds(4.10)
≤1
4TEZ
0
|xn+1 −xn|2ds + 4k2
1TE Z
0
|xn(τ(s)) −xn−1(τ(s))|2ds
≤1
4E·sup
0≤θ≤
|xn+1(θ)−xn(θ)|2¸+ 4k2
1TZ
0
E·sup
0≤θ≤s
|xn(θ)−xn−1(θ)|2¸ds .
Bu kholde –Da is–Gundy’s inequali y implies
2E·sup
0≤θ≤ ¯¯¯¯Z
0¡xn+1 −xn,(C(xn
ρ)−C(xn−1
ρ)) dws¢¯¯¯¯¸
(4.11)
≤6 (W)E·µsup
0≤θ≤
|xn+1(θ)−xn(θ)|2¶Z
0
|C(xn
ρ)−C(xn−1
ρ)|2ds¸1/2
≤1
4E·sup
0≤θ≤
|xn+1(θ)−xn(θ)|2¸+ 36k2
2 (W)Z
0
E·sup
0≤θ≤s
|xn(θ)−xn−1(θ)|2¸ds .
I we se
(4.12) ϕn( ) = E·sup
0≤θ≤
|xn+1(θ)−xn(θ)|2¸,
hen, (4.7) −(4.11) yield
(4.13) ϕn( )≤3
4ϕn( ) + (λ2T+k2
2 (W) + 4k2
1T+ 36k2
2)Z
0
ϕn−1(s)ds ,
and so, he e exis s k > 0 such ha
(4.14) ϕn( )≤kZ
0
ϕn−1(s)ds .
By i e a ion om (4.14), we ge
(4.15) ϕn( )≤kn−1Tn−1
(n−1)! ϕ1(T),∀n > 1,∀ ∈[0, T].
The e o e,
(4.16) E·sup
0≤θ≤T
|xn+1(θ)−xn(θ)|2¸≤kn−1Tn−1
(n−1)! ϕ1(T),∀n > 1.
8
Ob iously, (4.16) implies ha {xn}is a Cauchy sequence in L2(Ω; C(0, T ;H)) .
STEP 2.– The sequence {xn}is bounded in Ip(0, T;V).
Indeed, I ˆo’s o mula o |xn|2,wi h n≥2, yields
E|xn(T)|2+ 2EZT
0
hA(xn), xnids +λE ZT
0
|xn|2ds(4.17)
=E|x0|2−2EZT
0
(B(xn−1
τ), xn)ds −2EZT
0
( , xn)ds
+λE ZT
0
(xn, xn−1)ds +EZT
0
h¡C(xn−1
ρ) + g¢W¡C(xn−1
ρ) + g¢∗ids .
The e o e,
2EZT
0
hA(xn), xnids +λE ZT
0
|xn|2ds(4.18)
≤E|x0|2+ 2EZT
0
|B(xn−1
τ)||xn|ds + 2EZT
0
| ||xn|ds
+|λ|EZT
0
|xn||xn−1|ds + (W)EZT
0
|C(xn−1
ρ) + g|2ds .
Since {xn}is con e gen in L2(Ω; C(0, T;H),i will be bounded in his space. Now, i is no
di icul o check ha he e exis s a posi i e cons an , k0,such ha he igh -hand side o (4.18)
is bounded by his cons an . As an example, we will es ima e one o hose e ms:
2EZT
0
|B(xn−1
τ)||xn|ds ≤2k1EZT
0
|xn−1(τ(s))||xn(s)|ds
≤k1EZT
0£|xn−1(τ(s))|2+|xn(s)|2¤ds
≤k1EZT
0·sup
0≤θ≤T
|xn−1(θ)|2+ sup
0≤θ≤T
|xn(θ)|2¸ds
≤Tk1·Eµsup
0≤θ≤T
|xn−1(θ)|2¶+Eµsup
0≤θ≤T
|xn(θ)|2¶¸
≤Tk1³kxnk2
L2(Ω;C(0,T ;H)) +kxn−1k2
L2(Ω;C(0,T ;H))´.
This ac , (4.18) and (a.1) lead o he ollowing inequali ies:
(4.19) αZT
0
Ekxn(s)kpds ≤2EZT
0
hA(xn), xnids +λE ZT
0
|xn|2ds ≤k0,
and S ep 2 is p o ed.
STEP 3.– We can ake limi s in (4.4).
Indeed, om S ep 1, xn→x , o some xin L2(Ω; C(0, T ;H)) .Since (b.2) and
(c.2) hold, we also ha e B(xn
τ)→B(xτ) (in L2(Ω; L∞(0, T;H)) ), and C(xn
ρ)→C(xρ)
(in L2(Ω; L∞(0, T;L(K, H))) ).
On he o he hand, by S ep 2, {xn}has a subsequence which is weakly con e gen in
Ip(0, T;V) . Bu , since xn→xin L2(Ω; C(0, T;H)) ,we can assu e ha xn→xweakly in
9
such ha
du( , x)−∆u( , x)d +u3( , x)d +ϕ1(u(τ( ), x)) d
=ϕ2(u(ρ( ), x)) dw ,in (0, T)× O ,
u( , x) = ψ( , x) in (−h, 0) × O ,
u( , x) = 0 in (−h, T)×∂O.
Example 6.3.– A s ochas ic non–linea mono one pa abolic equa ion.
Le p > 2 . Now, we conside V=W1,p
0(O),and we de ine A:V→V0by
hA(u), i=
N
X
i=1 ZO¯¯¯¯
∂u
∂xi¯¯¯¯
p−2∂u
∂xi
∂
∂xi
dx +ZO
|u|p−2u dx , ∀u, ∈V ,
and le B , C , ρ , τ , w as in example 6.2. I is easy o check ha (a.1)–(a.5) hold, wi h α=
1/2, λ = 0 and β=N+ 1 . Consequen ly, gi en ψ∈Ip(−h, 0; V)∩L2(Ω; C(−h, 0; H)) and
, g ∈I2(0, T;H) , he e exis s a unique p ocess uin Ip(−h, T;V)∩L2(Ω; C(−h, T;H)) solu ion
o (PC)0. In o he wo ds,
du( , x) = ÃN
X
i=1
∂
∂xiﯯ¯
∂u( , x)
∂xi¯¯¯¯
p−2∂u( , x)
∂xi!d +|u( , x)|p−2u( , x)!d
+ (ϕ1(u(τ( ), x)) + ( , x)) d + (ϕ2(u(ρ( ), x)) + g( , x)) dw ,in (0, T)× O ,
u( , x) = ψ( , x),in (−h, 0) × O ,
u( , x) = 0 ,in (−h, T)×∂O.
Re e ences
[1] A. Balak ishnan, S ochas ic bilinea pa ial di e en ial equa ions, U.S.–I aly Con e ence on
Va iable S uc u e Sys ems, O egon (1974).
[2] A. Bensoussan, Fil age op imal des sys emes lin´eai es, Dunod.
[3] A. Bensoussan and R. Temam, Equa ions aux d´e i ´ees pa ielles s ochas iques non lin´eai es,
Is ael J. Ma h.,11 (1972), 95–129.
[4] A. Bensoussan and R. Temam, Equa ions s ochas iques du ype Na ie –S okes, J. Func ional
Analysis, 13,2 (1973), 195–222.
[5] P.L. Chow, S ochas ic Pa ial Di e en ial Equa ions: Tu bulence and Rela ed P oblems, P ob.
Analysis and Rela ed Topics, 1, A.T. Bha ucha-Reid, Academic P ess, New Yo k (1978).
16

[6] R. Cu ain, S ochas ic di e en ial equa ions in Hilbe spaces, Ph. D. Thesis, B own Uni e si y
(1969).
[6] D. Dawson, S ochas ic e olu ion equa ion, Ma h. Biosc.,15 (1972)
[7] W.H. Fleming, Dis ibu ed Pa ame e S ochas ic Sys ems in Popula ion Biology, Lec u e No es
in Economics and Ma hema ical Sys ems, ol. 107, Sp inge , Be lin-New Yo k (1975).
[8] A. Ichikawa, S abili y o Semilinea S ochas ic E olu ion Equa ions, J.Ma h.Anal.Appl.90
(1982), 12-44.
[9] J.L. Lions, Quelque m´e hodes de ´esolu ion des p oblemes aux limi es non lineai es, Dunod
Gau hie –Villa s, Pa is (1969).
[10] R. Ma cus, Pa abolic I o equa ions, T ans. Am. Ma h. Soc.,198 (1974), 177–190.
[11] M. M´e i ie and J. Pellaumail, S ochas ic In eg a ion, Academic P ess, New Yo k, (1980).
[12] E. Pa doux, ´
Equa ions aux D´e i ´ees Pa ielles S ochas iques non Lin´eai es Mono ones, Thesis,
Uni e si y o Pa is XI (1975).
[13] E. Pa doux, S ochas ic Pa ial Di e en ial Equa ions and Fil e ing o Di usion P ocesses,
S ochas ics 3, (1979) 127-167.
[14] J. Real, Con ibuci´on al es udio de una clase de Ecuaciones en De i adas Pa ciales Es-
oc´as icas con Re a do, Thesis, Uni e si y o Se illa (1980).
[15] J. Real, S ochas ic Pa ial Di e en ial Equa ions wi h Delays, S ochas ics 8, 2 (1982-83),
81-102.
[16] M. Vio , Solu ions aibles d’´equa ions aux d´e i ´ees pa ielles s ochas iques non lin´eai es, The-
sis, Uni e si y o Pa is VI (1976).
17