Non–Linea S ochas ic Pa ial Di e en ial Equa ions wi h delays:
Exis ence and uniqueness o solu ions
Tom´as Ca aballo Ga ido
Dp o. de An´alisis Ma em´a ico. Facul ad de Ma em´a icas (Uni e sidad de Se illa).
Apa ado de co eos 1.160. 41080–Se illa
Clasi icaci´on A.M.S.: 60H, 35K.
1. In oduc ion
The main aim o his pape is o s udy s ochas ic PDE’s 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(ρ( ))) + 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.
When he e a e no delays ( τ( ) = ρ( ) = 0 ), he equa ion (1) has been s udied: in he case
B=C= 0, o Anon linea , in Bensoussan [2] and Cu ain [5], and o some ype o non linea
ope a o s A, in Bensoussan–Temam [3] and Ma cus [7]; 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 [8].
In he case wi h de ia ing a gumen s, Real [9] 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:
he mono onici y me hod. Pa doux [8] 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.
2. S a emen o he p oblem and he main esul s
The heo y o s ochas ic in eg als in Hilbe spaces is well de eloped (see [8], o example).
We conside he classical pai o eal sepa able Hilbe spaces V , H sa is ying V ,→H
(injec ion con inuous and dense).
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 ) ≥0.
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 < +∞.
Fo sho , we shall w i e L2(Ω; C(−h, T;H)) ins ead o L2(Ω,F, dP ;C(−h, T;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 ∈R: 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: ∃k1:|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: ∃k2:|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
( .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 ].
The main esul we p o e is he ollowing heo em
Theo em 1
Assume he p eceden condi ions. Then, he e exis s a unique solu ion o (P C)in
Ip(0, T ;V)∩L2(Ω; C(0, T;H)) .
P oo . (See [4]) Uniqueness ollows om I o’s o mula and G onwall’s inequali y. Fo he exis ence,
we conside he equa ions
(∗)x1( ) + Z
0·A(s, x1(s)) + λ
2x1(s)¸ds +Z
0
(s)ds =x0+Z
0
g(s)dws
xn+1( ) + Z
0·A(s, xn+1(s)) + λ
2xn+1(s)¸ds +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, ...
and we p o e ha he e exis s a sequence o solu ions o (∗)−(∗∗) , {xn}n≥1⊂Ip(0, T;V)∩
L2(Ω; C(0, T;H)) .
Las , we p o e ha he sequence {xn}is con e gen in Ip(0, T ;V)∩L2(Ω; C(0, T;H)) , and
he limi p ocess is he solu ion o (P C).
Rema k 1.– We obse e ha heo em 1 also holds when Vis a sepa able and e lexi e Banach
space wi h V ,→H .
Rema k 2.– We no e ha heo em 1 holds when ρ , τ ake nega i e alues.
Theo em 2
Assume he hypo heses in heo em 1, bu changing (ρ.τ)by he ollowing:
∃h > 0such ha −h≤τ( ), ρ( )≤ , ∀ ∈[0, T ],
and le ψbe a p ocess such ha ψ∈Ip(−h, 0; V)∩L2(Ω; C(−h, 0; H)) (whe e hese spaces
a e de ined in he ob ious manne , se ing F =F0,∀ ∈[−h, 0] ). Then, he e exis s a unique
p ocess x∈Ip(−h, T ;V)∩L2(Ω; C(−h, T ;H)) such ha ,
(PC)0


x( ) + R
0[A(s, x(s)) + B(s, x(τ(s))) + (s)] ds
=ψ(0) + R
0[C(s, x(ρ(s))) + g(s)] dws, P −a.s., ∀ ∈[0, T ],
x( ) = ψ( ), ∈(−h, 0]
P oo . See Ca aballo [4]
Rema k 3.– Some examples a e gi en in Ca aballo [4] in o de o jus i y he esul s.
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] T. Ca aballo, Exis ence and uniqueness o solu ions o non–linea sx ochas ic PDE’s, o
appea in Collec anea Ma hema ica.
[5] 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] R. Ma cus, Pa abolic I o equa ions, T ans. Am. Ma h. Soc.,198 (1974), 177–190.
[8] 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).
[9] 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.
To appea in Collec anea Ma hema ica