No e CRAS p ojec
´
Equa ions aux d´e i ´ees pa ielles/Pa ial Di e en ial Equa ions
(Sys `emes dynamiques/Dynamical Sys ems)
Global a ac o s o mul i alued andom semi lows gene -
a ed by andom di e en ial inclusions wi h addi i e noise
Tom´as CARABALLO a, Jos´e A. LANGA a, Jos´e VALERO b
aDp o. Ecuaciones Di e enciales y An´alisis Num´e ico, Uni e sidad de Se illa, Apdo. Co eos 1160,
41080-Se illa (Spain) E-mails : ca aball@nume .us.es ; langa@nume .us.es
bUni e sidad Ca denal He e a CEU, Comisa io 3, 03203-Elche, Alican e (Spain)
Abs ac . We in oduce he concep o mul i alued andom dynamical sys em (MRDS) as a mea-
su able mul i alued low sa is ying he cocycle p ope y. We show how his is a sui -
able amewo k o he s udy o he asymp o ic beha iou o some mul i alued s ochas ic
pa abolic equa ions by gene alizing he concep o global andom a ac o o he case o
a MRDS. c
°Acad´emie des Sciences/Else ie , Pa is
A ac eu s globaux pou des semi- lo s al´ea oi es mul i alu´es
engend ´es pa des inclusions di ´e en ielles s ochas iques a ec
un b ui addi i
R´esum´e. Dans ce e No e, on p ´esen e la no ion de Sys `eme Dynamique Al´ea oi e Mul i alu´e
(MRDS) comme un lo mesu able mul i alu´e qui sa is ai la p op i´e ´e du cocycle. Pa
une g´en´e alisa ion du concep d’a ac eu global al´ea oi e dans le cas d’un MRDS, on
mon e que ce e no ion es bien adap ´ee pou l’´e ude du compo emen asymp o ique
de quelques equa ions s ochas iques pa aboliques mul i alu´ees. c
°Acad´emie des Sci-
ences/Else ie , Pa is
F ench Ab idged Ve sion
Ce e No e a comme objec i ondamen al l’analyse du compo emen asymp o ique de quelques
inclusions di ´e en ielles s ochas iques. D’abo d, nous d´e eloppons une h´eo ie similai e `a celle de
C auel e Flandoli [8] pou le cas des sys `emes dynamiques uni alu´es. Nous in oduisons la no ion
de Sys `eme Dynamique Al´ea oi e Mul i alu´e (MRDS), ou Semi- lo Al´ea oi e Mul i alu´e, comme
une mul i-applica ion mesu able qui sa is ai la p op i´e ´e du cocycle (De ini ion 1). Ensui e, nous
d´emon ons que l’exis ence d’un ensemble al´ea oi e abso ban e compac , ainsi que la semicon i-
nui ´e sup´e ieu e du MRDS, impliquen l’exis ence d’un d’a ac eu global al´ea oi e (Th´eo `eme 2
e De ini ion 4).
Au pa ag aphe 3, nous ´e udions une inclusion di ´e en ielle s ochas ique a ec un b ui addi i
( oi (1)), e nous d´emon ons que, sous ce aines hypo h`eses su les op´e a eu s Ae F(su ou ,
m-dissipa i i ´e pou Ae ca ac `e e lipschi zien pou F), l’inclusion di ´e en ielle (1) engend e un
MRDS. La cons uc ion es bas´ee su un changemen de a iables ad´equa , qui an o me l’inclusion
s ochas ique dans une au e inclusion d´e e minis e d´ependan d’un pa am`e e, `a laquelle on peu
applique les ´esul a s de la h´eo ie d´e e minis e. U ilisan la no ion de solu ion in ´eg ale du
T. Ca aballo, J.A. Langa, J. Vale o
sys `eme ans o m´e ( oi De ini ions 5-7), nous mon ons commen on peu cons ui e un semi- lo
al´ea oi e mul i alu´e.
Pou e mine , sous une hypo h`ese de dissipa i i ´e, e une condi ion de compaci ´e su le MRDS,
nous d´emon ons l’exis ence d’un ensemble al´ea oi e abso ban e compac , ce qui assu e l’exis ence
d’un a ac eu global al´ea oi e.
Une ´e ude plus de aill´ee, a ec des applica ions `a des inclusions s ochas iques de ´eac ion-di usion,
se a publi´ee p ochainemen ( oi [6]).
1. In oduc ion
The s udy o he quali a i e beha iou o o dina y and pa ial di e en ial equa ions is one o
he mos de eloped b anches in his ield. When a phenomenon om Physics, Chemis y, Biology,
Economics can be desc ibed by a sys em o di e en ial equa ions whe e he exis ence o global
solu ions can be assu ed, one o he mos in e es ing p oblems is o know wha is he asymp o ic
beha iou o he sys em when ime g ows o in ini e. The s udy o he asymp o ic beha iou o he
sys em is gi ing us ele an in o ma ion abou “ he u u e” o he phenomenon desc ibed in he
model. In his con ex , he concep o global a ac o has become a e y use ul ool o desc ibe he
long- ime beha iou o many impo an di e en ial equa ions (see, among o he s, Ladyzhenkaya
[11], Hale [10], Temam [14]).
Howe e , some di icul ies appea when we ha e o wo k wi hou uniqueness o solu ions in he
sys em o when he model is be e desc ibed by, o ins ance, a di e en ial inclusion. In hese
cases, i has been shown ha he heo y o mul i alued lows makes sui able he ea men o he
asymp o ic beha iou o hese di e en ial equa ions and inclusions (Melnik and Vale o [12], Vale o
[16]).
A new and di e en di icul y appea s when a andom e m is added o he de e minis ic equa ion,
a whi e noise o ins ance, so ha he co esponding s ochas ic pa ial di e en ial equa ion mus be
ea ed in a di e en way. Fi s ly, he equa ion becomes nonau onomous, which makes necessa y
he in oduc ion o a p ocess ins ead o a semig oup. Mo eo e , he s ong dependence on he
andom e m adds ano he di icul y. The new and apidly g owing heo y o andom dynamical
sys ems (A nold [2]) has become he app op ia e ool o he s udy o many impo an andom
and s ochas ic equa ions. In his amewo k, C auel and Flandoli [8] (see also Schmal uss [13])
in oduced he concep o andom a ac o as a p ope gene aliza ion (see Ca aballo e al. [5]) o
he co esponding (de e minis ic) global a ac o .
The join ea men o mul i alued unc ions and s ochas ic e ms in a di e en ial equa ion makes
di icul e en he exis ence and uniqueness o solu ions o hese sys ems (Ahmed [1], Da P a o and
F ankowska [9], among o he s). In his pape we show how some s ochas ic di e en ial inclusions
gene a e mul i alued andom semi lows o mul i alued andom dynamical sys ems (MRDS) and
s udy he asymp o ic beha iou o some s ochas ic di e en ial inclusions by p e iously in oducing
he co esponding concep o andom a ac o o his case. The p oo s o he esul s in his No e
and a de ailed analysis o he p oblem (including some applica ions) can be ound in [6].
2. Mul i alued andom dynamical sys ems and a ac o s
Le (X, dX) be a comple e and sepa able me ic space wi h he Bo el σ-algeb a B(X). Le
(Ω,F,P) be a p obabili y space and θ : Ω →Ω a measu e p ese ing g oup o ans o ma ions in
Ω such ha he map ( , ω)7→ θ ωis measu able and sa is ying
θ +s=θ ◦θs=θs◦θ ;θ0=Id.
2
Global a ac o s o mul i alued semi lows
The pa ame e akes alues in Rendowed wi h he Bo el σ-algeb a B(R).
De ini ion 1. – A se alued map G:R+×Ω×X→C(X) (C(X) deno es he se o
non-emp y closed subse s o X) is called a mul i alued andom dynamical sys em (MRDS) o a
mul i alued andom semi low i is measu able (see Aubin and F ankowska [4], de ini ion 8.1.1) and
i sa is ies
i) G(0, ω) = Id on X;
ii) G( +s, ω)x=G( , θsω)G(s, ω)x(cocycle p ope y) o all , s ∈R+, x ∈X, ω ∈Ω.
Rema k. When ii) holds iden ically, we call Gape ec cocycle. We call Gac ude cocycle i ii)
holds o ixed sand all ∈R+, x ∈X, P−a.s. (whe e he excep ional se Nscan depend on s).
We call Ga e y c ude cocycle i ii) holds o ixed s, ∈R+, o all x∈X, P−a.s. (whe e he
excep ional se Ns, can depend on bo h sand ).
De ini ion 2. – The MRDS Gis said o be uppe semicon inuous i o all ∈R+and ω∈Ω
i ollows ha gi en x∈Xand a neighbou hood o G( , ω)x,O(G( , ω)x), he e exis s δ > 0 such
ha i dX(x, y)< δ, hen G( , ω)y⊂ O(G( , ω)x).
On he o he hand, Gis called lowe semicon inuous i o all ∈R+and ω∈Ω,gi en xn→x
(n→+∞) and y∈G( , ω)x, he e exis s yn∈G( , ω)xnsuch ha yn→y.
I is said o be con inuous i i is uppe and lowe semicon inuous.
Now, we ex end he concep o andom a ac o o he case o a MRDS and show a gene al
esul o he exis ence and uniqueness o a ac o s. Fi s ly we need some de ini ions.
De ini ion 3. – Aclosed andom se Dis a measu able map D: Ω →C(X) in he sense o
Cas aing and Valadie [7], ha is, gi en x∈X he map ω∈Ω7→ dis (x, D(ω)) is measu able.
A closed andom se D(ω) is said o be nega i ely ( esp. s ic ly)in a ian o he MRDS Gi
D(θ ω)⊂G( , ω)D(ω) ( esp. D(θ ω) = G( , ω)D(ω)),∀ ∈R+, ω ∈Ω.
Le us assume he ollowing condi ions o he MRDS G:
(H1) The e exis s an abso bing andom compac se B(ω), ha is, o P−almos all ω∈Ω and
e e y bounded se D⊂X, he e exis s D(ω) such ha G( , θ− ω)D⊂B(ω), o all ≥ D(ω).
(H2) G( , ω) : X→C(X) is uppe semicon inuous, o all ∈R+and ω∈Ω.
De ine he limi se o a bounded se D⊂Xas
ΛD(ω) = ∩T≥0∪ ≥TG( , θ− ω)D.
We now ha e he ollowing p oposi ion conce ning some p ope ies o he se ΛD(ω):
P oposi ion 1. – Assume condi ions (H1) and (H2) hold. Then, o P−almos all ω∈Ωand
e e y D⊂Xbounded, i ollows:
i) ΛD(ω)⊂B(ω)is non oid and compac .
ii) ΛD(ω)is nega i ely in a ian . I in addi ion Gis lowe semicon inuous, hen ΛD(ω)is s ic ly
in a ian .
iii) ΛD(ω)a ac s D, i.e.,
lim
→+∞dis (G( , θ− ω)D, ΛD(ω)) = 0.
De ini ion 4. – The closed andom se ω7→ A(ω) is a global andom a ac o o he MRDS
Gi P−a.s.
a) G( , ω)A(ω) = A(θ ω), o all ≥0,( ha is, i is s ic ly in a ian );
3
T. Ca aballo, J.A. Langa, J. Vale o
b) o all bounded D⊂X,
lim
→+∞dis (G( , θ− ω)D, A(ω)) = 0;
c) A(ω) is compac .
We can now s a e he ollowing heo em on he exis ence o andom a ac o s o MRDS:
Theo em 2. – Le (H1)−(H2) hold, he map ( , ω)∈R+×Ω7→ G( , ω)Dbe measu able o all
de e minis ic bounded se s D⊂X, and he map x∈X7→ G( , ω)xha e compac alues. Then,
A(ω) = ∪boundedD⊂XΛD(ω)
is a global andom a ac o o G(measu able wi h espec o F). I is unique and he minimal
closed a ac ing se .
3. MRDS gene a ed by a s ochas ic di e en ial inclusion wi h addi i e noise
3.1 Gene a ion
Le Xbe a eal sepa able Hilbe space wi h he scala p oduc h·,·i and he no m k·k. Conside
he ollowing s ochas ic di e en ial inclusion





du
d ∈Au ( ) + F(u( )) +
m
X
i=1
φi
dwi( )
d , ∈(0, T ),
u(0) = u0,
(1)
whe e A:D(A)→Xis a linea ope a o , φi∈D(A) and wi( ) a e independen wo-sided, i.e.
∈R, eal Wiene p ocesses wi h wi(0) = 0, i= 1, ..., m.
Le us in oduce he nex condi ions:
(A) The ope a o Ais m-dissipa i e, i.e. ∀y∈D(A),hAy, yi ≤ 0,and Im(A−λI) = X, ∀λ > 0.
(F1) F:X→C (X), whe e C (X) is he se o all non-emp y, bounded, closed, con ex subse s o
X.
(F2) The map Fis Lipschi z on D(A), i.e. ∃C≥0 such ha ∀y1, y2∈D(A)
dis H(F(y1), F (y2)) ≤Cky1−y2k,
whe e dis H(·,·) deno es he Hausdo me ic o bounded se s.
Le ζ( ) = Pm
i=1 φiwi( ). Le us conside he Wiene p obabili y space (Ω,F,P) de ined by
Ω = {ω= (w1(·), ..., wm(·)) ∈C(R,Rm)|ω(0) = 0},equipped wi h he Bo el σ−algeb a F, he
Wiene measu e P,and he usual uni o m con e gence on bounded se s o R. Each ω∈Ω gene a es
a map ζ(·) = Pm
i=1 φiwi(·)∈C(R, X) such ha ζ(0) = 0.
Making he change o a iable ( ) = u( )−ζ( ), inclusion (1) u ns in o





d
d ∈A ( ) + F( ( ) + ζ( )) +
m
X
i=1
Aφiwi( ),
(0) = 0=u0.
(2)
De ining he mul i alued map e
F: [0, T ]×Ω×X→C (X),by e
F( , ω, x) = F(x+ζ( ))+Aζ ( ),
i is easy o ob ain om (F2) ha e
Fsa is ies (F1), (F2) and he nex p ope y:
4
Global a ac o s o mul i alued semi lows
(F3) Fo any x∈X he e exis s n(·)∈L1(0, T ) depending on xand ωsuch ha °
°
°e
F( , ω, x)°
°
°
+
≤
n( ), a.e. in (0, T ).
Conside also he p oblem 




d ( )
d =A ( ) + ( ),
(0) = 0,
(3)
whe e (·)∈L1([0, T ], X).
De ini ion 5. – The unc ion u: [0, T ]→Xis called a s ong solu ion o p oblem (3) i :
i) u(·) is con inuous on [0, T ] and u(0) = u0;
ii) u(·) is absolu ely con inuous on any compac subse o (0, T ) and almos e e ywhe e (a.e.)
di e en iable on (0, T );
iii) u(·) sa is ies (3) a.e. on (0, T ).
De ini ion 6. – The con inuous unc ion : [0, T ]→Xis called an in eg al solu ion o p oblem
(3) i (0) = 0and
k ( )−ξk2≤ k (s)−ξk2+ 2 Z
s
h (τ) + Aξ, (τ)−ξidτ, ≥s, ∀ξ∈D(A).(4)
I is well known (see Ba bu [4, p.124]) ha any s ong solu ion o p oblem (3) is an in eg al
solu ion.
De ini ion 7. – The p ocess : [0, T ]×Ω→Xis said o be an in eg al solu ion o p oblem
(2) i o any ω∈Ω he map (·) = (·, ω) : [0, T ]→Xis con inuous, (0) = 0,and o some
selec ion ∈L1([0, T ], X), ( )∈e
F( , ω, ( )) a.e. on (0, T ), he inequali y (4) holds.
In wha ollows, we will omi ωi no con usion is possible.
I condi ion (A) holds and ∈L1([0, T ], X), hen ∀ 0∈D(A) he e exis s a unique in eg al
solu ion (·) o (3) o each T > 0 (see Ba bu [4, p.124]). We shall deno e his solu ion by
(·) = I( 0) (·). I (A),(F1) −(F3) hold, hen ∀ 0∈D(A) he e exis s a leas one in eg al
solu ion (·) = I( 0) (·) o (2) o each T > 0 (see Tols onogo [15], Theo em 3.1), so each
solu ion can be ex ended on [0,∞). Le us deno e by D( 0, ω) he se o all in eg al solu ions o
(2) such ha (0) = 0. We de ine he maps G:R+×Ω×D(A)→P(D(A)), θs: Ω →Ω as
ollows
G( , ω) 0={ ( ) + ζ( )| (·)∈ D( 0, ω)},
θsω= (w1(s+·)−w1(s), ..., wm(s+·)−wm(s)) ∈Ω.
Then he unc ion e
ζco esponding o θsωis e
ζ(τ) = ζ(s+τ)−ζ(s) = Pm
i=1 φi(wi(s+τ)−wi(s)) .
Theo em 3. – Le (A),(F1),(F2) hold. Then Gsa is ies he cocycle p ope y. Mo eo e , i
he semig oup S( , ·)gene a ed by he ope a o Ais compac , hen Ggene a es a MRDS.
3.2 Exis ence o a andom a ac o
As in he de e minis ic case, a dissipa i e assump ion will imply he exis ence o a compac
abso bing se and, as a consequence o Theo em 2, he exis ence o he andom a ac o o he
mul i alued andom semi low. Howe e , o his end mo e egula i y o he in eg al solu ions is
needed.
P oposi ion 4. – Le (A),(F1),(F2) hold. Suppose ha each in eg al solu ion o (2), (·) =
I(u0) (·)is a s ong solu ion o (3). Le he e exis cons an s δ > 0, M ≥0such ha ∀u∈D(A),
y∈F(u),
hy, ui ≤ (−δ+ε)kuk2+M,
5

T. Ca aballo, J.A. Langa, J. Vale o
whe e ε≥0is he bigges cons an such ha
hAu, ui ≤ −εkuk2,∀u∈D(A).
Then he e exis s a andom adius (ω)>0such ha o P−almos all ω∈Ωand any bounded
se B⊂D(A)we can ind T(B) = T(B, ω)≥1 o which
kG(−1 + 0, θ− 0ω)u0k+≤ (θ−1ω),∀ 0≥T(B),∀u0∈B.
Theo em 5. – Le he condi ions o P oposi ion 4 hold, he semig oup S( , ·)gene a ed by he
ope a o Abe compac and he mul i alued map G(1, ω)be compac ( ha is, i maps bounded se s
in o p ecompac ones). Then, Ghas he minimal global andom a ac o A(ω). Mo eo e , i is
measu able wi h espec o F.
Rema k. This abs ac esul is applied in [6] o s ochas ic eac ion-di usion inclusions.
Acknowledgemen . The au ho s wish o hank P o . J. Real o help ul sugges ions on his
wo k and o his in aluable help wi h he F ench e sion.
Re e ences
[1] Ahmed N.U., Nonlinea s ochas ic di e en ial inclusions on Banach space, S och. Anal. Appl. 12 (1994), 1-10.
[2] A nold L., Random Dynamical Sys ems, Sp inge Monog aphs in Ma hema ics (1998).
[3] Aubin J.P., F ankowska H., Se -Valued Analysis, Bi kh¨ause , Bos on (1990).
[4] Ba bu V., Nonlinea Semig oups and Di e en ial Equa ions in Banach Spaces, Edi u a Academiei, Bucu es i
(1976).
[5] Ca aballo T., Langa J.A., Robinson J., Uppe semicon inui y o a ac o s o small andom pe u ba ions o
dynamical sys ems, Comm. Pa ial Di e en ial Equa ions 23 (1998), 1557-1581.
[6] Ca aballo T., Langa J.A., Vale o J., Global a ac o s o mul i alued andom dynamical sys ems, o appea .
[7] Cas aing C., Valadie M., Con ex Analysis and Measu able Mul i unc ions, LNM Vol. 580, Sp inge (1977).
[8] C auel H., Flandoli F., A ac o s o andom dynamical sys ems, P ob. Theo y Rela ed Fields 100 (1994),
365-393.
[9] Da P a o G., F ankowska H., A s ochas ic Filippo heo em, S och. Anal. Appl. 12 (1994), no. 4, 409–426.
[10] Hale J., Asymp o ic Beha io o Dissipa i e Sys ems, Ma h. Su eys and Monog aphs, AMS, P o idence (1988).
[11] Ladyzhenskaya O., A ac o s o Semig oups and E olu ion Equa ions, Accademia Nazionale dei Lincei, Cam-
b idge Uni e si y P ess, Camb idge (1991).
[12] Melnik V., Vale o J., On A ac o s o mul i alued semi- lows and di e en ial inclusions, Se -Valued Anal. 6
(1998), 83-111.
[13] Schmal uss B., Backwa d cocycle and a ac o s o s ochas ic di e en ial equa ions, in V. Rei mann, T. Red ich
and N. JKosch (eds.), In e na ional Semina on Applied Ma hema ics-Nonlinea Dynamics: A ac o App oxi-
ma ion and Global Beha iou (1992), 185-192.
[14] Temam R., In ini e Dimensional Dynamical Sys ems in Mechanics and Physics, Sp inge -Ve lag, New Yo k (1988).
[15] Tols onogo A.A., On solu ions o e olu ion inclusions.I, Sibi sk. Ma . Zh. 33, 3 (1992), 161-174 (English
ansla ion in Sibe ian Ma h. J., 33, 3 (1992)).
[16] Vale o J., Fini e and in ini e-dimensional a ac o s o mul i alued eac ion-di usion equa ions, Ac a Ma h. Hun-
ga . Vol. 88, no. 3 (2000), 239-258.
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