On he ela ionship be ween solu ions o s ochas ic and andom
di e en ial inclusions
T.Ca aballo1, J.A.Langa1, J.Vale o2
1Depa 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.
e-mails: [email p o ec ed] ; langa@nume .us.es
2Uni e sidad Ca denal He e a CEU,
Comissa i 3, 03203 Elche, Alican e. Spain.
e-mail: [email p o ec ed]
Abs ac
Some esul s on he ela ionship o he solu ions o a s ochas ic di e en ial inclusion and
he co esponding andom di e en ial inclusion ob ained a e a change o a iable a e p o ed.
As a consequence, we ob ain he pullback con e gence o he solu ions o he s ochas ic
inclusion o a compac andom se . The cases o a eac ion-di usion inclusion pe u bed by
addi i e and mul iplica i e noises a e conside ed.
1 In oduc ion
In ou p e ious wo ks Ca aballo e al. [4, 5, 6], he concep o mul i alued andom dynamical
sys em (MRDS) has been in oduced and some applica ions o s ochas ic di e en ial inclusions
ha e been conside ed. In ac , a eac ion-di usion inclusion pe u bed by addi i e and mul i-
plica i e noise is conside ed. As in he single- alued case, he cons uc ion o he MRDS elies
on he possibili y o pe o ming a sui able change o a iable d i ing he s ochas ic inclusion
o a de e minis ic one depending on a andom pa ame e (a andom inclusion), so ha he
de e minis ic ideas can be p ope ly adap ed o his si ua ion.
Howe e , he e exi s a esul due o Da P a o and F ankowska [7] which gua an ees he
exis ence o s ochas ic solu ions o a mo e gene al noise han he ones conside ed in [4, 5, 6].
In bo h si ua ions, we do no ha e uniqueness o solu ions. To de ine he MRDS in he
i s case, we ake he union o all he solu ions o he p oblem. Consequen ly, one can hink
abou he ela ionship be ween he solu ions o he s ochas ic inclusion and he andom inclusion
ob ained a e he change o a iable. This is ou main aim in his pape . Indeed, we shall p o e
ha in ou wo pa icula si ua ions (i.e., addi i e and mul iplica i e noises) each solu ion o he
s ochas ic inclusion co espondes o a solu ion in he andom one. The con e se is ue unde
some addi ional condi ions on he solu ions o he andom inclusion.
I is shown in [4, 5, 6] ha he andom inclusion gene a es a pe ec cocycle ha ing a compac
global andom a ac o . As a consequence, all solu ions o he s ochas ic inclusion wi h he
ini ial condi ion in a bounded se o he phase space con e ge uni o mly (in he sense o pull-
back a ac ion) o his a ac o in he Hausdo semidis ance. This jus i ies he in e es o he
1
esul s o [4, 5, 6], since hey gi e some in o ma ion on he asymp o ic beha iou o he solu ions
o he s ochas ic inclusion.
Hope ully, i will be possible in he u u e o gi e some ligh on he possibili y o conside ing
mo e gene al si ua ions han he ones ea ed up o da e. As a as we know, his p oblem
emains in e es ing e en in he single- alued case (see also Capinski and Cu land [3] o ano he
especial case o Na ie -S okes equa ion, and Imkelle and Schmal uss [8] o a gene al change o
a iables ela ing s ochas ic and andom di e en ial equa ions).
2 The addi i e case
Le Hbe a eal sepa able Hilbe space wi h he no m k·k and he scala p oduc h·,·i. The
linea ope a o A:D(A)⊂H→His called m-dissipa i e i
< A(y), y >≤0, o all y∈D(A)
and Im(A−λI) = H, o any λ > 0.
Le Abe a linea , m-dissipa i e, sel -adjoin and densely de ined ope a o (i.e. D(A) = H)
. I ollows om hese p ope ies ha −A=∂ϕ o some p ope , lowe semicon inuous con ex
unc ion ϕ:H→(−∞,+∞] (see Ba bu [2, p.60]).
Mo eo e , Philips-Lume Theo em implies ha Ais he in ini esimal gene a o o a semi-
g oup o class C0deno ed by S( ).Since Ais m-dissipa i e, i is known ha kS( )k ≤ 1.
The ope a o S( ) is assumed o be compac o any > 0.
Conside he equa ion



du ( )
d =Au ( ) + ( ),0≤ ≤T,
u(0) = u0∈H.
(1)
The unc ion u: [0, T]→His called a s ong solu ion o (1) i u(·)∈C([0, T], H), u(0) =
u0,uis absolu ely con inuous on compac se s o (0, T), u is a.e. di e en iable in (0, T ), and
du ( )
d =Au ( ) + ( ) o a.a. ∈(0.T).
I is well known ha o any (·)∈L2(0, T ;H) he equa ion has a unique s ong solu ion
(see Ba bu [2, p.189]). Mo eo e , i is well known ha
u( ) = S( )u0+Z
0
S( −s) (s)ds, o any ≥0.
Conside he s ochas ic di e en ial inclusion (see Ca aballo e al. [4, 6] o mo e de ails)



du ( )
d ∈Au ( ) + F(u( )) + Pm
i=1 φi
dwi( )
d ,0≤ ≤T,
u(0) = u0∈H,
(2)
whe e F:H→2Hsa is ies:
(F1) Fhas closed, bounded, con ex, non-emp y alues.
2
(F2) The e exis s C > 0 such ha
dis H(F(u), F ( )) ≤Cku− k,∀u, ∈H,
whe e ‘dis H’ deno es he Hausdo me ic.
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, H) such ha
ζ(0) = 0.
Conside he maps ρs, : Ω →Ω
ρs, (ω) (τ) = 


ω(s),si τ≤s,
ω(τ),si s≤τ≤ ,
ω( ) , si τ≥ ,
and de ine he σ-algeb as Fs, =ρ−1
s, F. I is clea ha Fs, ⊂ Fs0, 0⊂ F,∀s0≤s≤ ≤ 0.
A p ocess u( , ω) : [0,∞)×Ω→His said o be adap ed i u( , ·) is F0, measu able o any
≥0.
Now, Theo em 2.1 om [7] p o ides he exis ence o a leas one solu ion u(·, ω, u0) o (2)
o any u0∈H, ha is, an adap ed p ocess wi h alues in Hsuch ha :
1. u(·, ω, u0) is con inuous o P-a.a. ω∈Ω.
2. u(0, ω, u0) = u0.
3. Fo any ∈[0, T ]
u( ) = S( )u0+Z
0
S( −s) (s)ds +
m
X
i=1 Z
0
S( −s)φidwi(s),
being (s, ω) an adap ed p ocess such ha
(s, ω)∈F(u(s, ω)) , o a.a. (s, ω)∈(0, T )×Ω,
EZT
0
k (s)k2ds<∞.
No e ha , since T > 0 is a bi a y, we can ex end any solu ion o he whole [0,+∞).
Conside now he change o a iable ( ) = u( )−ζ( ). Then, inclusion (2) becomes
( o mally)



d ( )
d ∈A ( ) + F( ( ) + ζ( )) + Aζ ( ),0≤ ≤T,
(0) = u0.
(3)
Fo any u0∈Hand ω∈Ω he e exis s a s ong solu ion (·, ω, u0) o (3), i.e., a s ong
solu ion o (1) o some (·)∈L2(0, T;H) such ha ( )∈F( ( ) + ζ( )) + Aζ ( ), a.e. in
(0, T).
Le us deno e he solu ions o (2) and (3) by u(·) = Is(u0) (·) and (·) = I (u0) (·),
espec i ely. Then, we can p o e he ollowing esul s.
3
P oposi ion 1 I u(·) = Is(u0) (·), hen he unc ion de ined as (·) = u(·)−ζ(·)sa is ies
(·) = I (u0) ( (·) + Aζ (·)) .In o he wo ds, i u(·)is a solu ion o he s ochas ic inclusion
(2) in he sense o Da P a o and F ankowska, and we de ine he unc ion (·)by he equali y
(·) = u(·)−ζ(·), hen (·)is a s ong solu ion o (2) bu subs i u ing he o iginal ( )in he
igh -hand side o (1) by ( ) + Aζ ( ).
P oo . Since u(·) is a solu ion o (2) we ha e
u( ) = S( )u0+Z
0
S( −s) (s)ds +
m
X
i=1 Z
0
S( −s)φidwi(s).(4)
Le Y(s) = Pm
i=1 S( −s)φiwi(s). Since dS ( )u0=AS ( )u0d , we easily ge
dY (s) = S( −s)
m
X
i=1
φidwi(s)−AS ( −s)
m
X
i=1
φiwi(s)ds.
Hence, using he ac ha Aand S( ) commu e, we ha e
m
X
i=1
φiwi( ) =
m
X
i=1 Z
0
S( −s)φidwi(s)−
m
X
i=1 Z
0
S( −s)Aφiwi(s)ds. (5)
Now, combining (4) and (5), we ob ain
( ) = u( )−ζ( ) = S( )u0+Z
0
S( −s) ( (s) + Aζ (s)) ds.
I ollows (·) = I (u0) ( (·) + Aζ (·)) .
P oposi ion 2 I (·) = I (u0) (·), being ( )an adap ed p ocess such ha ERT
0k (s)k2ds<
∞, hen u(·) := (·) + ζ(·) = Is(u0) ( (·)−Aζ (·)) .
P oo . Since (·) is a s ong solu ion o (1), we ha e
( ) = S( )u0+Z
0
S( −s) (s)ds
=S( )u0+Z
0
S( −s)g(s)ds +
m
X
i=1 Z
0
S( −s)Aφiwi(s)ds,
whe e g(s) = (s)−Aζ (s)∈F( (s) + ζ(s)), a.e. on (0, T ).
Using (5) we ob ain
u( ) = ( ) + ζ( ) = S( )u0+Z
0
S( −s)g(s)ds +
m
X
i=1 Z
0
S( −s)φidwi(s).
Hence, u(·) = Is(u0)g(·).(Obse e ha u(·) is adap ed since and ζalso a e.)
Now, no ice ha he andom di e en ial inclusion (3) gene a es a pe ec mu i alued cocycle
G:R+×Ω×H→C(H) by means o
G( , ω)u0=[
(·)∈D(u0,ω)
{ ( ) + ζ( )},
4
whe e
D(u0, ω) = { (·) : (·) is a s ong solu ion o (3)}.
Mo eo e , Ghas compac alues (see Ca aballo e al. [4]).
De ini ion 3 The closed andom se ω7→ A(ω)( ha is, a measu able map wi h closed alues)
is called a global andom a ac o o Gi :
i) G( , ω)A(ω) = A(θ ω), o all ≥0,P−a.s ( ha is, i is s ic ly in a ian );
ii) Fo all bounded D⊂X,
lim
→+∞dis (G( , θ− ω)D, A(ω)) = 0,P−a.s.;
iii) A(ω)is compac P−a.s.
P ope y ii) means ha he ini ial momen o ime goes o -∞and he inal momen is 0.
This is called he pullback con e gence in he li e a u e (Kloeden and Schmal uss [9]) .
I we assume he ollowing condi ions:
(H1) The e exis cons an s δ > 0, M ≥0 such ha
hy, ui ≤ (−δ+ε)kuk2+M, o all u∈D(A), y ∈F(u),(6)
whe e ε≥0 is he bigges cons an such ha
hAu, ui≤−εkuk2.
(H2) The le el se s
MR={u∈D(ϕ)| kuk ≤ R, ϕ(u)≤R}
a e compac in H o any R > 0.
Then, i is p o ed in [4, Theo em 16] ha Ghas he global andom a ac o A(ω), which is
measu able wi h espec o he σ−algeb a F.
As a consequence o P oposi ion 1 we ha e u( , ω, u0)∈G( , ω)u0, o all ( , u0)∈R+×
H, ω ∈Ω and u(·, ω, u0)∈ L (u0, ω),whe e
L(u0, ω) = {u(·, ω, u0) : u(·, ω, u0) is a solu ion o (2)}.
We hen ob ain ha all he solu ions o inclusion (2) wi h ini ial condi ions on a bounded se
con e ge uni o mly o A(ω) in he sense o De ini ion 3.
Co olla y 4 Assuming (H1) −(H2) , o any bounded se Bi holds
lim
→+∞sup
u∈L(u0,ω)
sup
u0∈B
dis (u( , θ− ω, u0),A(ω)) = 0.
5

These esul s can be applied o he ollowing eac ion-di usion inclusion



∂u
∂ ∈∆u+ (u) + h+Pm
i=1 φidwi( )
d ,on O × (0, T),
u= 0, on ∂O × (0, T),
u(x, 0) = u0(x) on O,
(7)
whe e O ⊂ Rnis an open bounded subse wi h smoo h bounda y ∂O,h(·)∈L2(O) and :R→
2Ris a mul i alued map wi h non-emp y, compac con ex alues. Assume ha is Lipschi z,
i.e. he e exis s C≥0 such ha
dis H( (x), (z)) ≤Ckx−zk, o all x, z ∈R.(8)
De ine he ope a o s A:D(A)→H, F :H→2H, H =L2(O),
Au = ∆u,
F(u) = {y∈H:y(x)∈ (u(x)) + h(x),a.e. on O} ,
wi h D(A) = H2(O)∩H1
0(O). I is assumed ha φi∈D(A).The map −Ais he subdi e en ial
o a p ope , con ex, lowe semicon inuous unc ion ϕand he map Fsa is ies (F1) −(F2).
Mo eo e , condi ion (H2) is sa is ied and D(ϕ) = H(see Melnik and Vale o [10, Sec ion 3.2.2.]).
On he o he hand, i is well known ha he ope a o Asa is ies all he condi ions assumed
be o e.
Hence, P oposi ions 1-2 hold.
I we also assume he exis ence o M≥0, δ > 0 such ha
zs ≤(λ1−2δ)|s|2+M1, o all s∈R, z ∈ (s),(9)
whe e λ1is he i s eigen alue o −∆ in H1
0(Ω), hen (H1) is also sa is ied (see [4, Theo em
17]). I ollows ha Co olla y 4 holds.
3 The mul iplica i e case
Le us now conside he mul iplica i e case om Ca aballo e al. [5]. Tha is, le us conside
he ollowing s ochas ic di e en ial inclusion in he S a ono ich sense



du ( )
d ∈Au ( ) + F(u( )) + σu( )◦dw ( )
d ,0≤ ≤T,
u(0) = u0∈H,
(10)
whe e σ∈R,F:H→2Hsa is ies (F1)-(F2) and now he Wiene p obabili y space (Ω,F,P)
is de ined by
Ω = {ω=w(·)∈C(R,R)|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.
The change o a iable which akes he s ochas ic inclusion (10) in o a andom one is di e en
om he one in he p e ious sec ion. Indeed, se α( ) = α( , ω) = e−σw (ω).Using he change
( ) = α( )u( ),inclusion (10) becomes ( o mally)



d ( )
d ∈A ( ) + α( )Fα−1( ) ( ),0≤ ≤T,
(0) = u0∈H.
(11)
6
Thus, as in Sec ion 2 and aking in o accoun he ela ion be ween he I o and S a ono ich
in eg al in his special case, Theo em 2.1 in [7] also ensu es he exis ence o a leas one solu ion
o (10) sa is ying:
1. u(·, ω, u0) is con inuous o P-a.a. ω∈Ω.
2. u(0, ω, u0) = u0.
3. Fo any ∈[0, T ]
u( ) = S( )u0+Z
0
S( −s) (s)ds +Z
0
S( −s)u(s)◦dw (s),
whe e (s, ω) is an adap ed p ocess such ha
(s, ω)∈F(u(s, ω)) , o a.a. (s, ω)∈(0, T )×Ω,
EZT
0
k (s)k2ds<∞.
On he o he hand, gi en u0∈Hand ω∈Ω, he e exis s a s ong solu ion o (11), i.e. a
s ong solu ion o (1) o some (·)∈L2(0, T;H) such ha ( )∈α( )Fα−1( ) ( )a.e. in
(0, T).
Using again a simila no a ion o ha one in Sec ion 2, i.e. u(·) = Js(u0) (·) o he solu ions
o (10), and (·) = J (u0) (·) o he solu ions o (11), we can p o e he ollowing esul s.
P oposi ion 5 I u(·) = Js(u0) (·), hen (·) = α(·)u(·) = J (u0) (α(·) (·)) .In o he wo ds,
i u(·)is a solu ion o he s ochas ic inclusion (10) in he sense o Da P a o and F ankowska,
and we de ine he unc ion (·)by (·) = α(·)u(·), hen (·)is a s ong solu ion o (11) bu
subs i u ing he o iginal ( )in he igh -hand side o (1) by α( ) ( ).
P oo . Le u(·) be a solu ion o (10). Then, we ha e
u( ) = S( )u0+Z
0
S( −s) (s)ds +σZ
0
S( −s)u(s)◦dw (s),(12)
whe e (s)∈F(u(s)) a.e. in (0, T)×Ω.
Le Y(s) = α(s)S( −s)u(s),0≤s≤ . Then, i ollows
dY (s) = −σα(s)S( −s)u(s)◦dw (s) + α(s)d[S( −s)u(s)] ,
and, by a di ec in eg a ion
Y( )−Y(0) = −σZ
0
α(s)S( −s)u(s)◦dw (s)
+Z
0
α(s)d[S( −s)u(s)] .(13)
Now, we wo k wi h he las e m in (13).
7
Obse e ha
d[S( −s)u(s)]
=dS( −s)S(s)u0+Zs
0
S(s−τ) (τ)dτ +σZs
0
S(s−τ)u(τ)◦dw (τ)
=dS( )u0+Zs
0
S( −τ) (τ)dτ +σZs
0
S( −τ)u(τ)dw (τ)
=S( −s) (s)ds +σS ( −s)u(s)◦dw (s),
and, consequen ly, (13) u ns in o
Y( )−Y(0) = −σZ
0
α(s)S( −s)u(s)◦dw (s)
+Z
0
S( −s)α(s) (s)ds
+Z
0
σα(s)S( −s)u(s)◦dw (s)
=Z
0
S( −s)α(s) (s)ds.
Taking in o accoun ha Y( ) = ( ) and Y(0) = S( )u0,i holds
( ) = S( )u0+Z
0
S( −s)˜
(s)ds,
whe e ˜
(s) = α(s) (s).
P oposi ion 6 I (·) = J (u0) (·), being ( )an adap ed p ocess such ha ERT
0k (s)k2ds<
∞, hen u(·) = α−1(·) (·) = Js(u0)˜
(·),whe e ˜
(s) = α−1(s) (s).
P oo . Since (·) is a s ong solu ion o (11), we ha e
( ) = S( )u0+Z
0
S( −s) (s)ds,
whe e (s)∈α(s)F(α−1(s) (s)), a.e. on (0, T).
Le us deno e Y(s) = α−1(s)S( −s) (s) = eσws(ω)S( −s) (s),0≤s≤ . Then, a guing as
in he p eceding p oo , i ollows
dY (s) = σeσwsS( −s) (s)◦dws+ eσwsd[S( −s) (s)]
=σeσwsS( −s) (s)◦dws+ eσwsS( −s) (s)ds,
and, a e in eg a ion, i yields
Y( )−Y(0) = Z
0
S( −s)α−1(s) (s)ds
+σZ
0
S( −s)α−1(s) (s)◦dws.
8
Se ing u( ) = α−1( ) ( ),˜
( ) = α−1( ) ( ), i holds
u( ) = S( )u0+Z
0
S( −s)˜
(s)ds +σZ
0
S( −s)u(s)◦dws.
The e o e he p oo is comple e since he adap edness o uis immedia e.
Now, as in he addi i e case, di e en ial inclusion (11) gene a es a pe ec mu i alued cocycle
GM:R+×Ω×H→C(H) by se ing
GM( , ω)u0=[
(·)∈D(u0,ω)
α−1( ) ( ),
whe e
D(u0, ω) = { (·) : (·) is a solu ion o (11)}.
Mo eo e , GMhas compac alues (see Ca aballo e al. [5] o mo e de ails).
I we assume (H1)−(H2) , hen i is p o ed in [5, Theo em 26] ha Ghas he global andom
a ac o A(ω), which is measu able wi h espec o he σ−algeb a F.
As a consequence o P oposi ion 5 we ha e u( , ω, u0)∈GM( , ω)u0, o all ( , u0)∈R+×
H, ω ∈Ω and u(·, ω, u0)∈ LM(u0, ω),whe e
LM(u0, ω) = {u(·, ω, u0) : u(·, ω, u0) is a solu ion o (10)}.
We ob ain hen ha all he solu ions o inclusion (10) wi h ini ial condi ions on a bounded se
con e ge uni o mly o A(ω) in he sense o De ini ion 3.
Co olla y 7 Assuming (H1) −(H2) , o any bounded se Bi holds
lim
→+∞sup
u∈LM(u0,ω)
sup
u0∈B
dis (u( , θ− ω, u0),A(ω)) = 0.
These esul s can be applied o he ollowing eac ion-di usion inclusion





∂u
∂ ∈∆u+ (u) + h+σu( )◦dw ( )
d ,on O × (0, T),
u= 0, on ∂O × (0, T),
u(x, 0) = u0(x) on O,
(14)
whe e O ⊂ Rn, h, , F and Aa e de ined as o (7).
Hence, P oposi ions 5-6 hold.
I we assume also he exis ence o M≥0, δ > 0 such ha
zs ≤(λ1−2δ)|s|2+M1, o all s∈R, z ∈ (s),
whe e λ1is he i s eigen alue o −∆ in H1
0(Ω), hen (H1) is also sa is ied (see [5, Theo em
27]). I ollows ha Co olla y 7 holds.
9