Vector-valued stochastic delay equations - a weak solution and its Markovian representation

0.34 MB·English

by  Mariusz Górajski

VECTOR-VALUED STOCHASTIC DELAY EQUATIONS - A WEAK SOLUTION
AND ITS MARKOVIAN REPRESENTATION
MARIUSZGO´RAJSKI
3
1 Abstract. A class of stochastic delay equations in Banach space E driven by cylindrical Wiener
0 process is studied. We investigate two concepts of solutions: weak and generalised strong, and
2 give conditions under which they are equivalent. We present an evolution equation approach in a
n Banach space Ep := E×Lp(−1,0;E) proving that the solutions can be reformulated as Ep-valued
a Markov processes. Based on the Markovian representation we prove the existence and continuity
J of the solutions. The results are applied to stochastic delay partial differential equations with an
applicationtoneutral networksandpopulationdynamics.
2
2

] AMS 2000 subject classification: 34K50, 60H15, 60H30, 47D06
R
P
1. Introduction
.
h
t LetH beaseparableHilbertspace. ForaBanachspaceE andp 1define p :=E Lp( 1,0;E).
a Let W be an H-cylindrical Wiener process on a given probability≥space (Ω,(E ) , ×,P).−We shall
m H Ft t≥0 F
consider stochastic delay equation in a Banach space E of the form:
[
dX(t)=(BX(t)+ΦXt+φ(X(t),Xt))dt+ψ(X(t),Xt)dWH(t), t>0;
1
(SDE) X(0)=x ;
v  0
X =f ,
0  0 0
0
3 for initial conditions [x0,f0] ∈ L0((Ω,F0);Ep), where (Xt)t≥0 is a segment process formed from
5 (X(t))t≥0 in the following way:
.
1 X (s):=X(t+s), s [ 1,0].
t
∈ −
0
3 Let us consider (SDE) with the following hypotheses:
1 (H1) B :D(A) E E is a linear operator and generates a C -semigroup (S(t)) on E,
0 t≥0
: ⊂ →
v (H2) Φ is given by Riemann-Stieltjes integral
i
X 0
Φf = dηf dla f C([ 1;0];E),
r ∈ −
a Z−1
where η :[ 1,0] (E) is of bounded variation,
(H3) φ : D(φ) − →ELis densely defined mapping and there exists a Lp (0, ) such that for
⊂Ep → ∈ loc ∞
all t>0 and , D(φ),
X Y ∈
S(t)φ( ) a(t)(1+ ),
k X kE ≤ kXkEp
S(t)(φ( ) φ( ) a(t) ,
k X − Y kE ≤ kX −YkEp
Keywords and phrases. Stochasticpartialdifferentialequationswithfinitedelay,Stochasticevolutionequation,UMD
Banachspaces,Type2Banachspaces.
1
2 MARIUSZGO´RAJSKI
(H4) ψ :D(ψ) (H,E)isdenselydefinedmappingsuchthatforallt>0and , D(ψ),
p
⊂E →L X Y ∈
S(t)ψ( ) belongs to γ(H,E) and there exists b Lp∨2(0, ) such that
X ∈ loc ∞
S(t)ψ( ) b(t)(1+ ),
k X kγ(H,E) ≤ kXkEp
S(t)(ψ( ) ψ( ) b(t) ,
k X − Y kγ(H,E) ≤ kX −YkEp
where γ(H,E) is the space of γ-radonifying operators from H to E (see Section 2 in [9] or [27]).
We use the evolutionequationapproachto the delay equationas givenin the monographof Batkai
and Piazzera [3]. Thus, one can define a closed operator on by
p
A E
D( )= [x,f]′ D(B) W1,p( 1,0;E) : f(0)=x ;
A { ∈ × − }
B Φ
(1) = .
A 0 d
(cid:20) dθ (cid:21)
Under assumptions (H1)-(H2) generates a C -semigroup ( (t)) on (see [3], Theorem 3.29).
0 t≥0 p
A T E
Hence we can consider the following stochastic Cauchy problem corresponding to (SDE):
dY(t)= Y(t)dt+F(Y(t))dt+G(Y(t))dW (t), t 0;
(SDCP) A H ≥
Y(0)=[x ,f ]′,
0 0
(cid:26)
where
(2) F(Y(t)):=[φ(Y(t)),0]′, G(Y(t)):=[ψ(Y(t)),0]′.
Recall the classical result [7], where equivalence of solutions to the stochastic delay equation and
the corresponding abstract Cauchy problem has been shown by Chojnowska-Michalik for p = 2 and
E finite-dimensional. For a general class of spaces including the spaces the variation of constants
p
E
formula for finite-dimensional delay equations with additive noise and a bounded delay operator is
discussed in Riedle [23]. For more references see [9]. We complement and extend result from [9]
concerning existence and uniqueness of solution to (SDE) by adding non-linear part and introducing
weak concept of solution to (SDE). The line of thought we take is to prove existence and continuity
of a weak solution to the stochastic Cauchy problem (SDCP) and then using the equivalence between
weaksolutionsto(SDCP)and(SDE)weobtaincorrespondingresultsforthestochasticdelayequation
(SDE).
Alargeclassofstochasticpartialdifferentialequations(stochasticPDEs)withdelaycanberewritten
as stochastic ordinary equations with delay (SDE) in infinite dimensional space E. Moreover, PDEs
with delay are used in modelling phenomena inter alia in bioscience (see [2], [5] and [18]) or in neural
networks (see [20]). For some stochastic PDEs with delay e.g. stochastic delay reaction-diffusion
equationswithnon-linearitiesgivenbytheNiemyckioperator(seeSection3.3Examples)ageneralized
strong solution may not exist, whereas one can prove the existence and uniqueness of weak solution.
In the Da Prato and Zabczyk monograph [10] an extensive treatment of the stochastic Cauchy
problem in Hilbert spaces is given. In the Banach space framework stochastic Cauchy problem has
beenconsideredbyBrze´zniak[6]andVanNeerven,VeraarandWeis[28]. Theybothconsiderthecase
that generates an analytic semigroup.
A
Following the semigroup approach let us consider the following variation of constants formula:
t t
(3) Y(t)= (t)Y(0)+ (t s)F(Y(s))ds+ (t s)G(Y(s))dW (s),
H
T T − T −
Z0 Z0
where the precise definition of the stochastic integral above and the relevant theory on vector-valued
stochastic integrals can be found in [28]. A process satisfying (3) is usually referred to as a mild
VECTOR-VALUED STOCHASTIC DELAY EQUATIONS 3
solution. The existence of a mild solution to (SDCP) is proved by the Banach fixed-point theorem in
Section 3 (Theorem 3.7).
In Theorems 3.2 and 3.6 we show that a mild solution of (SDCP) is equivalent to a weak solution
of (SDCP) and under some additional assumption they are equivalent to generalized strong solution
of (SDCP). Finally,usingthese theoremsinTheorem3.9westatethatweaksolutionsto (SDCP)and
(SDE) areequivalent. Combining allthese results we obtainexistence andcontinuityof weaksolution
of (SDE) (see Corollaries 3.12 and 3.13).
The correspondence between strong, weak and mild concept of solution to stochastic linear delay
equations in Hilbert space has been considered by Liu using the properties of the Green operator in
[19].
Theequivalenceofsolutionsto(SDE)andto(SDCP)isusefulbecausethelatterisaMarkovprocess
and can be studied in the framework of the stochastic abstract Cauchy problem; and one can answer
questions concerning e.g. invariant measures of the solutions to (SDE) (see [8] and [4] and reference
therein), Feller property (see [22]) and regularity of solutions (see Corollary 3.13 and [28]).
2. The Stochastic Cauchy Problem
In the introduction we have mentioned that the stochastic delay equation (SDE) can be rewritten
as a stochastic Cauchy problem. In this section we recall the result concerning different concept of
solution to (SCP) form [16]. Let E be a Banach space and H be a separable Hilbert space, and let
A : D(A) E E be the generator of a C -semigroup (T(t)) on E. The sun dual semigroup
0 t≥0
⊂ →
(T⊙(t)) defined as subspace semigroup by T⊙(t) = T∗(t) defined on E⊙ = D(A∗) is strongly
t≥0 |E⊙
continuous (see Section 2.6 in [14] and Chapter 1 in [24]). A generator (A⊙,D(A⊙)) of the sun dual
semigroup is given by A⊙ =A∗ and D(A⊙)= x∗ D(A∗):A∗x∗ E⊙ .
|E⊙ { ∈ ∈ }
LetW be anH-cylindricalBrownianmotionandfollowingthe monographofPeszatandZabczyk
H
(see Section 9.2 and Remark 9.3 in [21]) let F :D(F) E E and G:D(G) E (H,E) satisfy
⊂ → ⊂ →L
the following conditions:
(HA) D(F) is dense in E and there exists a L1 (0, ) such that for all t > 0 and x,y D(F)
∈ loc ∞ ∈
we have
T(t)F(x) a(t)(1+ x ),
E E
k k ≤ k k
T(t)(F(x) F(y)) a(t) x y ,
E E
k − k ≤ k − k
(HB) D(G) is dense in E and there exists b L2 (0, ) such that for all t > 0 and x,y D(G)
∈ loc ∞ ∈
we have
T(t)G(x) b(t)(1+ x ),
γ(H,E) E
k k ≤ k k
T(t)(G(x) G(y)) b(t) x y .
γ(H,E) E
k − k ≤ k − k
Let us consider the following stochastic Cauchy problem in E:
dY(t) =AY(t)dt+F(Y(t))dt+G(Y(t))dW (t), t 0;
(SCP) H ≥
Y(0) =Y .
0
(cid:26)
Definition 2.1. An H-strongly measurable adapted process Y is called a weak solution to (SCP) if
Y is a.s. (almost surely) locally Bochner integrable and for all t>0 and all x∗ D(A⊙):
∈
(i) F(Y),x∗ is integrable on [0,t] a.s.;
h i
(ii) G∗(Y)x∗ is stochastically integrable on [0,t];
4 MARIUSZGO´RAJSKI
(iii) for almost all ω
t t t
Y(t) Y ,x∗ = Y(s),A⊙x∗ ds+ F(Y(s)),x∗ ds+ G∗(Y(s))x∗dW (s).
0 H
h − i h i h i
Z0 Z0 Z0
In the next theorem we need stochastic integral for (H,E)-valued process (for a definition and
L
the following characterisationsee [27]). Recall that umd property stands for Unconditional Martigale
DifferencepropertyanditsaysthatallLp(Ω;E)-convergence,E-valuedsequenceofmartingledifference
are unconditionally convergent (see [15] and [27]). It turns out that for a Banach space with umd
property we may characterise stochastic integrability in terms of γ-radonifying norm. More precisely
a H-strongly measurable adapted process Ψ : [0,t] Ω (H,E) is stochastically integrable with
× → L
respect to cylindrical Wiener process W if and only if Ψ represents γ(L2(0,t;H);E)-valued random
H
variable R given by
Ψ
t
(4) R f,x∗ = Ψ(s)f(s),x∗ ds a.s,
Ψ
h i h i
Z0
for every f L2(0,t;H) and for all x∗ E∗. In this situation one has also the following Burkh¨older-
∈ ∈
Gundy-Davies type inequalities :
s
(5) E sup Ψ(u)dW p h E R p
k HkE p k ψkγ(L2(0,t;H),E)
s∈[0,t] Z0
forallp>01. TosimplifyterminologywesaythatprocessΨisinγ(L2(0,t;H);E)a.s. iffΨrepresents
a random variable R given by (4).
Φ
In [15] Garling has characterisedumd property in terms of two properties: umd− and umd+.
Definition 2.2. A Banachspace E has umd− property,if for all 1<p< there exists β− >0 such
∞ p
that for all E-valued sequence of Lp-martingle difference (d )N and for all Rademacher sequence
n n=1
(r )N independent from (d )N we have the following inequality:
n n=1 n n=1
p p
N N
(umd−) E d β−E r d .
(cid:13) n(cid:13) ≤ p (cid:13) n n(cid:13)
(cid:13)nX=1 (cid:13)E (cid:13)nX=1 (cid:13)E
(cid:13) (cid:13) (cid:13) (cid:13)
A Banach space E has umd+ pro(cid:13)perty, if(cid:13)the reverse(cid:13)inequalit(cid:13)y to (umd−) holds. Recall that class
(cid:13) (cid:13) (cid:13) (cid:13)
of umd Banach spaces is in class of reflexive spaces and includes Hilbert spaces and Lp spaces for
p (1, ). Moreover,class of umd− Banach spaces includes also non-reflexive L1 spaces.
∈ ∞
To integrate processes with values in L1 one needs a weakened notion of stochastic integral. In a
BanachspaceE withumd− propertythefollowingcondition: Ψisinγ(L2(0,t;H),E)a.s. issufficient
for stochastic integrability of Ψ (cf. [27] and [28]).
Theorem 2.3 ([16]). Assume that E has umd− property and conditions (HA) and (HB) are satisfied.
Let Y be an E-valued H-strongly measurable adapted process with almost all locally Bochner square
integrable trajectories. If for all t>0 the process:
(6) u T(t u)G(Y(u))
7→ −
1ForrealsA,BweusethenotationA.pBtoexpressthefactthatthereexistsaconstantC>0,dependingonp,such
thatA≤CB. WewriteAhpB ifA.pB.pA.
VECTOR-VALUED STOCHASTIC DELAY EQUATIONS 5
is in γ(L2(0,t;H),E) a.s. Then Y is a weak solution to (SCP) if and only if Y is a mild solution to
(SCP) i.e. Y satisfies, for all t 0,
≥
t t
(7) Y(t)=T(t)Y + T(t s)F(Y(s))ds+ T(t s)G(Y(s))dW (s) a.s.
0 H
− −
Z0 Z0
Remark 2.4. Fix x∗ D(A∗). Let Y be a E-valued, strongly measurable adapted process with almost
∈
all locally Bochner square integrable trajectories. Then we have the following:
(i) condition (HA) implies that E x F(x),x∗ R is a Lipschitz function by Lemma 2.3 in
∋ 7→h i∈
[16].
(ii) if (HB) holds then Lemma 2.3 in [16] implies that the function E x G∗(x)x∗ H is
∋ 7→ ∈
Lipschitz-continuous. Hence the process G∗(Y)x∗ is strongly measurable adapted with almost
all locally square integrable trajectories. In particular, G∗(Y)x∗ is stochastically integrable on
[0,t] for all t>0.
(iii) by (HA) and (HB) the functions E x T(s)F(x) E, E x T(s)G(x) γ(H,E) are
∋ 7→ ∈ ∋ 7→ ∈
continuous. Hence processes T(t )F(Y()), T(t )G(Y()) are adapted, strongly and H-
−· · −· ·
strongly measurable, respectively, and T(t )F(Y()) has trajectories locally Bochner square
−· ·
integrable a.s.
(iv) since process in (6) represent element from γ(L2(0,t;H),E) a.s. and E has umd− property,
stochastic integral in (7) is well defined.
A generalisedstrong solutionto (SCP) is defined and its equivalence to a mild solution of (SCP) is
proven in [9].
Definition 2.5. A strongly measurable adapted process Y is called a generalized strong solution to
(SCP) if Y is, almost surely, locally Bochner integrable and for all t>0:
t
(i) Y(s)ds D(A) a.s.,
0 ∈
(ii) F(Y) is Bochner integrable in [0,t] a.s.,
R
(iii) G(Y) is stochastically integrable on [0,t],
and
t t t
Y(t) Y =A Y(s)ds+ F(Y(s))ds+ G(Y(s))dW (s) a.s.
0 H
−
Z0 Z0 Z0
Theequivalenceofmild,weakandgeneralisedstrongsolutionto(SCP)hasbeenestablishedin[16].
First we recall the hypotheses
(HA’) Assume that (HA) is satisfied and for all t > 0 and all g L1(0,t;E) the function F(g) is
∈
Bochner integrable on [0,t].
If F is a Lipschitz function then (HA’) is satisfied.
Theorem 2.6([16]). Assumethat E has umd− propertyandconditions (HA’)and(HB) aresatisfied.
Let Y be an E-valued H-strongly measurable adapted process with locally Bochner square integrable
trajectories a.s. If for all t>0 the processes:
t−u
(8) u G(Y(u)), u T(t u)G(Y(u)), u T(s)G(Y(u,ω))ds
7→ 7→ − 7→
Z0
are in γ(0,t;H,E) a.s. Then the following condition are equivalent:
(i) Y is a generalised strong solution of (SCP).
(ii) Y is a weak solution of (SCP).
(iii) Y is a mild solution of (SCP).
6 MARIUSZGO´RAJSKI
3. The Stochastic Delay Equation
3.1. The variation of constants formula. We now turn to the stochastic delay equation (SDE) as
presented in the introduction and to the related stochastic Cauchy problem (SDCP) on page 2.
We assume (H1)-(H2). Then, the operator (cf. (1)) generates the strongly continuous semigroup
A
1
( (t)) on with a Lp-norm given by [x,f] = x p + f p p(see Theorem 3.29 in
T t≥0 Ep k kEp k kE k kLp(−1,0;E)
[3]). We shall define the projections π : E and π(cid:16): Lp( 1,0;E) a(cid:17)s follows: π [x,f]′ =x;
1 p 2 p 1
π [x,f]′ =f. E → E → −
2
The following property of ( (t)) is intuitively obvious and useful in the following:
t≥0
T
x x
(9) π (t) (u)=π (t+u) ,
2T f 1T f
(cid:18) (cid:20) (cid:21)(cid:19) (cid:20) (cid:21)
for [x,f]′ ,u [ 1,0],t> u (for a proof see [3], Proposition 3.11).
p
∈E ∈ − −
Lemma 3.1. Assume that (H1) and (H3)-(H4) hold. Then for F : and G : (H, )
p p p p
E → E E → L E
given by (2) there exist a˜ Lp (0, ), ˜b L2∨p(0, ) such that for all t>0 we have
∈ loc ∞ ∈ loc ∞
(i) if (H2) holds, then
(10) π (t)F( ) a˜(t)(1+ ),
k 1T X kE ≤ kXkEp
(11) π (t)(F( ) F( )) a˜(t)
k 1T X − Y kE ≤ kX −YkEp
for all , D(φ), and
X Y ∈
(12) π (t)G( ) ˜b(t)(1+ ),
k 1T X kL(H,E) ≤ kXkEp
(13) π (t)(G( ) G( )) ˜b(t) ,
k 1T X − Y kL(H,E) ≤ kX −YkEp
for all , D(ψ).
X Y ∈
(ii) if (H2) holds, E is a Hilbert space, then one can replace the (H,E)-norm in (12) and (13)
L
with the γ(H,E)-norm 2.
(iii) ifthedelayoperator isboundedi.e. Φ (Lp( 1,0;E),E),thenonecanreplacethe (H,E)-
∈L − L
norm in (12) and (13) with the γ(H,E)-norm.
Proof. We only prove (13). The same proofs works for (10)-(12).
S(t) 0
(i). The following formula defines a semigroup on : (t)= for every t 0, where
Ep T0 t Tl(t) ≥
(cid:20) S (cid:21)
(T (t)) is the left translation semigroup on Lp( 1,0;E) and (E,Lp( 1,0;E)) is given by
l t≥0 − Ss ∈L −
0 θ ( 1, s 1)
(Ssx)(θ)= S(θ+s)x θ ∈(−s − 1∨,−0) .
(cid:26) ∈ − ∨−
B 0
for all s 0 and all x E (cf. Theorem 3.25 in [3]). Let = ,D( )=D( ) be the
≥ ∈ A0 0 d A0 A
(cid:18) (cid:20) dθ (cid:21) (cid:19)
generatorof( (t)) . Recallthatthedelaysemigroup( (t)) canbebuiltbytheMiyadera-Voight
T0 t≥0 T t≥0
perturbationtheorem as a semigroupgeneratedby additive perturbation of generator of the form:
0
A
2If H and E are Hilbert spaces then we have γ(H,E) = L2(H,E), where L2(H,E) is a space of Hilbert-Schmidt
operators.
VECTOR-VALUED STOCHASTIC DELAY EQUATIONS 7
0 Φ
= + (cf. Theorem 1.37 in [3]). Moreover,we have the variation of constant formula:
A A0 0 0
(cid:20) (cid:21)
t
0 Φ
(14) (t) = (t) + (t s) (s) ds, D( ).
T X T0 X T − 0 0 T0 X X ∈ A
Z0 (cid:20) (cid:21)
Then, for all t>0 and for every =[x,g]′ D( ) we have:
X ∈ A
t Φ( x+T (s)g)
(15) kπ1T(t)XkE ≤kS(t)xkE +k π1T(t−s) Ss 0 l dskE.
Z0 (cid:20) (cid:21)
Since Φ is given by Riemann-Stieltjes integral (cf. (H2)), one can apply the Fubini theorem and the
Ho¨lder inequality to
t 1∧t t∨1
(16) Φ( x+T (s)g) ds= Φ( x+T (s)g) ds+ Φ( x+T (s)g) ds
k Ss l kE k Ss l kE k Ss l kE
Z0 Z0 Z1
1∧t 0 t∨1
η ( 1,0) g + S(s)x ds + S(s+θ)x dsdη (θ)
≤| | − k kLp(−1,0;E) k kE k kE | |
(cid:18) Z0 (cid:19) Z−1Z1
t
η ( 1,0) g + S(s)x ds .
≤| | − k kLp(−1,0;E) k kE
(cid:18) Z0 (cid:19)
Thus
t Φ( x+T (s)g) t
π1 T(t−s) Ss 0 l ds ≤MT(t)|η|(−1,0) kgkLp(−1,0;E)+ kS(s)xkEds ,
(cid:13) Z0 (cid:20) (cid:21) (cid:13)E (cid:18) Z0 (cid:19)
(cid:13) (cid:13)
wh(cid:13)ere M (t)=sup (u) . On(cid:13)substituting the above estimation into (15) we obtain
(cid:13) T s∈[0,t]kT kL(Ep) (cid:13)
t
(17) π (t) S(t)x +M (t)η ( 1,0) g + S(s)x ds ,
k 1T XkEp ≤k kE T | | − k kLp(−1,0;E) k kE
(cid:18) Z0 (cid:19)
for all =[x,g]′ D( ). Since D( ) is dense in and (t) is bounded, (17) holds for all .
p p
X ∈ A A E T X ∈E
Inparticular,by secondinequality in(H4) andthe inequality one hasthat, for
L(H,E) γ(H,E)
k·k ≤k·k
all , ,
p
X Y ∈E
π (t)(G( ) G( )) S(t)(ψ( ) ψ( ))
k 1T X − Y kL(H,E) ≤k X − Y kL(H,E)
t
(18) +M (t)η ( 1,0) S(s)(ψ( ) ψ( )) ds
T | | − k X − Y kL(H,E)
Z0
t
b(t)+M (u)η ( 1,0) b(s)ds .
≤ T | | − kX −YkEp
(cid:18) Z0 (cid:19)
Let us notice that the function ˜b defined as ˜b(t) := b(t)+M (t)η ( 1,0) tb(s)ds for a.e. (almost
T | | − 0
every) t 0 belongs to L2 (0, ). The proof of (10)-(11) follows between the same lines with
≥ loc t ∞ R
a˜(t)=a(t)+M (t)η ( 1,0) a(s)ds for a.e. t 0.
T | | − 0 ≥
(ii). Assume now that E is a Hilbert space. Let (h ) be an orthonormalsystem in H. By (17)
n n≥1
R
and (H4) we obtain, for all , ,
p
X Y ∈E
1
∞ p
π (t)(G( ) G( )) = π (t)(G( ) G( )h 2
k 1T X − Y kL2(H,E) k 1T X − Y nkE!
n=1
X
8 MARIUSZGO´RAJSKI
1
∞ 2
S(t)(ψ( ) ψ( ))h 2
≤ k X − Y nkE!
n=1
X
1
∞ t 2 2
+M (t)η ( 1,0) S(s)(ψ( ) ψ( ))h ds
T | | − n=1(cid:18)Z0 k X − Y nkE (cid:19) !
X
S(t)(ψ( ) ψ( ))
≤k X − Y kL2(H,E)
t
+M (t)η ( 1,0) S(s)(ψ( ) ψ( )) ds
T | | − k X − Y kL2(H,E)
Z0
t
b(t)+M (t)η ( 1,0) b(s)ds ,
≤ T | | − kX −YkE2
(cid:18) Z0 (cid:19)
where in the second inequality we use the Minkowski integral inequality.
(iii). Assume that Φ (Lp( 1,0;E),E). First, let us observethat if Φ (Lp( 1,0;E),E) and
∈L − ∈L −
(H1) hold, then , defined by (1) generates strongly continuous semigroup on (cf. [3] ). For any
p
A E
, let Λ=ψ( ) ψ( ) denote anoperatorfrom (H,E). Then using (14) we obtainthat for
p
X Y ∈E X − Y L
all t>0
t
(19) π (t)(G( ) G( )) S(t)Λ + π (t s)[Φ Λ,0]′ds .
k 1T X − Y kγ(H,E) ≤k kγ(H,E) 1 T − Ss
(cid:13) Z0 (cid:13)γ(H,E)
(cid:13) (cid:13)
By boundedness of Φ and due to the ideal property of th(cid:13)e γ-radonifying operators (cid:13)and therefore
(cid:13) (cid:13)
by γ-Fubini isomorphism (cf. Proposition 2.6 in [27]) between the spaces: Lp( 1,0;γ(H,E)) and
−
γ(H,Lp( 1,0;E)) we can estimate the second term on right hand side of (19) as follows
−
t t
(20) π (t s)[Φ Λ,0]′ds π (t s)[Φ Λ,0]′ ds
1 T − Ss ≤ 1 kT − Ss kγ(H,E)
(cid:13) Z0 (cid:13)γ(H,E) Z0
(cid:13) (cid:13)
(cid:13) t (cid:13)
(cid:13) (t s) Φ (cid:13) Λ ds
≤ kT − kL(Ep)k kL(Lp(−1,0;E),E)kSs kγ(H,Lp(−1,0;E))
Z0
1
0 p
M (t) Φ S(s+θ)Λ p dθ .
≤ T k kL(Lp(−1,0;E),E) k kγ(H,E) kX −YkEp
(cid:18)Z−s∨−1 (cid:19)
t s 1
p
M (t) Φ bp(r)dr ds ,
≤ T k kL(Lp(−1,0;E),E) kX −YkEp
Z0 (cid:18)Z0 (cid:19)
where in the last inequality we use assumption (H4). Finally, using (19)-(20) we obtain
t p1
(t)(G( ) G( )) b(t)+ bp(s)ds
kT X − Y kγ(H,Ep) ≤ (cid:18)Z0 (cid:19)
t p1
+M (t)η ( 1,0)t bp(s)ds .
T | | − (cid:18)Z0 (cid:19) !kX −YkEp
for all t>0. (cid:3)
Now we shall consider (SDCP) in = E Lp( 1,0;E) where E is a type 2 Banach space with
p
E × −
umd− property. Recall that a Banach space E is said to have type p [1,2] if there exists a constant
∈
VECTOR-VALUED STOCHASTIC DELAY EQUATIONS 9
C 0 such that for all finite choices x ,...,x F we have
1 k
≥ ∈
k 2 1 k 1
E γ x 2 C x p p,
j j E ≤ k jkE
(cid:16) (cid:13)Xj=1 (cid:13) (cid:17) (cid:16)Xj=1 (cid:17)
(cid:13) (cid:13)
where (γ ) is a sequence of ind(cid:13)ependent(cid:13)standard Gaussians. We note that Hilbert spaces have
j j≥1
type 2 and Lp-spaces with p [1, ) have type min p,2 . For more details we refer the reader to [1].
∈ ∞ { }
In the next theorem we will need the following embedding:
(21) L2(0,t;γ(H,E))֒ γ(L2(0,t;H),E),
→
which holds for a type 2 Banach space E (see p. 1460 in [27]). The first result concerning stochas-
tic Cauchy problem for delay equation (SDCP) says that its weak solutions and mild solutions are
equivalent.
Theorem 3.2. Let E be a type 2 umd− Banach space and let p [1, ). Assume that (H1) and
∈ ∞
(H3)-(H4) hold and that one of the following is satisfied:
(a) Φ (Lp( 1,0;E),E);
∈L −
(b) (H2) holds and either H has a finite dimension or E is a Hilbert space.
Let us consider (SDCP); i.e. let defined by (1) be the generator of the C -semigroup ( (t)) on
0 t≥0
A T
=E Lp( 1,0;E). LetF : andG: (H, )begivenby (2). LetY :[0, ) Ω
p p p p p p
E × − E →E E →L E ∞ × →E
be a strongly measurable, adapted process satisfying 3
(22) sup E Y(s) 2 < for all t>0.
k kEp ∞
s∈[0,t]
Then Y is a weak solution to (SDCP) if and only if Y is a solution to:
x t t
(23) Y(t)= (t) 0 + (t s)F(Y(s))ds+ (t s)G(Y(s))dW (s),
T (cid:20) f0 (cid:21) Z0 T − Z0 T − H
a.s. for all t 0.
≥
Proof. Let Y : [0, ) Ω be a strongly measurable, adapted process satisfying (22). We
p
∞ × → E
apply Theorem 2.3 to obtain the above assertion. Thus we need to check conditions (HA) and (HB)
with F and G defined by (2), and that the processes given by (6) in that theorem is an element of
γ(L2(0,t;H), ) a.s. for all t>0. Let t>0 be fixed.
p
E
First let us notice that if Φ (Lp( 1,0;E),E) or (H2) holds, E is a Hilbert space, then by
∈ L −
Lemma 3.1.(iii) or Lemma 3.1.(ii), respectively, conditions (HA) i (HB) from Theorem 2.3 hold. If H
is a finite dimensional space, then γ(H,E) is isomorphic with (H,E). Hence by Lemma 3.1.(i) it
follows that conditions (HA) i (HB) from Theorem 2.3 are satiLsfied. Since Y SL2(0,t; ) for all
∈ F Ep
t>0,inparticularY has,almostsurely,trajectoriessquareintegrable. Itisenoughtocheckcondition
(6) in Theorem 2.3. In the proof we use the following inequality:
(24) C π + π ,
kGkγ(L2(0,t;H),Ep) ≤ k 1GkL2(0,t;γ(H,E)) k 2GkLp(−1,0;L2(0,t;γ(H,E)))
(cid:16) (cid:17)
whichholdsbyembedding (21)andbyγ-Fubiniisomorphismbetweenγ(L2(0,t,H),Lp( 1,0;E))and
−
Lp( 1,0;L2(0,t;γ(H,E))) (cf. Proposition 2.6 in [27]).
−
3From now on we denote by SLqF(0,t;Ep) for some t > 0 and q ≥ 1 a Banach space of strongly measurable, adapted
processY withthenormkYkSLqF(0,t;Ep)=sups∈[0,t](cid:16)EkY(s)kqEp(cid:17)q.
10 MARIUSZGO´RAJSKI
By Lemma 3.1 we obtain
1 t 12
E u π (t u)G(Y(u)) 2 2 E ˜b2(t u)(1+ Y(u) )2du
k 7→ 1T − kL2(0,t;γ(H,E)) ≤ − k kEp
(cid:16) (cid:17) (cid:18) Z0 (cid:19)
(25) ˜b (1+ Y )< .
≤ L2(0,t) k kSL2F(0,t;Ep) ∞
(cid:13) (cid:13)
Next, by (9) and by Lemma 3.1 we see that for almost a(cid:13)ll(cid:13)ω
(cid:13) (cid:13)
(26) (θ,u) (π (t u)G(Y(u)))(θ)
k 7→ 2T − kLp(−1,0;L2(0,t;γ(H,E)))
0 t+θ p2 p1
= π (t u+θ)G(Y(u)) 2 du dθ
Z−1 Z0 k 1T − kγ(H,E) ! 
 0 t+θ 2 p2 p1
˜b2(t u+θ) 1+ Y(u) du dθ
≤Z−1 Z0 − (cid:16) k kEp(cid:17) ! 
  1
t 0 p∨22 2 2
gp∨2(t u+θ)dθ 1+ Y(u) du ,
≤ Z0 (cid:18)Z−1 − (cid:19) (cid:16) k kEp(cid:17) !
where g(u) = 1 ˜b(u),u R. In the last inequality in (26) if p 2 we apply Jensen’s inequality
{u≥0}
∈ ≤
for integral with respect to θ and then Fubini’s theorem, and if p > 2 we use Minkowski’s integral
inequality. Moreover,notice that for a.e. u [0,t] we obtain
∈
2
0 p∨2 2
gp∨2(t u+θ)dθ ˜b .
(cid:18)Z−1 − (cid:19) ≤(cid:13) (cid:13)Lp∨2(0,t)
We conclude from the above inequality and (26) that (cid:13) (cid:13)
(cid:13) (cid:13)
(27) (θ,u) (π (t u)G(Y(u)))(θ)
k 7→ 2T − kLp(−1,0;L2(0,t;γ(H,E)))
˜b √t+ Y < a.s.,
≤ Lp∨2(0,t) k kL2(0,t;Ep) ∞
(cid:13) (cid:13) (cid:16) (cid:17)
whereinthelastinequalityweuseSL2(0,t; )(cid:13) (cid:13)L2((0,t) Ω; ). Consequently,from(24)itfollows
F Ep(cid:13)⊂(cid:13) × Ep
that u (t u)G(Y(u)) belongs to γ(L2(0,t,H); ) a.s.
p
7→T − E
Havingcheckedconditions(HA),(HB)andthatprocess (t u)G(Y(u))isinγ(L2(0,t;H),E)a.s.
T −
we may apply Theorem 2.3 to obtain the desired result. (cid:3)
In the sequel we need Lemma 3.3 (cf. Lemma 4.1 in [9]). The proof of the following lemma is left
to the reader.
Lemma 3.3. Let t>0, p [1, ) and g Lp( 1,t;E). For all s [0,t] let us denote y(s)=g the
s
∈ ∞ ∈ − ∈
segments of function g. Then the following hold:
(i) the functions y : [0,t] Lp( 1,0;E), [0,t] s sy(r)dr W1,p([ 1,0];E) are continu-
7→ − ∋ 7→ 0 ∈ −
ous, and
R
(28) sup y(s) g ,
k kLp(−1,0;E) ≤k kLp(−1,t;E)
s∈[0,t]
d t
(29) y(s)ds=y(t) y(0),
dθ −
Z0