Escher J., Guidotti P., Hieber M. et al. (editors) Parabolic Problems: The Herbert Amann Festschrift

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138 W. Desch and S.-O. Londen

Theorem 3.6. Let the probability space (Ω, F, P) and the Wiener processes





∞

k=1

be as in Hypothesis 3.2.Letp ∈ [2, ∞),letthemeasurespace(B,A, Λ) and the

operator A : D(A) ⊂ L

(B; R) → L

(B; R) satisfy Hypothesis 3.1.Letα ∈ (0, 2),

β>

and γ>0.LetT>0 and assume that F :[0,T]×Ω×D((−A)

) →D((−A)



)

and G :[0,T] × Ω ×D((−A)

) → L

(B; l

) satisfy Hypotheses 3.4 and 3.5 with

suitable θ,  ∈ [0, 1].Letu

∈ L

(Ω; D((−A)

)), u

∈ L

(Ω; D((−A)

)),with

suitable δ

∈ [0, 1],bothu

measurable with respect to F

. Suppose that the following

inequalities hold:

αθ < γ + α, (3.5)

+ αθ < β, (3.6)

αθ <

+ αδ

, (3.7)

αθ < 1+

+ αδ

. (3.8)

Then there exists a unique function u ∈ L

([0,T] × Ω; D((−A)

)) such that for

almost all t ∈ [0,T]



(t −s)u(s, ω, ·) ds ∈D(A) for a.e. ω ∈ Ω,

and (1.1) is satisﬁed for almost all t ∈ [0,T] and almost all ω ∈ Ω.

The following theorem provides additional regularity results in terms of frac-

tional power domains D((−A)

) and of fractional time derivatives. However, de-

pending on the parameters, u(t) −u

may sometimes exhibit more regularity than

u(t) itself. Similar considerations hold for u(t) − tu

and u(t) −a

∗F (·,ω,u). To

handle all cases in one term, we will introduce the function v in (3.13).

Theorem 3.7. Let the assumptions of Theorem 3.6 hold. Moreover, assume that

η ∈ (−1, 1), ζ ∈ [0, 1] are such that

η + αζ < γ + α, (3.9)

+ η + αζ < β, (3.10)

η + αζ <

+ αδ

, (3.11)

η + αζ < 1+

+ αδ

. (3.12)

With the notation 1

{a>b}

=1if a>band 1

{a>b}

=0if a ≤ b,weput

v(t)=u(t) − 1

{δ

>ζ}

−1

{δ

>ζ}

− 1

{>ζ}



(t − s)F (s, ω, u(s)) ds.

(3.13)

Stochastic Integral Equations 139

(a) Then, if η>0, the function v, considered as a Banach space valued function

v :[0,T] → L

(Ω; D((−A)

)), has a fractional derivative of order η.

(b) If η<0, the function v :[0,T] → L

(Ω; L

(B; R)) has a fractional integral

of order −η. Moreover, D

v takes values in L

(Ω; D((−A)

)).

v = v.

In any case, there exists a constant M

, depending on A, p, T , α, β, γ, δ

, , ζ,

η, θ, M

, M

such that

D

v

([0,T ]×Ω;D((−A)

))

(3.14)

≤ M

u



(Ω;D((−A)

))

+ u



(Ω;D((−A)

))

+ M

F,0

+ M

G,0

Corollary 3.8. Let the Assumptions of Theorem 3.6 hold. Let ζ ∈ [0, 1].Letu be

the solution of (1.1) and v be deﬁned by (3.13).

(1) Let p<q<∞ be such that

−

+ αζ < γ + α,

−

+ αζ < β,

αζ −

<αδ

,αζ−

< 1+αδ

Then v ∈ L

([0,T]; L

(Ω; D((−A)

))).

(2) Let μ ∈ (0, 1 −

) be such that

+ μ + αζ < γ + α,

+ μ + αζ < β,

μ + αζ < αδ

,μ+ αζ < 1+αδ

Then v ∈ h

0→0

([0,T]; L

(Ω; D((−A)

))).

Hypothesis 3.9. Let F

:[0,T] × Ω ×D((−A)

) →D((−A)



)satisfyHypoth-

esis 3.4, G

:[0,T] × Ω ×D((−A)

) → L

(B; l

) satisfy Hypothesis 3.5, and

suppose that there are nonnegative functions μ

ΔF

, μ

ΔG

∈ L

([0,T] × Ω; R)such

that for all t ∈ [0,T]andu ∈D((−A)

), and almost all ω ∈ Ω

F

(t, ω, u) − F

(t, ω, u)

D((−A)



)

≤ μ

ΔF

(t, ω), (3.15)

G

(t, ω, u) − G

(t, ω, u)

(B;l

)

≤ μ

ΔG

(t, ω). (3.16)

Remark 3.10. The standard example of F

satisfying Hypothesis 3.9 is (for

i =1, 2):

(t, ω, u)=F (t, ω, u)+f

(t, ω),

(t, ω, u)=G(t, ω, u)+g

(t, ω),

where F and G satisfy Hypotheses 3.4 and 3.5, respectively, and f

∈ L

([0,T] ×

Ω; D((−A)



)), g

∈ L

([0,T] × Ω; L

(B; l

)). Here we take

ΔF

(t, ω)=f

(t, ω) − f

(t, ω)

D((−A)



)

ΔG

(t, ω)=g

(t, ω) −g

(t, ω)

(B;l

)

140 W. Desch and S.-O. Londen

Theorem 3.11. Let the probability space (Ω, F, P) and the Wiener processes w

be as in Hypothesis 3.2.Letp ∈ [2, ∞),letthemeasurespace(B,A, Λ) and the

operator A : D(A) ⊂ L

(B; R) → L

(B; R) satisfy Hypothesis 3.1.

Let T>0, α ∈ (0, 2), β>

, γ>0,andδ

,δ

, ∈ [0, 1] be such that

(3.5), (3.6), (3.7),and(3.8) hold. Let η ∈ (−1, 1) and ζ ∈ [0, 1] be such that (3.9),

(3.10), (3.11), (3.12) hold. Then there exists a constant M

Δu

> 0, dependent on

p, T, α, β, γ, δ

,δ

,,ζ,M

, such that the following Lipschitz estimate holds:

Let F

satisfy Hypotheses 3.4, 3.5 and 3.9 with , θ as above. For

i =1, 2 let the initial data u

0,i

∈ L

(Ω; D((−A)

)) and u

1,i

∈ L

(Ω; D((−A)

))

be F

-measurable, and let u

(t, ω, x), u

(t, ω, x) be the solutions of (1.1) with

F, G, u

replaced by F

0,i

1,i

.Letv

be deﬁned according to (3.13) with

u replaced by u

.Then

D

−D



([0,T ]×Ω;D((−A)

))

(3.17)

≤ M

Δu

u

0,1

− u

0,2



(Ω;D((−A)

))

+ u

1,1

− u

1,2



(Ω;D((−A)

))

+ μ

ΔF

(t, ω)+μ

ΔG

(t, ω)

([0,T ]×Ω;R)

4. Stochastic lemmas

Lemma 4.1 ([21], Theorem 3.10). Let (Ω, F, P) satisfy Hypothesis 3.2.LetY be a

dense subspace of L

(B; R), 0 <T ≤∞,andg ∈ L

([0,T] × Ω; L

(B; l

)).Then

there exists a sequence of functions g

∈ L

([0,T] × Ω; L

(B; l

)) converging to g

in L

([0,T] × Ω,L

(B; l

)) such that each g

=(g

)

∞

k=1

is of the form

(t, ω, x)=



i=1

i−1

(ω)<t≤τ

(ω)

(t)g

j,i

(x) if k ≤ j,

0 else,

(4.1)

where τ

≤ τ

≤···τ

are bounded stopping times with respect to the ﬁltration F

and g

j,i

∈ Y . (Here, for any set A, I

denotes its indicator function.)

Remark 4.2. We will apply Lemma 4.1 with Y = D(A) ∩ L

(B; R) ∩ L

∞

(B; R).

Lemma 4.3. Let (Ω, F, P) and the Wiener processes w

be as in Hypothesis 3.2.

Let p ∈ [2, ∞).LetY be a dense subspace of L

(B; R),letT>0,andletg

∈

([0,T] × Ω; L

(B; l

)) be of the simple structure given in (4.1).Fort ∈ [0,T],

let V (t):Y → L

(B; R) be a linear operator such that the function t → V (t)y is

in L

([0,T]; L

(B; R)) for each y ∈ Y . Then there exists a constant M, depending

only on p and T , such that for all t ∈ (0,T]



k=1



[V (t − s)g

(s, ω)](x) dw

dP(ω) dΛ(x)

≤ M







|[V (t − s)g

(s, ω)](x)|



dP(ω) dΛ(x).

(4.2)

Stochastic Integral Equations 141

Proof. First ﬁx some t ∈ (0,T]. For x ∈ B, r>0 we deﬁne

(r, ω, x)=

k=1



[V (t −s)g

(s, ω)](x) dw

By the elementary structure of g



[V (t −s)g

(s, ω)](x)

ds < ∞

for almost all x ∈ B,sothatY

(r, ω, x) is well deﬁned as an Ito integral for such

x, and it is a martingale. Since the Wiener processes w

are independent, the

quadratic variation of Y

(·, ·,x)is

k=1



[V (t − s)g

(s, ω)](x)

ds.

Now the Burkholder-Davis-Gundy inequality (see [18, p. 163]) yields for r ∈ [0,t]

and each x ∈ B,



k=1



[V (t − s)g

(s, ω)](x) dw

dP(ω)

≤ M



k=1

|[V (t − s)g

(s, ω)](x)|

dP(ω)

= M







|V (t − s)g

(s, ω)](x)|



dP(ω).

(4.3)

In (4.3), take r = t and integrate over B:



k=1



[V (t − s)g

(s, ω)](x) dw

dP(ω) dΛ(x)

≤ M







|[V (t − s)g

(s, ω)](x)|



dP(ω) dΛ(x). 

Lemma 4.4. Let (Ω, F, P) and the Wiener processes w

satisfy Hypothesis 3.2.Let

T>0, 2 ≤ p<∞,andg ∈ L

([0,T] × Ω; L

(B; l

)), moreover, let {g

} be a

sequence approximating g in the sense of Lemma 4.1.Letβ>

, η ∈ [0, 1) such

that β − η>

. Then the functions

∞

k=1



(t − s)g

(s, ω, x) dw

(ω)

converge in L

([0,T] × Ω; L

(B; R)),asj →∞.

142 W. Desch and S.-O. Londen

Proof. Put h

i,j

:= g

− g

. The stochastic Fubini theorem implies that

−η



β−η

(t −s)h

i,j

(s, ω, x) dw



(t − s)h

i,j

(s, ω, x) dw

i.e.,



β−η

(t −s)h

i,j

(s, ω, x) dw

= D



(t − s)h

i,j

(s, ω, x) dw

We use Lemma 4.3 and the fact that a

β−η

∈ L

([0,T]; R):



∞

k=1



(t −s)h

i,j

(s, ω, x) dw

dP(ω) dΛ(x) dt



∞

k=1



β−η

(t −s)h

i,j

(s, ω, x) dw

dP(ω) dΛ(x) dt

≤ M







β−η

(t −s)h

i,j

(s, ω, x)|



dP(ω) dΛ(x) dt

≤ M



β−η

(s) ds



i,j

(s, ω, x)|

dP(ω) dΛ(x)

≤ Mh

i,j



([0,T ]×Ω;L

(B;l

))

As i, j →∞,wehaveh

i,j

→ 0inL

([0,T]×Ω; L

(B; l

)), thus D

∞

k=1



(t−

s)g

(s, ω, x) dw

(ω) is a Cauchy sequence in L

([0,T] × Ω; L

(B; R)). 

Deﬁnition 4.5. Let (Ω, F, P) and the Wiener processes w

satisfy Hypothesis 3.2.

Let T>0, 2 ≤ p<∞,andg ∈ L

([0,T] × Ω; L

(B; l

)), moreover, let {g

} be a

sequence approximating g in the sense of Lemma 4.1. Let β>

. Then we deﬁne

g)(t, ω):=

∞

k=1



(t − s)g

(s, ω, x) dw

(ω)

:= lim

j→∞

∞

k=1



(t −s)g

(s, ω, x) dw

(ω).

5. Linear theory

In this section we replace the semilinear inhomogeneity in (1.1) by inhomogeneities

independent of u, so that we obtain a linear integral equation:

u(t, ω, x)=A



(t −s)u(s, ω, x) ds +

∞

k=1



(t −s)g

(s, ω, x) dw



(t − s)f(s, ω, x) ds + u

(ω, x)+tu

(ω, x). (5.1)

Stochastic Integral Equations 143

We will prove the following propositions by a chain of lemmas:

Proposition 5.1. Let the probability space (Ω, F, P) and the Wiener processes w

be as in Hypothesis 3.2.Letp ∈ [2, ∞),letthemeasurespace(B,A, Λ) and the

operator A : D(A) ⊂ L

(B; R) → L

(B; R) satisfy Hypothesis 3.1. Assume that

T>0 and let f ∈ L

([0,T] × Ω; L

(B; R)),andg ∈ L

([0,T] × Ω; L

(B,l

)).Let

∈ L

(Ω; L

(B; R)) and u

∈ L

(Ω; L

(B; R)) be F

-measurable.

Let α ∈ (0, 2), β>

, γ>0. Then there exists a unique function u ∈

([0,T] × Ω; L

(B,R)) such that for almost all t ∈ [0,T]



(t −s)u(s, ω, ·) ds ∈D(A) for a.e. ω ∈ Ω,

and (5.1) holds for almost all ω ∈ Ω and almost all t ∈ [0,T].

Proposition 5.2. Let the assumptions of Proposition 5.1 hold. Suppose that f ∈

([0,T]×Ω; D((−A)



)), u

∈ L

(Ω; D((−A)

)) and u

∈ L

(Ω; D((−A)

)) with

suitable , δ

,δ

∈ [0, 1).Letu be as in Proposition 5.1.Letη ∈ (−1, 1), ζ ∈ [0, 1]

satisfy

η + αζ < γ + α, (5.2)

+ η + αζ < β, (5.3)

η + αζ <

+ αδ

, (5.4)

η + αζ < 1+

+ αδ

. (5.5)

With the notation 1

{a>b}

=1if a>band 1

{a>b}

=0else, we put

v(t)=u(t) − 1

{δ

>ζ}

− 1

{δ

>ζ}

−1

{>ζ}



(t − s)f(s) ds.

(a) Then, if η>0, the function v, considered as a Banach space valued function

v :[0,T] → L

(Ω; D((−A)

)), has a fractional derivative of order η.

(b) If η<0, the function v :[0,T] → L

(Ω; L

(B; R)) has a fractional integral

of order −η. Moreover, D

v takes values in L

(Ω; D((−A)

)).

v = v.

In either case, there exist constants M

init

, M

T,Leb

,andM

T,Ito

depending on p, T ,

α, β, γ, δ

, δ

, , ζ, η such that

D

v(t)

([0,T ]×Ω;D((−A)

))

(5.6)

≤ M

init

u



(Ω;D((−A)

))

+ u



(Ω;D((−A)

))

+ M

T,Leb

f

([0,T ]×Ω;D((−A)



))

+ M

T,Ito

g

([0,T ]×Ω;L

(B,l

))

Moreover, the constants M

T,Leb

and M

T,Ito

can be made arbitrarily small by choos-

ing the time interval [0,T] suﬃciently short.

144 W. Desch and S.-O. Londen

The proof of the propositions above relies on the concept of a resolvent op-

erator (see [29]), introduced by the following deﬁnition:

Deﬁnition 5.3. Let A satisfy Hypothesis 3.1, let α ∈ (0, 2) and β>0. For t>0

we deﬁne the resolvent operator S

α,β

(t):L

(B; R) → L

(B; R)by

α,β

(t)x :=

2πi



ρ,φ

λt

α−β

(λ

− A)

−1

xdλ (5.7)

along the contour

ρ,φ

(t)=

⎧

⎪

⎨

⎪

⎩

(t − φ + ρ)e

iφ

for t>φ,

ρe

for t ∈ (−φ, φ),

(−t − φ + ρ)e

−iφ

for t<−φ,

with ρ>0, φ>

, αφ + φ

<π.

For β = 1, this deﬁnition coincides with the known notion of a resolvent op-

erator, c.f. [29]. For β>1, S

α,β

could be obtained by fractional integration of S

α,1

Equation (5.1) is formally solved by the variation of parameters formula

u(t)=S

α,1

(t)u

+ S

α,2

(t)u



α,γ

(t −s)f(s) ds +



∞

k=1

α,β

(t −s)g(s) dw

(5.8)

The task of the proof is to make sense of this formal expression in suitable function

spaces, and to show that it gives a solution of (5.1). Moreover, the estimates

claimed in Proposition 5.2 need to be veriﬁed. Since the equation is linear, all terms

,f,g can be treated separately. This is done in the following Lemmas 5.6, 5.7,

and 5.9. Uniqueness can be proved by the standard reduction to a deterministic

homogeneous equation with zero initial data, which has only the zero solution by

the well-known theory of deterministic evolutionary integral equations (see [29]).

First we collect some basic facts about the resolvent operator:

Lemma 5.4. Let A satisfy Hypothesis 3.1,letα ∈ (0, 2) and β>0. The resolvent

operator deﬁned above has the following properties:

1) For al l t>0 and all ζ ∈ [0, 1], the operator S

α,β

(t) is a bounded linear

operator L

(B,R) →D((−A)

2) For al l x ∈ L

(B; R), the function t → S

α,β

(t)x can be extended analytically

to some sector in the right half-plane.

Stochastic Integral Equations 145

3) For al l x ∈ L

(B; R) and all t>0, we have



(t −s)S

α,β

(s)x

(B;R)

ds < ∞,



(t −s)S

α,β

(s)xds ∈D(A),

α,β

(t)x = A



(t − s)S

α,β

(s)xds+ a

(t)x. (5.9)

4) Let T>0, δ, ζ ∈ [0, 1],andη ∈ (−1, 1) such that

η + αζ < β + αδ. (5.10)

Let x ∈D((−A)

) and put

v(t)=



α,β

(t)x if δ ≤ ζ,

α,β

(t)x − a

(t)x if δ>ζ.

(a) Then, if η>0, the function v, considered as a Banach-space valued

function v :[0,T] →D((−A)

), admits a fractional derivative of order

η.

(b) If η<0, the function v :[0,T] → L

(B; R), has a fractional integral of

order −η. Moreover, D

v takes values in D((−A)

v = v.

In either case, there exists some M>0 (dependent on A, α, β, ζ, δ, η)such

that for all t ∈ (0,T] and all x ∈D((−A)

D

v(t)

D((−A)

)

≤ Mt

(β+αδ)−(η+αζ)−1

x

D((−A)

)

. (5.11)

Remark 5.5. In fact, if x ∈D((−A)

)withδ ≥ ζ and β>η, the function t →

(t)x admits a fractional derivative D

x = a

β−η

x in D((−A)

). In this case,

(5.10) holds, and both functions, S

α,β

(t)x and S

α,β

(t)x − a

(t)x admit fractional

derivatives of order η in D((−A)

). On the other hand, evidently, if β ≤ η or

x ∈D((−A)

), at most one of the two functions above can have a fractional

derivative of order η in D((−A)

Proof. All these results come out of standard estimates of the contour integral,

along with the usual analyticity arguments. Since the estimate (5.11) is crucial in

the sequel, we give a more detailed proof.

First we consider the case δ ≤ ζ where we can utilize Lemma 2.2 with θ =0

for ρ in a suitable sector:

(ρ − A)

−1

x

D((−A)

)

≤ M|ρ|

ζ−δ−1

x

D((−A)

)

Formally, the Laplace transform of D

α,β

x is λ

η+α−β

(λ

−A)

−1

x. We show that

the contour integral

w(t):=

2πi



ρ,φ

λt

η+α−β

(λ

−A)

−1

xdλ

146 W. Desch and S.-O. Londen

exists in D((−A)

), if (5.10) holds.



ρ,φ

λt

η+α−β

(λ

− A)

−1

xdλ

D((−A)

)



tρ,φ





η+α−β

(





−A)

−1

dμ

D((−A)

)

= t

β−α−η−1



1,φ

η+α−β

(





− A)

−1

xdμ

D((−A)

)

≤ t

β−α−η−1



1,φ

(μ)

|μ|

α+η−β





− A



−1

D((−A)

)

|dμ|

≤ t

β−α−η−1



1,φ

(μ)

|μ|

α+η−β

α(ζ−δ−1)

x

D((−A)

)

|dμ|

= t

β−η−αζ+αδ−1

Mx

D((−A)

)



1,φ

(μ)

|μ|

η−β+α(ζ−δ)

|dμ|.

Because of (5.10), w is locally integrable and admits a Laplace transform. It re-

quires some standard complex analysis, to show that ˆw(λ)=λ

η+α−β

(λ

−A)

−1

Nowwehavetoshowthatinfactw = D

α,β

First consider the case η>0: By the convolution theorem for Laplace trans-

forms we have



−η

w](λ)=λ

α−β

(λ

−A)

−1

whence w = D

α,β

x.Incaseη<0, the convolution theorem yields



α,β

x(λ)=λ

α−β

(λ

− A)

−1

x =ˆw(λ).

To handle the case δ>ζ, we will use Lemma 2.2 with θ =1:

A(ρ −A)

−1

x

D((−A)

)

≤ Mρ

ζ−δ

x

D((−A)

)

Notice ﬁrst that ˆa

(λ)=λ

−β

,and

(t)=

2πi



ρ,φ

λt

−β

dλ.

Therefore,

α,β

(t)x −a

(t)x =

2πi



ρ,φ

λt

[λ

α−β

(λ

−A)

−1

x − λ

−β

x] dλ

2πi



ρ,φ

λt

−β

A(λ

− A)

−1

xdλ.

Stochastic Integral Equations 147

Now we estimate similarly as above



ρ,φ

λt

η−β

A(λ

− A)

−1

xdλ

D((−A)

)

≤



1,φ

(μ)

η−β

α(ζ−δ)

x

D((−A)

)

|dμ|

= Mx

D((−A)

)

−η+β−αζ+αδ−1



1,φ

(μ)

|μ|

η−β+αζ−αδ

|dμ|.

Thus, for t>0, the following integral exists in D((−A)

(t):=

2πi



ρ,φ

λt

η−β

A(λ

− A)

−1

xdλ,

w

(t)

D((−A)

)

≤ Mt

(β+αδ)−(η+αζ)−1

In the end one veriﬁes again that in fact w

(t)=D

v(t). 

Lemma 5.6 (Contribution of the initial conditions u

, u

). Let A satisfy Hypo-

thesis 3.1,let(Ω, F, P) be a probability space, p ∈ [2, ∞).Letα ∈ (0, 2), 0 <T<

∞,andu

∈ L

(Ω; L

(B; R)). We deﬁne u(t):=S

α,1

(t)u

+ S

α,2

(t)u

1) The function u exists in L

∞

([0,T]; L

(Ω × B; R)). For all t>0 we have



(t −s)u(s) ds ∈D(A),

u(t)=A



(t −s)u(s) ds + u

+ tu

2) Moreover, suppose that u

∈ L

(Ω; D((−A)

)) with some δ

∈ [0, 1].Let

ζ ∈ [0, 1], η ∈ (−1, 1) be such that

η + αζ <

+ αδ

, (5.12)

η + αζ < 1+

+ αδ

, (5.13)

Put

v(t)=u(t) −1

ζ<δ

− 1

ζ<δ

Then v has a fractional derivative of order η (if η<0: a fractional integral

of order −η) which is in L

([0,T] × Ω; D((−A)

)) and satisﬁes

D

v(t, ω)

([0,T ]×Ω;D((−A)

))

≤ M

u



(Ω;D((−A)

))

+ u



(Ω;D((−A)

))

with a constant M depending on p, A, T, α, δ

,δ

,ζ,η.