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Maximal regularity for stochastic convolutions driven by Levy noise

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Abstract

We show that the result from Da Prato and Lunardi is valid for stochastic convolutions driven by L\'evy processes.
arXiv:0709.3179v1 [math.PR] 20 Sep 2007
MAXIMAL REGULARITY FOR STOCHASTIC
CONVOLUTIONS DRIVEN BY LEVY NOISE
ZDZIS LAW BRZE
´
ZNIAK AND ERIKA HAUSENBLAS
Abstract. We show that the result from Da Prato and Lunardi
is valid for stochastic convolutions driven by evy processes.
1. Introduction
The aim of the article is to investigate the maximal regularity of the
Ornstein-Uhlenbeck driven by purely discontinuous noise. In particu-
lar, let (S, S) be a measurable space, E be a Banach space of martingale
type p, 1 < p 2, a nd A be an infinitesimal generator of an analytic
semigroup (e
tA
)
0t<
in E. We consider the following SPDE written
in the Itˆo-form
du(t) = Au(t) dt +
R
S
ξ(t; x)˜η(dx; dt),
u(0) = 0,
(1)
where ˜η is a S-valued time homog eneous compensated Poisson random
measure defined on a filtered probability space (Ω; F; (F
t
)
0t<
; P)
with evy measure ν on S, specified later, and ξ : × S E is a pre-
dictable process satisfying certain integrability conditions also specified
later. The solution to (1) is given by the so called Ornstein-Uhlenbeck
process
u(t) :=
Z
t
0
Z
S
e
A(tr)
ξ(r, x) ˜η(dx; dr), t > 0.
Date: February 2, 2008.
Key words and phrases. Stochastic convolution and time homogeneous Poisson
random measure and maximal regularity and martingale type p Banach spaces.
The research of the second named author was supported by a grant P17273
of the Austrian Science Foundation. The research of the first named author was
supported by a grant. He would like to thank the Department of Mathematics,
University of Salzburg, for the hospitality. The research on this paper was initiated
during a visit of both authors to the Centro di Ricerca Matematica Ennio de Giorgi
in Pisa (Italy), in July 2006.
1
2 ZB AND EH FEBRUARY 2, 2008
Suppose 1 q p. Our main result will be the following inequality
(2)
E
Z
T
0
|u(t)|
p
D
A
(θ+
1
p
,q)
dt CE
Z
T
0
Z
S
|ξ(t, z)|
p
D
A
(θ,q)
dt,
where D
A
(θ, p), θ (0, 1), denotes the real interpolation space of order
δ between E and D(A).
As mentioned in the beginning, if the Or nstein-Uhlenbeck process
is driven by a scalar Wiener process, the question of maximal regu-
larity was answered by Da Prato in [7] or Da Prato and Lunardi [8].
We transfer these results to the Ornstein-Uhlenbeck process driven by
purely discontinuous noise.
Notation 1. By N we deno te the set of natural numbers, i.e. N =
{0, 1, 2, · · · } and by
¯
N we denote the set N {+ }. Whenever we
speak about N (or
¯
N)-valued measurable functions we impl i c itly assume
that that set is equipped with the trivial σ-field 2
N
(or 2
¯
N
). By R
+
we
will denote the interval [0, ). If X is a topologica l spa ce, then by
B(X) we will denote the B orel σ-field on X. B y λ we will denote the
Lebesgue measure on (R, B(R)). Fo r a measurable space (S, S) let M
+
S
be the set of a ll non negative measures on (S, S).
2. Main results
Suppose that p (1, 2] and that E is a Banach space of martingale
type p. Let (S, S) be a measurable space and ν M
+
S
. Suppose that
P = (Ω, F, (F
t
)
t0
, P) is a filtered probability space, η : S×B(R
+
)
¯
N
is time homogeneous Poisson random measure with intensity measure ν
defined over (Ω, F, P) and adapted to filtration (F
t
)
t0
. We will denote
by ˜η = η γ the to η associated compensated Poisson random measure
where γ is given by
B(R
+
) × S (A, I) 7→ γ(A, I) = ν(A)λ(I) R
+
.
We denote by P the σ field on × R
+
generated by all sets A
F
ˆ
×B(R
+
), where A is of the form A = F × (s, t], with F F
s
and
s, t R
+
. If ξ : × R
+
S is P measurable, ξ is called predictable.
It is then known, see e.g. appendix B, that there exists a unique
continuous linear operator associating with each predictable process
ξ : R
+
× S × E with
(3) E
Z
T
0
Z
S
|ξ(r, x)|
p
ν(dx) dr < , T > 0,
an adapted adl´ag process, denoted by
R
t
0
R
S
ξ(r, x)˜η(dx, dr ) , t 0
such that if ξ satisfies the above condition (3) and is a step process
February 2, 2008MAXIMAL REGULARITY 3
with representation
ξ(r) =
n
X
j=1
1
(t
j1
,t
j
]
(r)ξ
j
, 0 r,
where {t
0
= 0 < t
1
< . . . < t
n
< ∞} is a partition o f [0, ) and for all
j, ξ
j
is an F
t
j1
measurable random variable, then
(4)
Z
t
0
Z
S
ξ(r, x)˜η(dx, dr) =
n
X
j=1
Z
S
˜
ξ
j
(x)η (dx, (t
j1
t, t
j
t]) .
The continuity mentioned above means that there exists a constant
C = C(E) independent of ξ such that
(5) E|
Z
t
0
Z
S
ξ(r, x)˜η(dx, dr ) |
p
CE
Z
t
0
Z
S
|ξ(r, x)|
p
ν(dx) dr, t 0.
One can prove
1
, see e.g. the proof of Proposition 3.3 in [12], or The-
orem 3.1 in [3] for the case q < p, and Corollary B.6 in Appendix B,
that for any q [1, p] there exists a constant C = C
q
(E) such that for
each process ξ as above and for all t 0,
(6)
E|
Z
t
0
Z
S
ξ(r, x)˜η(dx, dr ) |
q
CE
Z
t
0
Z
S
|ξ(r, x)|
p
ν(dx) dr
q/p
.
Remark 1. Let us denote
I(ξ)(t) :=
Z
t
0
Z
S
ξ(r, x)˜η(dx, dr), t 0
kξk :=
Z
S
|ξ(x)|
p
ν(dx)
1/p
, ξ L
p
(S, ν; E).
Then the inequality (6 ) takes the followin g form
E|I(ξ)(t)|
q
C
q
(E)E
Z
t
0
kξ(r)k
p
dr
q/p
.
This should be (and will be) compared with the Gaussian case. Note that
in this case kξk is simply the L
p
(S, ν, E) norm of ξ. In the Gaussian
case the situation is different.
Let us also point out that the inequality (6) for q < p follows from the
same inequality for q = p. In fact, using Proposition IV.4.7 from [20],
see the proof of Theo rem 3. 1 in [3], one can prove a stronger result.
Namely that if inequality (6) holds true for q = p, then for q [1 , p )
1
The case q (p, ) is different and will be discussed later.
4 ZB AND EH FEBRUARY 2, 2008
there exists a constant K
q
> 0 such that for each accessible stopping
time τ > 0,
(7) E sup
0tτ
|I(ξ)(t)|
q
K
q
E
Z
τ
0
kξ(t)k
p
dt
q/p
.
Assume further that A is an infinitesimal generator of an analytic
semigroup denoted by (e
tA
)
t0
on E.
Define the stochastic convolution of the semigroup (e
tA
)
t0
and a n
E-valued process ξ as above by the following formula
(8) SC(ξ)(t) =
Z
t
0
Z
S
e
(tr)A
ξ(r, x)˜η(dx, dr), t 0.
Let us recall that there exist constants C
0
and ω
0
such that
ke
tA
k C
0
e
0
, t 0.
Without loss of generality, we will assume from now on that ω
0
< 0.
Let us also recall the following characterization of the real interpola-
tion
2
spaces (E, D(A
m
))
θ,q
= (D(A
m
), E)
1θ,q
, where m N, between
D(A
m
) and E with parameters θ (0, 1) and q [1, ), see section
1.14.5 in [21] or [7]. If δ (0, ] then
(D(A
m
), E)
1θ,q
=
x E :
Z
δ
0
|t
m(1θ)
A
m
e
tA
x|
q
dt
t
<
.
(9) (E, D(A
m
))
ϑ,q
=
x E :
Z
δ
0
|t
m(1ϑ)
A
m
e
tA
x|
q
dt
t
<
.
The norms defined by the equality (9) for different values of δ are
equivalent.
The space (D(A
m
), E)
1θ,q
= (E, D(A
m
))
θ,q
is often denoted by
D
A
m
(θ, p) and we will use the fo llowing notation
(10)
|x|
q
D
A
m
(θ,q);δ
=
Z
δ
0
|t
m(1θ)
A
m
e
tA
x|
q
dt
t
.
2
In order to fix the notation let me point out that the interpolation functor
(X
0
, X
1
)
θ,q
, θ (0, 1), q [1, ], between two Banach spaces X
1
and X
0
such that
both are continuously embedded into a common topological Hausdorff vector space,
satisfies the following properties: (i)(X
1
, X
0
)
θ,q
= (X
0
, X
1
)
1θ,q
, (ii) if X
0
X
1
,
0 < θ
1
< θ
2
< 1 and p, q [1, ], then (X
0
, X
1
)
θ
1
,p
(X
0
, X
1
)
θ
2
,q
. Roughly
speaking, (ii) implies that, if X
0
X
1
, then (X
0
, X
1
)
ϑ,p
ց X
0
as ϑ ց 0 and
(X
0
, X
1
)
θ,p
ր X
1
as ϑ ր 0. Or equivalently, if X
0
X
1
, then (X
1
, X
0
)
θ,p
ց X
0
as θ ր 1 and (X
1
, X
0
)
θ,p
ր X
1
as θ ց 1. See Proposition 1.1.4 in [15] and section
1.3.3 in [21].
February 2, 2008MAXIMAL REGULARITY 5
In the general case, one has the following equality but only for δ
(0, ):
(11)
(E, D(A
m
))
θ,q
=
x E :
Z
δ
0
|t
m(1θ)
(ω
0
I + A)
m
e
t(ω
0
+A)
x|
p
dt
t
<
.
In this case, the formula (11) takes the following form
(12) |x|
q
D
A
m
(θ,q);δ
=
Z
δ
0
|t
m(1θ)
A
m
e
tA
x|
q
dt
t
+ |x|
q
.
Let us finally recall that if 0 < k < m N, p [1, ] and θ
(0, 1), then (E, D(A
k
))
θ,p
= (E, D(A
m
)) k
m
θ,p
, see [21] Theorem 1.15.2
(f). Therefore, if p [1, ) and θ [0, 1
1
p
), then
(13) D
A
(θ +
1
p
, q) = D
A
2
(
θ
2
+
1
2p
, q)
with equivalent norms.
Our main result in this note is the following
Theorem 2.1. Under the above assumptions, for all θ ( 0 , 1
1
p
),
there e xists a constant C =
ˆ
C
θ
(E) such that for any process ξ described
above and all T 0, the followin g inequality holds
(14)
E
Z
T
0
|SC(ξ)(t)|
p
D
A
(θ+
1
p
,q)
dt CE
Z
T
0
Z
S
|ξ(t, z)|
p
D
A
(θ,q)
ν(dx) dt.
In the Gaussian case and q = p = 2, and E being a Hilbert space,
the above result was proved by Da Prato in [7]. This result was then
generalized to a class of so called Banach spaces of martingale type 2
in [1], see also [2], for nuclear Wiener process and in [4], to the case
of cylindrical Wiener process. Finally, Da Prato and Lunardi studied
in [8] the case when p = 2 and q 2 for a one dimensional Wiener
process. However, a generalisation of the last result to a cylindrical
Wiener process does not cause any serious problems. We will state
corresponding result at the end of this Note.
Theorem 2.1 will be deduced from a more general result whose idea
can be tra ced back t o R emark 1.
Theorem 2.2. Let (Ω, F, (F
t
)
t0
, P) is a filtered probability spa ce, p
(1, 2] and q [p, ). Le t E
p
be a class of se parable Banach spaces
satisfying the following properties.
6 ZB AND EH FEBRUARY 2, 2008
(R1) With each space E belonging to the class E
p
we associate a
separable Banach space R = R(E) such that there is a fam-
ily (I
t
)
t0
of linear opera tors from the class M
p
loc
(R(E)) of all
predictable R (E)-valued processes to L
p
(Ω, F
t
, P; E) such that
for some con s tant C = C
p
> 0
(15) E|I
t
(ξ)|
p
E
C
p
E
Z
t
0
kξ(r)k
p
R(E)
dr
.
(R2) If E E
p
and E
1
isomorphic to E, then E
1
belongs to E
p
as
well.
(R3) If E
1
, E
2
E
p
and Φ : E
1
E
2
is a bounded linear operator,
then
kΦξk
R(E
2
)
|Φ|kξk
R(E
1
)
, ξ R(E
1
).
(R4) If (E
0
, E
1
) is an interpolation couple such that E
1
, E
2
E
p
, then
the real interpolation s paces (E
0
, E
1
)
θ,p
, θ (0, 1), belongs to E
p
as well.
(R5) For every δ > 0 here exists a constant K
δ
> 0 such that
(16)
Z
δ
0
kr
1θ
Ae
rA
ξk
p
R(E)
dr
r
K
p
δ
kξk
p
R(D
A
(θ,p))
, ξ R(E).
(R6) There exis ts a constant
ˆ
C
q
> 0 such that for all t > 0
(17) E|I
t
(ξ)|
q
E
ˆ
C
q
E
Z
t
0
kξ(r)k
p
R(E)
dr
q/p
, ξ M
p
loc
(R(E)).
Define another f amily (SC
t
)
t0
of linear operators from M
p
loc
(R(E)) to
L
p
(Ω, F
t
, P; E) by the following formula
(18) SC
t
(ξ) = I
t
e
(t−·)A
ξ(·)
, t 0.
Then, for every θ (0, 1
1
p
), there exists a constant
ˆ
C
q
(E) such
that for all T > 0 the following inequality holds
(19) E
Z
T
0
|SC
t
(ξ)|
q
D
A
(θ+
1
p
,q)
dt
ˆ
C
q
(E)E
Z
T
0
kξ(s)k
q
R(D
A
(θ+
1
p
,q))
dt.
Remark 2. It follows from (i) that if ξ(r) = η(r) a.s. for a.a. r [0, t],
then I
t
(ξ) = I
t
(η).
Now we shall present two basic examples.
Example 2.3. Let (Ω, F, (F
t
)
t0
, P) be a filtered probability spa ce, p =
2. Let H be a separable Hilbert space and let E
2
be a class of all
2-smoothable Banach spaces. With E E
2
we associate the space
R(E) := R(H, E) of all γ-radonifying operators from H to E. It is
February 2, 2008MAXIMAL REGULARITY 7
known, see [17] that R (H, E) is a separable Banach space equipped with
any of the followin g equivale nt norms
3
, 2 q < ,
kϕ k
q
R(H,E);q
:= E|
X
j
β
j
ϕ e
j
|
q
E
, ϕ R(H, E),(20)
{e
k
}
k
be an ONB of H and {β
k
}
k
a sequence of i.i.d. Gaussian N(0,1)
random variables .
Example 2.4. Let (Ω, F, (F
t
)
t0
, P) be a filtered probability space, p
(1, 2]. Let (S, S) be a measurable space an d η : S
ˆ
×B(R
+
) N
+
be a time homogeneous, compensa ted Poisson random meas ure over
(Ω; F; P) adapted to filtration (F
t
)
t0
with intensity ν M
+
S
. Let E
p
be the set of all separable Banach spaces of martingale type p. With
E E
p
we associated a measurab l e transformation ξ : S E such that
Z
S
|ξ(x)|
p
E
ν(dx) < .
Then for q [p, ) let
kξk
q
R(E)
:= E
Z
1
0
Z
S
ξ(x) ˜η(dx, dr)
q
E
.
3. Proof of Theorem 2.2
We begin with the case q = p. Without lo ss of generality the norm
| · |
D
A
(θ+
1
p
,p);1
, defined by formula (10), will be denoted by | · |
D
A
(θ+
1
p
,p)
.
Also, we may assume that A
1
exists and is bounded so that the graph
norm in D(A) is equivalent to the norm |A · |.
By the equality (1 3), definition (10), the Fubini Theorem and formula
(18) we have
E
Z
T
0
|SC
t
(ξ)|
p
D
A
(θ+
1
p
,p)
dt CE
Z
T
0
|SC
t
(ξ)|
p
D
A
2
(
θ
2
+
1
2p
,p)
dt
= C
Z
T
0
Z
1
0
E|r
2(1
θ
2
1
2p
)
A
2
e
rA
SC
t
(ξ)|
p
dr
r
dt
= C
Z
T
0
Z
1
0
r
p(2θ)1
E|A
2
e
rA
I
t
e
(t−·)A
ξ(·)
|
p
dr
r
dt
= C
Z
T
0
Z
1
0
r
p(2θ)1
E|I
t
A
2
e
rA
e
(t−·)A
ξ(·)
|
p
dr
r
dt · · ·
3
Equivalence of the norms is a consequence of Khinchin-Kahane inequality.
8 ZB AND EH FEBRUARY 2, 2008
By applying next the inequality (15), the property (R3), the Fubini
Theorem, the fact that |Ae
r
2
A
| Cr
1
, r > 0, f or some constant
C > 0 as well as by observing that
1
tu+r
1
r
for t [u, T ], r > 0, we
infer that
· · · C
p
Z
1
0
r
p(2θ)1
Z
T
0
E
Z
t
0
kA
2
e
(tu+r)A
ξ(u)k
p
R(E)
du dt
dr
r
C
p
Z
1
0
r
p(2θ)1
Z
T
0
E
Z
t
0
|Ae
tu+r
2
A
|
p
kAe
tu+r
2
A
ξ(u)k
p
R(E)
du dt
dr
r
C
p
E
Z
1
0
r
p(2θ)1
sup
0ut
(t u + r)
p
Z
T
0
Z
T
u
kAe
tu+r
2
A
ξ(u)k
p
R(E)
dt
du
dr
r
C
p
E
Z
T
0
Z
T +1ρ
0
kAe
σ
2
A
ξ(ρ)k
p
R(E)
Z
ρ+σ
ρ(σ+ρ1)
(σ + ρ τ )
p(1θ)2
C
p
E
Z
T
0
Z
T +1ρ
0
kAe
σ
2
A
ξ(ρ)k
p
R(E)
Z
ρ+σ
ρ
(σ + ρ τ )
p(1θ)2
= C
p
E
Z
T
0
Z
T +1ρ
0
kAe
σ
2
A
ξ(ρ)k
p
R(E)
Z
σ
0
τ
p(1θ)2
= C
p
E
Z
T
0
Z
T +1ρ
0
σ
p(1θ)1
kAe
σ
2
A
ξ(ρ)k
p
R(E)
C
′′
p
E
Z
T
0
Z
T/2
0
kσ
1θ
Ae
σA
ξ(ρ)k
p
R(E)
σ
ˆ
C
′′′
p
K
p
T
E
Z
T
0
kξ(r)k
p
R(D
A
(θ,p))
dr,
where the last inequality is a consequence o f the assumption (R5).
The proof in the case q > p follows the same ideas. Note also that the
above prove resembles closely the proof from [8]. We give full details
below.
February 2, 2008MAXIMAL REGULARITY 9
We consider now the case q > p. We use the same notation as in
the previous case. But we will make some (or the same) additional as-
sumptions. By the equality (13), definition (10), the Fubini Theorem
and f ormula (18) we have
E
Z
T
0
|SC
t
(ξ)|
q
D
A
(θ+
1
p
,q)
dt CE
Z
T
0
|SC
t
(ξ)|
q
D
A
2
(
θ
2
+
1
2p
,q)
dt
= C
Z
T
0
Z
1
0
s
q(2θ)
q
p
E|A
2
e
sA
SC
t
(ξ)|
q
ds
s
dt
= C
Z
T
0
Z
1
0
s
q(2θ)
q
p
E|A
2
e
sA
I
t
e
(t−·)A
ξ(·)
|
q
ds
s
dt
= C
Z
T
0
Z
1
0
s
q(2θ)
q
p
E|I
t
A
2
e
sA
e
(t−·)A
ξ(·)
|
q
ds
s
dt · · ·
Before we continue, we formulate the following simple Lemma.
Lemma 3.1. There exists a constant C > 0 s uch that for all t > 0,
s (0, 1)
Z
t
0
1
(t r + s)
pq
qp
dr
q
p
1
C
1
s
q(1
1
p
)+1
Proof of Lemma 3.1. Denote α =
pq
qp
and observe that α > 1. Since
R
t
0
1
(tr+s)
α
dr =
R
t
0
1
(r+s)
α
dr
R
0
1
(r+s)
α
dr =
1
α1
1
s
α1
and (α 1)(
q
p
1) = q(1
1
p
) + 1 , the result follows.
As in the earlier case, by applying the inequality (15), the property
(R3), the Fubini Theorem, the fact that |Ae
s
2
A
| Cs
1
, s > 0, for
some constant C > 0 as well as older inequality and Lemma 3.1 we
infer that
· · ·
ˆ
C
q
Z
1
0
s
q(2θ)
q
p
Z
T
0
E
Z
t
0
kA
2
e
(tr+s)A
ξ(r)k
p
R(E)
dr
q/p
dt
ds
s
C
ˆ
C
q
E
Z
1
0
s
q(2θ)
q
p
Z
T
0
Z
t
0
|Ae
tr+s
2
A
|
p
kAe
tr+s
2
A
ξ(r)k
p
R(E)
dr
q/p
dt
ds
s
C
ˆ
C
q
E
Z
1
0
s
q(2θ)
q
p
Z
T
0
Z
t
0
|Ae
tr+s
2
A
|
pq
qp
dr
q
p
1
Z
t
0
kAe
tr+s
2
A
ξ(r)k
q
R(E)
dr
dt
ds
s
10 ZB AND EH FEBRUARY 2, 2008
C
ˆ
C
q
E
Z
1
0
s
q(2θ)
q
p
Z
T
0
1
s
q(1
1
p
)+1
Z
t
0
kAe
tr+s
2
A
ξ(r)k
q
R(E)
dr dt
ds
s
= C
ˆ
C
q
E
Z
1
0
s
q(1θ)1
Z
T
0
Z
T
r
kAe
tr+s
2
A
ξ(r)k
q
R(E)
dt
dr
ds
s
C
ˆ
C
q
E
Z
T
0
Z
T +1ρ
0
kAe
σ
2
A
ξ(ρ)k
q
R(E)
Z
ρ+σ
ρ(σ+ρ1)
(σ + ρ τ )
q(1θ)2
C
ˆ
C
q
E
Z
T
0
Z
T +1ρ
0
kAe
σ
2
A
ξ(ρ)k
q
R(E)
Z
ρ+σ
ρ
(σ + ρ τ )
q(1θ)2
= C
ˆ
C
q
E
Z
T
0
Z
T +1ρ
0
kAe
σ
2
A
ξ(ρ)k
q
R(E)
Z
σ
0
τ
q(1θ)2
=
ˆ
C
q
E
Z
T
0
Z
T +1ρ
0
σ
q(1θ)1
kAe
σ
2
A
ξ(ρ)k
q
R(E)
ˆ
C
′′
q
E
Z
T
0
Z
T/2
0
kσ
1θ
Ae
σA
ξ(ρ)k
q
R(E)
σ
ˆ
C
′′
q
K
p
T/2
E
Z
T
0
kξ(r)k
q
R(D
A
(θ,q))
dr
where the last inequality follows from Assumption R5. This completes
the proof.
4. Stochastic convolution in the cylindrical Gaussian
case
Assume now t hat W (t), t 0, is a cylindrical Wiener process defined
on some complete filtered probability space (Ω, F, (F
t
)
t0
, P). Let us
denote by H the RKHS of that process, i.e. H is equal to the RKHS
of W (1).
Theorem 4.1. Under the above assumptions there exists a constant
ˆ
C
q
(E) such that for any process ξ described above the followin g in-
equality holds
(21)
E
Z
T
0
|SC(ξ)(t)|
q
D
A
(θ+
1
p
,q)
dt
ˆ
C
q
(E)E
Z
T
0
kξ(t)k
q
R(H,D
A
(θ,q))
dt, T 0.
The proof of Theorem 4.1 will be preceded by the following useful
result.
Proposition 4.2. Let us ass ume that θ (0, 1), q 1 and T > 0.
Then there exis ts a con s tan t K
T
> 0 such that for each bounded linear
map ϕ : H E the following inequali ty holds
February 2, 2008MAXIMAL REGULARITY 11
K
1
T
kϕ k
q
R(H,(E, D(A))
θ,q
Z
T
0
t
(1θ)q
kAe
tA
ϕ k
q
R(H,E)
dt
t
K
T
kϕ k
q
R(H,(E, D(A))
θ,q
)
.(22)
In particular, ϕ R(H, (D( A), E)
θ,q
) iff (for some and/or all T >
0) the integral
R
T
0
t
(1θ)q
kAe
tA
ϕ k
q
R(H,E)
dt
t
is fi nite.
Proof of Proposition 4.2. Let {e
k
}
k
be a n ONB of H and {β
k
}
k
a se-
quence of i.i.d. Gaussian N(0,1) random variables. It is known, see
e.g. [13] that there exists a constant C
p
(E) such that for each linear
operator ϕ : H E the following inequality holds.
C
p
(E)
1
E|
X
j
β
j
ϕ e
j
|
p
E
kϕ k
p
R(H,E)
C
p
(X)E|
X
j
β
j
ϕ e
j
|
p
E
(23)
We have
Z
T
0
t
(1θ)q
kAe
tA
ϕ k
q
R(H,E)
dt
t
C
q
(E)
Z
T
0
t
(1θ)q
E|
X
k
β
k
Ae
tA
ϕ e
k
|
q
E
dt
t
= C
q
(E)E
Z
T
0
t
(1θ)q
|
X
k
β
k
Ae
tA
ϕ e
k
|
q
E
dt
t
= C
q
(E)E|
X
k
β
k
Ae
tA
ϕ e
k
|
q
D
A
(θ,q);T
C(T )C
q
(E)kϕ k
q
R(H,D
A
(θ,q))
.
Since D
A
(ϑ, q) = (E, D(A))
θ,q
with equivalent norms, this proves the
second inequality in (22). The first inequality follows the same lines.
Proof of Theorem 4.1. From Proposition 4.2 we infer that the assump-
tion (r5) in Theorem 2.2 is satisfied. Since it is well known that the
other assumptions are also satisfied, see e.g. [3], the result follows from
Theorem 2.2.
5. Proof of Theorem 2.1
We only need to prove a version of Proposition 4.2 with R(H, E)
being replaced by R(E) := L
p
(S, ν, E). We recall that here the measure
space (S, S, ν) is fixed for the whole section.
12 ZB AND EH FEBRUARY 2, 2008
Proposition 5.1. Let us a ssume that θ (0, 1), q 1 and T >
0. Then there exists a constant K
T
> 0 such that for each ϕ
L
p
(S, ν, E) = : R(E) the f ollowin g inequality holds
K
1
T
kϕ k
q
R((E,D(A))
θ,q
)
Z
T
0
t
(1θ)q
kAe
tA
ϕ k
q
R(E)
dt
t
K
T
kϕ k
q
R((E,D(A))
θ,q
)
.(24)
In particular, ϕ R((D(A), E)
θ,q
) iff (for some and/or all T > 0)
the integral
R
T
0
t
(1θ)q
kAe
tA
ϕ k
q
R(E)
dt
t
is finite.
Proof of Proposition 5.1. Follows by applying the Fubini Theorem.
Appendix A. Martingale type p, p [1, 2], Banach spaces
In this section we collect some basic information about the martin-
gale type p, p [1, 2], Banach spaces.
Assume also that p [1, 2] is fixed. A Banach space E is of mar-
tingale type p iff there exists a constant L
p
(E) > 0 such that for all
X-valued finite martingale {M
n
}
N
n=0
the following inequality holds
(25) sup
n
E|M
n
|
p
L
p
(E)
N
X
n=0
E|M
n
M
n1
|
p
,
where as usually, we put M
1
= 0.
Let us recall that a Banach space X is of type p iff there exists a
constant K
p
(X) > 0 for any finite sequence ε
1
, . . . , ε
n
: {−1, 1} of
symmetric i.i.d. random variables and for any finite sequence x
1
, . . . , x
n
of elements of X, the f ollowing inequality holds
(26) E|
n
X
i=1
ε
i
x
i
|
p
K
p
(X)
n
X
i=1
|x
i
|
p
.
It is known, see e.g. [14, Theorem 3.5.2 ], that a Banach space X
is of type p iff it is of G aussian type p, i.e. there exists a constant
˜
K
p
(X) > 0 such that for any finite sequence ξ
1
, . . . , ξ
n
of i.i.d. N(0, 1)
random variables and for any finite sequence x
1
, . . . , x
n
of elements of
X, the following inequality holds
(27) E|
n
X
i=1
ξ
i
x
i
|
p
˜
K
p
(X)
n
X
i=1
|x
i
|
p
,
It is now well known, see e.g. Pisier [18] and [19], that X is of mar-
tingale type p iff it is p-smooth, i.e. there exists an equivalent norm | · |
February 2, 2008MAXIMAL REGULARITY 13
on X and there exist a constant K >) such that ρ
X
(t) Kt
p
for a ll
t > 0, where ρ
X
(t) is the modulus of smoothness of (X, | · |) defined by
ρ
X
(t) = sup{
1
2
(|x + ty| + |x ty|) 1 : | x|, |y| = 1}.
In particular, all spaces L
q
for q p and q > 1, are of martingale
type p.
Let us also recall that a Banach space X it is an UMD space (i.e.
X has the unconditional martingale difference property) iff for any
p (1, ) there exists a constant β
p
(X) > 0 such that for any X
-valued martingale difference {ξ
j
} ( i.e.:
P
n
j=1
ξ
j
is a martingale), for
any ǫ : N {−1, 1} and for any n N
(28) E|
n
X
j=1
ǫ
j
ξ
j
|
p
β
p
(X)E|
n
X
j=1
ξ
j
|
p
.
It is known, see [5] and references therein, that for a Banach space
X the following conditions are equivalent: i) X is an UMD space, (ii)
X is ζ convex, (iii) the Hilbert transform for X-valued functions is
bounded in L
p
(R, X) for any (or some ) p > 1.
Finally, it is known, see e.g. [18 , Proposition 2.4], that if a Banach
space X is both UMD and of type p, then X is of martingale type p.
Appendix B. Proof of inequality 5
In this appendix we formulate and prove inequality 5. Our a pproach
is a sense similar to the approach used in the G aussian case by Neidhard
[17] and Brze´zniak [2] or in the Poisson random measure in Madrekar
and R¨udiger [16]. In fact, our main result below can be seen a gen-
eralisation of Theorem 3.6 from [16] to the case of martingale type p
Banach spaces.
Notation 2. By M
¯
N
S×R
+
we denote the family of all
¯
N-valued measures
on (S × R
+
, S B
R
+
) and M
¯
N
S×R
+
is the σ-field on M
¯
N
S×R
+
generated
by functions i
B
: M µ 7→ µ(B)
¯
N, B S B
R
+
.
Let us assume that (S, S) is a measurable space, ν M
+
S
is a
non-negative measure on (S, S) and P = (Ω, F, (F
t
)
t0
, P) is a fil-
tered probability space. We also assume that η is time homogeneous
Poisson random measure over P, with t he intensity measure ν, i.e.
η : (Ω, F) (M
¯
N
S
, M
¯
N
S×R
+
) is a measurable function satisfying the
following conditions
14 ZB AND EH FEBRUARY 2, 2008
(i) for each B S B
R
+
, η(B) := i
B
η :
¯
N is a Poisson random
variable with parameter
4
Eη(B);
(ii) η is independently scattered, i.e. if the sets B
j
S B
R
+
, j =
1, · · · , n are pair-wise disjoint, then the random variables η(B
j
), j =
1, · · · , n ar e pair-wise independent;
(iii) for all B S and I B
R
+
, E
η(I × B)
= λ(I)ν(B) , where λ is
the Lebesgue measure;
(iv) for each U S, the
¯
N-valued processes (N(t, U))
t>0
defined by
N(t, U) := η((0, t] × U), t > 0
is (F
t
)
t0
-adapted and its increments ar e independent of the past, i.e.
if t > s 0, then N(t, U) N(s, U) = η((s, t] × U) is independent of
F
s
.
By ˜η we will denote the compensated Poisson ra ndom measure, i.e. a
function defined by ˜η(B) = η(B) E(η(B)), whenever the difference
makes sense.
Lemma B.1. Let p (1, 2] and assume that E is a Ban ach s pace of
martingale type p. If a finitely-valued function f belongs to L
p
(Ω ×
S, F
a
S; P ν; E) for some a R
+
, then for any b > a,
(29) E|
Z
S
f(x)˜η(dx, (a, b])|
p
E
2
2p
L
p
(E)(b a)E
Z
S
|f(x)|
p
E
ν(dx)
Since the space of finitely-valued functions is dense in L
p
(Ω×S, F
a
S; P ν; E), see e.g. Lemma 1.2.14 in [6].
Corollary B.2. Under the assumptions of Lemma B.1 there exists a
unique bounded lin ear operator
˜
I
(a,b)
: L
p
(Ω × S, F
a
S; P ν; E) L
p
(Ω, F, E)
such that for a finitely-valued function f, we have
˜
I
(a,b)
(f) =
Z
S
f(x)˜η(dx, (a, b]).
In particular, for every f L
p
(Ω × S, F
a
S; P ν; E),
(30) E|
˜
I
(a,b)
(f)|
p
E
2
2p
L
p
(E)(b a)E
Z
S
|ξ(x)|
p
E
ν(dx).
In what follows, unless we in danger of ambiguity, for ev ery L
p
(Ω ×
S, F
a
S; P ν; E) we will write
R
S
ξ(x)˜η(dx, (a, b]) instead of
˜
I
(a,b)
(f).
4
If Eη(B) = , then obviously η(B) = a.s..
February 2, 2008MAXIMAL REGULARITY 15
Let X be any Banach space. Later on we will take X to be one of
the spaces E, R(H, E) or L
p
(S, ν, E). For a < b [0, ] let N (a, b; X)
be the space of (equivalence classes of) predictable functions ξ : (a, b]×
X.
For q ( 1, ) we set
N
q
(a, b; X) =
ξ N (a, b; X) :
Z
b
a
|ξ(t)|
q
dt < a.s.
,(31)
M
q
(a, b; X) =
ξ N (a, b; X) : E
Z
b
a
|ξ(t)|
q
dt <
.(32)
Let N
step
(a, b; X) be the space of all ξ N (a, b; X) for which there
exists a partition a = t
0
< t
1
< · · · < t
n
< b such tha t for k
{1, · · · , n}, for t (t
k1
, t
k
], ξ(t) = ξ(t
k
) is F
t
k1
-measurable and ξ(t) =
0 for t (t
n
, b). We put M
q
step
= M
q
N
step
. Note that M
q
(a, b; X)
is a closed subspace of L
q
([a, b) × Ω; X)
=
L
q
([a, b); L
q
(Ω; X)).
In what follows we put a = 0 and b = . For ξ M
p
step
(0, ; L
p
(S, ν; E))
we set
(33)
˜
I(ξ) =
n
X
j=1
Z
S
ξ(t
j
, x)˜η(dx, (t
j1
, t
j
]).
Obviously,
˜
I(ξ) is a F-measurable map from with values in E.
We have the following auxiliary results.
Lemma B.3. Let p (1, 2] and assume that E is a Ban ach s pace of
martingale type p. Then for any ξ M
p
step
(0, ; L
p
(S, ν; E)),
˜
I(ξ)
L
p
(Ω, E), E
˜
I(ξ) = 0 and
(34) E|
˜
I(ξ)|
p
L
2
p
(E)2
2p
Z
0
E
Z
S
|ξ(t, x)|
p
E
ν(dx) dt
Lemma B.4. S uppose that ξ Po i s s (λ), where λ > 0. Then, for all
p [1, 2 ],
E|ξ λ|
p
2
2p
λ.
Remark 3. One can easily calculate that
E(|ξ λ|) = 2λ e
λ
, if λ (0, 1).
Theorem B.5. Assume that p (1, 2] a nd E is a martingale type p
Banach space. Then there exists there exists a unique bounded linear
operator
˜
I : M
p
(0, , L
p
(S, ν; E)) L
p
(Ω, F, E)
16 ZB AND EH FEBRUARY 2, 2008
such that for ξ M
p
step
(0, , L
p
(S, ν; E)) we have I(ξ) =
˜
I(ξ). In
particular, for every ξ M
p
(0, , L
p
(S, ν; E)),
(35) E|I(ξ)|
p
E
2
2p
L
2
p
(E)E
Z
0
Z
S
|ξ(t, x)|
p
E
ν(dx) dt.
Proof of Theorem B.5. Follows from Lemma B.3 and the density of
M
p
step
(0, , L
p
(S, ν; E)) in the space M
p
(0, , L
p
(S, ν; E)).
In a natural way we can define spaces M
p
lo c
(0, , L
p
(S, ν; E)) and
M
p
(0, T, L
p
(S, ν; E)), where T > 0. Then for any ξ M
p
lo c
(0, , L
p
(S, ν; E))
we can in a standard way define the integral
R
t
0
R
S
ξ(r, x)˜η(dx, dr),
t 0, as the adl´ag modification of the process
I(1
[0,t]
ξ), t 0,(36)
where [1
[0,t]
ξ](r, x; ω) := 1
[0,t]
(r)ξ(r, x, ω), t 0, r R
+
, x S and
ω Ω. To show that this adl´ag modification exists we arg ue as
follows. First of all we can assume that M
p
(0, T, L
p
(S, ν; E)), for some
T > 0. Let {ξ
n
}
nN
be an M
p
step
(0, T, L
p
(S, ν; E))-valued sequence
that is convergent in M
p
(0, T, L
p
(S, ν; E)) to ξ. Hence, the sequence
{ξ
n
, n N} is uniformly integrable and so it follows that the condition
(a) in Remark 3.8.7 from [10] is satisfied. Similarly, the compact
containment condition, i.e. the condition (a) in Theorem 3.7.2 from
[10], holds true in view of the Prohorov Theorem, since for any t 0
the laws of the sequence {I(1
[0,t]
ξ
n
), n N} are tight in the set of all
probability measures over E, compare also with [9].
Similarly, for a stopping time τ we can define and process ξ
M
p
lo c
(0, , L
p
(S, ν; E)) and the integral
Z
τ
0
Z
S
ξ(r, x)˜η(dx, dr) := I(1
[0]
ξ),(37)
provided 1
[0]
ξ M
p
(0, , L
p
(S, ν; E)). Theorem B.5 implies that in
this case t he following inequality holds.
(38) E|
Z
τ
0
Z
S
ξ(r, x)˜η(dx, dr)|
p
E
C
p
E
Z
τ
0
Z
S
|ξ(r, x)|
p
E
ν(dx) dr.
with some constant C
p
> 0 independent of ξ.
Proof of Lemma B.3. Let us o bserve that the sequence (M
k
)
n
k=1
de-
fined by M
k
=
P
k
j=1
R
S
ξ(t
j
, x)˜η(dx, [t
j1
, t
j
)) is an E-valued martin-
gale (with respect to the filtration (F
t
k
)
n
k=1
). Therefore, by the mar-
tingale type p property of the space E and Lemma B.1 we have the
February 2, 2008MAXIMAL REGULARITY 17
following sequence of inequalities
E|(
˜
I(ξ))|
p
E
= E|M
n
|
p
E
L
p
(E)
n
X
k=1
E|
Z
S
ξ(t
k
, x)˜η(dx, [t
k1
, t
k
])|
p
E
L
2
p
(E)2
2p
n
X
k=1
(t
k
t
k1
) E
Z
S
|ξ(t
k
, x)|
p
E
ν(dx)(39)
= L
2
p
(E)2
2p
Z
0
E
Z
S
|ξ(t, x)|
p
E
ν(dx) dt.
This concludes the proof.
Proof of Lemma B.1. Put I = (a, b]. We may suppose that f =
P
i
f
i
1
A
i
×B
i
with f
i
E, A
i
F
a
and B
i
S, the finite family of sets (A
i
× B
i
)
being pair-wise disjoint and ν(B
i
) < . Let us notice that
Z
S
f(x)˜η(dx, I) =
X
i
1
A
i
˜η(B
i
× I)f
i
.
Since the random variables ˜η(B
i
× I) are independent from the σ-
field F
a
, the random varia bles 1
A
i
˜η(B
i
× I) conditioned on F
a
are in-
dependent and so by the martingale type p property of the space E
and Lemma B.1 we infer that
E|
Z
S
ξ(x)˜η(dx, I)|
p
E
= E
E
|
X
i
1
A
i
˜η(B
i
× I)f
i
|
p
E
|F
a

E
L
p
(E)
X
i
|f
i
1
A
i
|
p
E
E|˜η(B
i
× I)|
p
L
p
(E)E
X
i
|f
i
|
p
E
1
A
i
2
2p
λ(I)ν(B
i
)
= 2
2p
L
p
(E)
X
i
|f
i
|
p
E
ν(B
i
)λ(I)P(A
i
)
= 2
2p
L
p
(E)λ(I)
Z
×S
|
X
i
f
i
1
A
i
×B
i
|
p
d(P ν)
=
˜
L
p
(E)(b a)E
Z
S
|f(x)|
p
E
ν(dx).
The proof is complete.
Proof of Lemma B.4. The case p = 2 is well known. Since ξ 0 and
E(ξ) = λ, the case p = 1 follows by the triangle inequality. The
case p (1, 2) follows then by applying the older inequality. Indeed,
with α = 2(p 1) and β = 2 p we have the following sequence of
18 ZB AND EH FEBRUARY 2, 2008
inequalities, where η := |ξ λ|.
E(η
p
) = E(η
α
η
β
) [E((η
α
)
2
)]
α/2
[E((η
β
)
1
)]
β
= [E(η
2
)]
α/2
[E(η)]
β
(λ)
α/2
(2λ)
β
= 2
2p
λ.
We conclude with a result corresponding to inequality (6).
Corollary B.6. Assume that 1 < q p < 2 and E is a martingale
type p Banach space. Then there exists there exis ts a constant C > 0
such that for any process ξ M
p
lo c
(0, , L
p
(S, ν; E)) L
p
(Ω, F, E),
and any T > 0,
(40)
E| sup
t[0,T ]
Z
t
0
Z
S
ξ(r, x)˜η(dx, dr)|
q
CE
Z
T
0
Z
S
|ξ(r, x)|
p
ν(dx) dr
q/p
.
The proof of the above result will be based on Proposition IV.4.7
from the monograph B.7 by Revuz and Yor which we recall here for
the convenience of the reader.
Proposition B.7. Suppose that a positive, adapted right-continuous
process Z is dominated by an increasing process A, with A
0
, i.e. there
exists a constant C > 0 s uch that for every bounded stopping time τ,
EZ
τ
CEA
τ
. Then for any k (0, 1),
E sup
0t<
Z
k
t
C
k
2 k
1 k
EA
k
.
Proof of Corollary B.6. Let now fix q (1, p). Put k = q/p. We will
apply Proposition B.7 to the processes Z
t
= |
R
t
0
R
S
ξ(r, x)˜η(dx, dr)|
p
E
and A
t
=
R
t
0
R
S
|ξ(r, x)|
p
E
ν(dx) dr, t [0, T ]. Let us notice that in view
of inequality (38) , the process Z is dominated by the process A. Since
Z is right continuous, sup
0tT
Z
k
t
= sup
0tT
|
R
t
0
R
S
ξ(r, x)˜η(dx, dr)|
q
E
and A
k
=
R
T
0
R
S
|ξ(r, x)|
p
E
ν(dx) dr
q/p
, we get inequality (40). This
completes the proof of Corollary B.6.
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February 2, 2008MAXIMAL REGULARITY 19
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20 ZB AND EH FEBRUARY 2, 2008
E-mail address: zb500@york.ac.uk
Department of Mathematics, University of York, Heslington, York
YO10 5DD, UK
E-mail address: erika.hausenblas@sbg.ac.at
Department of Mathematics, University of Salzburg, Hellbrunner-
str. 34, 5020 Salzburg, Austria
... The major differences of the current work with respect to [10] are as follows. Firstly, the approach [10] heavily depends on the use of the theory of Itô integral in martingale type 2 Banach spaces with respect to a cylindrical Wiener process, while we use an approach which relies on stochastic integration in martingale type p Banach spaces with respect to a Poisson random measure, see [13]. Secondly, the compactness argument in [10] relies on the Hölder continuity of the trajectories of the corresponding stochastic convolution process. ...
... Our paper confirms an observation that has already been made in earlier papers [13,42] that the theory of stochastic integration with respect to a Poisson random measure in martingale type p Banach spaces is, to a large extend, analogous to the theory of stochastic integration with respect to a cylindrical Wiener process in martingale type 2 Banach spaces provided the space of γ -radonifying operators is replaced by the space L p (ν), where ν is the intensity measure of the Poisson random measure in question. We would also like to point out that the theory of Banach space valued stochastic integrals with respect to a Poisson random measure is richer than the corresponding Gaussian theory, see the recent paper [30] by S. Dirksen. ...
... We would also like to point out that the theory of Banach space valued stochastic integrals with respect to a Poisson random measure is richer than the corresponding Gaussian theory, see the recent paper [30] by S. Dirksen. It would be of great interest to develop a theory of inequalities for stochastic convolutions and generalise the paper [13] in the framework of [30]. ...
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