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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 L´evy processes.
1. Introduction
The aim of the article is to investigate the maximal regularity of the
OrnsteinUhlenbeck 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
)
0≤t<∞
in E. We consider the following SPDE written
in the Itˆoform
du(t) = Au(t−) dt +
R
S
ξ(t; x)˜η(dx; dt),
u(0) = 0,
(1)
where ˜η is a Svalued time homog eneous compensated Poisson random
measure deﬁned on a ﬁltered probability space (Ω; F; (F
t
)
0≤t<∞
; P)
with L´evy measure ν on S, speciﬁed later, and ξ : Ω × S → E is a pre
dictable process satisfying certain integrability conditions also speciﬁed
later. The solution to (1) is given by the so called OrnsteinUhlenbeck
process
u(t) :=
Z
t
0
Z
S
e
−A(t−r)
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 nsteinUhlenbeck 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 OrnsteinUhlenbeck 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 σﬁeld 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 σﬁeld 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
)
t≥0
, 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
)
t≥0
. 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 Ω × 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 : Ω × 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 × Ω → E with
(3) E
Z
T
0
Z
S
ξ(r, x)
p
ν(dx) dr < ∞, T > 0,
an adapted c´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
j−1
,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
j−1
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
j−1
∧ 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
EI(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
0≤t≤τ
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
)
t≥0
on E.
Deﬁne the stochastic convolution of the semigroup (e
−tA
)
t≥0
and a n
Evalued process ξ as above by the following formula
(8) SC(ξ)(t) =
Z
t
0
Z
S
e
(t−r)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
tω
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)
1−θ,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)
1−θ,q
=
x ∈ E :
Z
δ
0
t
m(1−θ)
A
m
e
−tA
x
q
dt
t
< ∞
.
(9) (E, D(A
m
))
ϑ,q
=
x ∈ E :
Z
δ
0
t
m(1−ϑ)
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)
1−θ,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(1−θ)
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
)
1−θ,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(1−θ)
(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(1−θ)
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
)
t≥0
, 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
)
t≥0
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) EI
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
)
≤ Φkξk
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
1−θ
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) EI
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
)
t≥0
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,θ
(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,θ
(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
)
t≥0
, 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
2smoothable Banach spaces. With E ∈ E
2
we associate the space
R(E) := R(H, E) of all γradonifying 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
)
t≥0
, 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
)
t≥0
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
Er
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(2−θ)−1
EA
2
e
−rA
I
t
e
−(t−·)A
ξ(·)

p
dr
r
dt
= C
Z
T
0
Z
1
0
r
p(2−θ)−1
EI
t
A
2
e
−rA
e
−(t−·)A
ξ(·)

p
dr
r
dt ≤ · · ·
3
Equivalence of the norms is a consequence of KhinchinKahane 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
t−u+r
≤
1
r
for t ∈ [u, T ], r > 0, we
infer that
· · · ≤ C
p
Z
1
0
r
p(2−θ)−1
Z
T
0
E
Z
t
0
kA
2
e
−(t−u+r)A
ξ(u)k
p
R(E)
du dt
dr
r
≤ C
p
Z
1
0
r
p(2−θ)−1
Z
T
0
E
Z
t
0
Ae
−
t−u+r
2
A

p
kAe
−
t−u+r
2
A
ξ(u)k
p
R(E)
du dt
dr
r
≤ C
p
E
Z
1
0
r
p(2−θ)−1
sup
0≤u≤t
(t − u + r)
−p
Z
T
0
Z
T
u
kAe
−
t−u+r
2
A
ξ(u)k
p
R(E)
dt
du
dr
r
≤ C
p
E
Z
T
0
Z
T +1−ρ
0
kAe
−
σ
2
A
ξ(ρ)k
p
R(E)
Z
ρ+σ
ρ∨(σ+ρ−1)
(σ + ρ − τ )
p(1−θ)−2
dτ
dσ dρ
≤ C
p
E
Z
T
0
Z
T +1−ρ
0
kAe
−
σ
2
A
ξ(ρ)k
p
R(E)
Z
ρ+σ
ρ
(σ + ρ − τ )
p(1−θ)−2
dτ
dσ dρ
= C
p
E
Z
T
0
Z
T +1−ρ
0
kAe
−
σ
2
A
ξ(ρ)k
p
R(E)
Z
σ
0
τ
p(1−θ)−2
dτ
dσ dρ
= C
′
p
E
Z
T
0
Z
T +1−ρ
0
σ
p(1−θ)−1
kAe
−
σ
2
A
ξ(ρ)k
p
R(E)
dσ dρ
≤ C
′′
p
E
Z
T
0
Z
T/2
0
kσ
1−θ
Ae
−σA
ξ(ρ)k
p
R(E)
dσ
σ
dρ
≤
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(2−θ)−
q
p
EA
2
e
−sA
SC
t
(ξ)
q
ds
s
dt
= C
Z
T
0
Z
1
0
s
q(2−θ)−
q
p
EA
2
e
−sA
I
t
e
−(t−·)A
ξ(·)

q
ds
s
dt
= C
Z
T
0
Z
1
0
s
q(2−θ)−
q
p
EI
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
q−p
dr
q
p
−1
≤ C
1
s
q(1−
1
p
)+1
Proof of Lemma 3.1. Denote α =
pq
q−p
and observe that α > 1. Since
R
t
0
1
(t−r+s)
α
dr =
R
t
0
1
(r+s)
α
dr ≤
R
∞
0
1
(r+s)
α
dr =
1
α−1
1
s
α−1
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 H¨older inequality and Lemma 3.1 we
infer that
· · · ≤
ˆ
C
q
Z
1
0
s
q(2−θ)−
q
p
Z
T
0
E
Z
t
0
kA
2
e
−(t−r+s)A
ξ(r)k
p
R(E)
dr
q/p
dt
ds
s
≤ C
ˆ
C
q
E
Z
1
0
s
q(2−θ)−
q
p
Z
T
0
Z
t
0
Ae
−
t−r+s
2
A

p
kAe
−
t−r+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(2−θ)−
q
p
Z
T
0
Z
t
0
Ae
−
t−r+s
2
A

pq
q−p
dr
q
p
−1
Z
t
0
kAe
−
t−r+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(2−θ)−
q
p
Z
T
0
1
s
q(1−
1
p
)+1
Z
t
0
kAe
−
t−r+s
2
A
ξ(r)k
q
R(E)
dr dt
ds
s
= C
′
ˆ
C
q
E
Z
1
0
s
q(1−θ)−1
Z
T
0
Z
T
r
kAe
−
t−r+s
2
A
ξ(r)k
q
R(E)
dt
dr
ds
s
≤ C
′
ˆ
C
q
E
Z
T
0
Z
T +1−ρ
0
kAe
−
σ
2
A
ξ(ρ)k
q
R(E)
Z
ρ+σ
ρ∨(σ+ρ−1)
(σ + ρ − τ )
q(1−θ)−2
dτ
dσ dρ
≤ C
′
ˆ
C
q
E
Z
T
0
Z
T +1−ρ
0
kAe
−
σ
2
A
ξ(ρ)k
q
R(E)
Z
ρ+σ
ρ
(σ + ρ − τ )
q(1−θ)−2
dτ
dσ dρ
= C
′
ˆ
C
q
E
Z
T
0
Z
T +1−ρ
0
kAe
−
σ
2
A
ξ(ρ)k
q
R(E)
Z
σ
0
τ
q(1−θ)−2
dτ
dσ dρ
=
ˆ
C
′
q
E
Z
T
0
Z
T +1−ρ
0
σ
q(1−θ)−1
kAe
−
σ
2
A
ξ(ρ)k
q
R(E)
dσ dρ
≤
ˆ
C
′′
q
E
Z
T
0
Z
T/2
0
kσ
1−θ
Ae
−σA
ξ(ρ)k
q
R(E)
dσ
σ
dρ ≤
≤
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
)
t≥0
, 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
(1−θ)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
(1−θ)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
(1−θ)q
kAe
−tA
ϕ k
q
R(H,E)
dt
t
≤ C
q
(E)
Z
T
0
t
(1−θ)q
E
X
k
β
k
Ae
−tA
ϕ e
k

q
E
dt
t
= C
q
(E)E
Z
T
0
t
(1−θ)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
(1−θ)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
(1−θ)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
Xvalued ﬁnite martingale {M
n
}
N
n=0
the following inequality holds
(25) sup
n
EM
n

p
≤ L
p
(E)
N
X
n=0
EM
n
− M
n−1

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
: Ω → {−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 psmooth, 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 Xvalued 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
¯
Nvalued measures
on (S × R
+
, S ⊗ B
R
+
) and M
¯
N
S×R
+
is the σﬁeld 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
nonnegative measure on (S, S) and P = (Ω, F, (F
t
)
t≥0
, 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 : Ω →
¯
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 pairwise disjoint, then the random variables η(B
j
), j =
1, · · · , n ar e pairwise 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
¯
Nvalued processes (N(t, U))
t>0
deﬁned by
N(t, U) := η((0, t] × U), t > 0
is (F
t
)
t≥0
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 ﬁnitelyvalued 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
2−p
L
p
(E)(b − a)E
Z
S
f(x)
p
E
ν(dx)
Since the space of ﬁnitelyvalued 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 ﬁnitelyvalued 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
2−p
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]×
Ω → 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
k−1
, t
k
], ξ(t) = ξ(t
k
) is F
t
k−1
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
j−1
, t
j
]).
Obviously,
˜
I(ξ) is a Fmeasurable map from Ω 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
2−p
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 ],
Eξ − λ
p
≤ 2
2−p
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) EI(ξ)
p
E
≤ 2
2−p
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 c´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 c´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
}
n∈N
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,τ]
ξ),(37)
provided 1
[0,τ]
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
j−1
, t
j
)) is an Evalued 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
= EM
n

p
E
≤ L
p
(E)
n
X
k=1
E
Z
S
ξ(t
k
, x)˜η(dx, [t
k−1
, t
k
])
p
E
≤ L
2
p
(E)2
2−p
n
X
k=1
(t
k
− t
k−1
) E
Z
S
ξ(t
k
, x)
p
E
ν(dx)(39)
= L
2
p
(E)2
2−p
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 pairwise 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
2−p
λ(I)ν(B
i
)
= 2
2−p
L
p
(E)
X
i
f
i

p
E
ν(B
i
)λ(I)P(A
i
)
= 2
2−p
L
p
(E)λ(I)
Z
Ω×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 H¨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
)]
α/2
[E((η
β
)
1/β
)]
β
= [E(η
2
)]
α/2
[E(η)]
β
≤ (λ)
α/2
(2λ)
β
= 2
2−p
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 rightcontinuous
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
0≤t<∞
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
0≤t≤T
Z
k
t
= sup
0≤t≤T

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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20 ZB AND EH FEBRUARY 2, 2008
Email address: zb500@york.ac.uk
Department of Mathematics, University of York, Heslington, York
YO10 5DD, UK
Email address: erika.hausenblas@sbg.ac.at
Department of Mathematics, University of Salzburg, Hellbrunner
str. 34, 5020 Salzburg, Austria