Elect. Comm. in Probab. 16 (2011), 689–705
ELECTRONIC
COMMUNICATIONS
in PROBABILITY
A MAXIMAL INEQUALITY FOR STOCHASTIC CONVOLUTIONS IN
2-SMOOTH BANACH SPACES
JAN VAN NEERVEN1
Delft Institute of Applied Mathematics, Delft University of Technology, P.O. Box 5031, 2600 GA Delft,
The Netherlands
email:
J.M.A.M.vanNeerven@TUDelft.nl
JIAHUI ZHU2
Delft Institute of Applied Mathematics, Delft University of Technology, P.O. Box 5031, 2600 GA Delft,
The Netherlands
email:
Jiahui.Zhu@TUDelft.nl
Submitted May 24, 2011, accepted in final form October 24,2011
AMS 2000 Subject classification: Primary 60H05; Secondary 60H15
Keywords: Stochastic convolutions, maximal inequality, 2-smooth Banach spaces, Itô formula.
Abstract
Let (e tA) t >0 be a C0 -contraction semigroup on a 2-smooth Banach space E, let (Wt ) t >0 be a cylindrical Brownian motion in a Hilbert space H, and let (g t ) t >0 be a progressively measurable process
with values in the space γ(H, E) of all γ-radonifying operators from H to E. We prove that for all
0 < p < ∞ there exists a constant C, depending only on p and E, such that for all T > 0 we have
Z t
p
Z T
p
2
E sup e(t−s)A gs dWs 6 C E
kg t k2γ(H,E) dt .
06 t 6 T
0
0
For p > 2 the proof is based on the observation that ψ(x) = kxk p is Fréchet differentiable and
its derivative satisfies the Lipschitz estimate kψ0 (x) − ψ0 ( y)k 6 C(kxk + k yk) p−2 kx − yk; the
extension to 0 < p < 2 proceeds via Lenglart’s inequality.
1
Introduction
Let (e tA) t >0 be a C0 -contraction semigroup on a 2-smooth Banach space E and let (Wt ) t >0 be
a cylindrical Brownian motion in a Hilbert space H. Let (g t ) t >0 be a progressively measurable
process with values in the space γ(H, E) of all γ-radonifying operators from H to E satisfying
Z T
0
kg t k2γ(H,E) dt < ∞ P-almost surely
1
RESEARCH SUPPORTED BY VICI SUBSIDY 639.033.604 OF THE NETHERLANDS ORGANISATION FOR SCIENTIFIC
RESEARCH (NWO)
2
RESEARCH SUPPORTED BY VICI SUBSIDY 639.033.604 OF THE NETHERLANDS ORGANISATION FOR SCIENTIFIC
RESEARCH (NWO)
689
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for all T > 0. As is well known (see [6, 15, 16]), under these assumptions the stochastic convolution process
Z t
e(t−s)A gs dWs ,
Xt =
t > 0,
0
is well-defined in E and provides the unique mild solution of the stochastic initial value problem
dX t = AX t dt + g t dWt ,
X 0 = 0.
In order to obtain the existence of a continuous version of this process, one usually proves a
maximal estimate of the form
Z T
p
2
p
p
(1.1)
E sup kX t k 6 C E
kg t k2γ(H,E) dt .
06 t 6 T
0
The first such estimate was obtained by Kotelenez [11, 12] for C0 -contraction semigroups on
Hilbert spaces E and exponent p = 2. Tubaro [19] extended this result to exponents p > 2 by
a different method of proof which applies Itô’s formula to the C 2 -mapping x 7→ kxk p . The case
p ∈ (0, 2) was covered subsequently by Ichikawa [10]. A very simple proof, still for C0 -contraction
semigroups on Hilbert spaces, which works for all p ∈ (0, ∞), was obtained recently by Hausenblas
and Seidler [9]. It is based on the Sz.-Nagy dilation theorem, which is used to reduce the problem
to the corresponding problem for C0 -contraction groups. Then, by using the group property, the
maximal estimate follows from Burkholder’s inequality. This proof shows, moreover, that the
constant C in (1.1) may be taken equal to the constant appearing in Burkholder’s inequality. In
particular, this constant depends only on p.
The maximal inequality (1.1) has been extended by Brzeźniak and Peszat [4] to C0 -contraction
semigroups on Banach spaces E with the property that, for some p ∈ [2, ∞), x 7→ kxk p is twice
continuously Fréchet differentiable and the first and second Fréchet derivatives are bounded by
constant multiples of kxk p−1 and kxk p−2 , respectively. Examples of spaces with this property,
which we shall call (C p2 ), are the spaces L q (µ) for q ∈ [p, ∞). Any (C p2 ) space is 2-smooth (the
definition is recalled in Section 2), but the converse doesn’t hold:
Example 1.1. Let F be a Banach space. The space `2 (F ) is 2-smooth whenever F is 2-smooth [8,
Proposition 17]. On the other hand, the norm of `2 (F ) is twice continuously Fréchet differentiable
away from the origin if and only if F is a Hilbert space [14, Theorem 3.9]. Thus, for q ∈ (2, ∞),
`2 (`q ) and `2 (L q (0, 1)) are examples of 2-smooth Banach spaces which fail property (C p2 ) for all
p ∈ [2, ∞).
To the best of our knowledge, the general problem of proving the maximal estimate (1.1) for C0 contraction semigroups on 2-smooth Banach space remains open. The present paper aims to fill
this gap:
Theorem 1.2. Let (e tA) t >0 be a C0 -contraction semigroup on a 2-smooth Banach space E, let (Wt ) t >0
be a cylindrical Brownian motion in a Hilbert space H, and let (g t ) t >0 be a progressively measurable
process in γ(H, E). If
Z T
0
kg t k2γ(H,E) dt < ∞
P-almost surely,
Rt
then the stochastic convolution process X t = 0 e(t−s)A gs dWs is well-defined and has a continuous
version. Moreover, for all 0 < p < ∞ there exists a constant C, depending only on p and E, such that
Z T
p
E sup kX t k p 6 C p E
06 t 6 T
0
kg t k2γ(H,E) dt
2
.
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691
For p > 2, the proof of Theorem 1.2 is based on a version of Itô’s formula (Theorem 3.1) which
exploits the fact (proved in Lemma 2.1) that in 2-smooth Banach spaces the function ψ(x) = kxk p
is Fréchet differentiable and satisfies the Lipschitz estimate
kψ0 (x) − ψ0 ( y)k 6 C(kxk + k yk) p−2 kx − yk.
The extension to exponents 0 < p < 2 is obtained by applying Lenglart’s inequality (see (4.1)).
We conclude this introduction with a brief discussion of some developments of the inequality (1.1)
into different directions in the literature. Seidler [18] has proved the inequality (1.1) with optimal
p
constant C = O( p) as p → ∞ for positive C0 -contraction semigroups on the (2-smooth) space
q
E = L (µ), q > 2. He also proved that the same result holds if the assumption ‘e tA is a positive
contraction semigroup’ is replaced by ‘−A has a bounded H ∞ -calculus of angle strictly less than
1
π’. The latter result was subsequently extended by Veraar and Weis [20] to arbitrary UMD spaces
2
E with type 2. In the same paper, still under the assumption that −A has a bounded H ∞ -calculus
of angle strictly less than 12 π, the following stronger estimate is obtained for UMD spaces E with
Pisier’s property (α):
p
E sup kX t k p 6 C p Ekgkγ(L 2 (0,T ;H),E)
(1.2)
06 t 6 T
with a constant C depending only on p and E. If, in addition, E has type 2, then the mapping
f ⊗(h⊗ x) 7→ ( f ⊗h)⊗ x extends to a continuous embedding L 2 (0, T ; γ(H, E)) ,→ γ(L 2 (0, T ; H), E)
and (1.2) implies (1.1).
Let us finally mention that, for p > 2, a weaker version of (1.1) for arbitrary C0 -semigroups on
Hilbert spaces has been obtained by Da Prato and Zabczyk [5]. Using the factorisation method
they proved that
Z T
p
E sup kX t k p 6 C p E
06 t 6 T
0
kg t kγ(H,E) dt
with a constant C depending on p, E, and T . The proof extends verbatim to C0 -semigroups on
martingale type 2 spaces. This relates to the above results for 2-smooth spaces through a theorem
of Pisier [17, Theorem 3.1], which states that a Banach space has martingale type p if and only if
it is p-smooth.
2
The Fréchet derivative of k · k p
Let 1 < q 6 2. A Banach space E is q-smooth if the modulus of smoothness
n
o
ρk·k (t) = sup 21 (kx + t yk + kx − t yk) − 1 : kxk = k yk = 1
satisfies ρk·k (t) 6 C t q for all t > 0.
It is known (see [17, Theorem 3.1]) that E is q-smooth if and only if there exists a constant K > 1
such that for all x, y ∈ E,
kx + ykq + kx − ykq 6 2kxkq + Kk ykq .
(2.1)
Lemma 2.1. Let E be a Banach space and let 1 < q 6 2 be given. For p > q set ψ p (x) := kxk p .
1. E is q-smooth if and only if the Fréchet derivative of ψq is globally (q − 1)-Hölder continuous
on E.
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2. If E is q-smooth, then for p > q the Fréchet derivative of ψ p is locally (q − 1)-Hölder continuous
on E.
Moreover, for all p > q and x, y ∈ E we have
kψ0p (x) − ψ0p ( y)k 6 C(kxk + k yk) p−q kx − ykq−1 ,
(2.2)
where C depends only on p, q and E.
Proof. If the Fréchet derivative of ψq is (q − 1)-Hölder continuous on E, then by the mean value
theorem we can find 0 6 θ , ρ 6 1 such that for all x, y ∈ E,
kx + ykq + kx − ykq − 2kxkq = (kx + ykq − kxkq ) + (kx − ykq − kxkq )
6 kψ0q (x + θ y) − ψ0q (x − ρ y)k k yk
6 Lk(x + θ y) − (x − ρ y)kq−1 k yk 6 2q−1 Lk ykq .
Hence the Banach space E is q-smooth.
Suppose now that the norm of E is q-smooth. Then for all x, y ∈ E with kxk, k yk = 1 and all t > 0
we have
kx + t yk + kx − t yk − 2kxk 6 Kkt ykq .
Thus
lim
kx + t yk + kx − t yk − 2kxk
kt yk
t→0
(2.3)
= 0,
which by [7, Lemma I.1.3] means that k · k is Fréchet differentiable on the unit sphere. Hence, by
homogeneity, k · k is Fréchet differentiable on E\{0}. Let us denote by f x its Fréchet derivative at
the point x 6= 0.
We begin by showing the (q − 1)-Hölder continuity of x 7→ f x on the unit sphere of E, following
the argument of [7, Lemma V.3.5]. We fix x 6= y ∈ E such that kxk, k yk = 1 and h ∈ E with
khk = kx − yk and x − y + h 6= 0. Since the norm k · k is a convex function,
f y (x − y) 6 kxk − k yk.
Similarly, we have
f x (h) 6 kx + hk − kxk,
f y ( y − x − h) 6 k2 y − x − hk − k yk.
By using above inequalities and the linearity of the function f x , we have
f x (h) − f y (h) 6 kx + hk − kxk − f y (h)
= kx + hk − k yk − f y (x + h − y) + k yk − kxk + f y (x − y)
6 kx + hk − k yk − f y (x + h − y)
= kx + hk − k yk + f y ( y − x − h)
6 kx + hk + k2 y − x − hk − 2k yk
x +h− y x +h− y = y + kx + h − yk ·
+ y − kx + h − yk ·
− 2k yk
kx + h − yk
kx + h − yk
6 Kkx + h − ykq 6 K(kx − yk + khk)q = 2q Kkx − ykq ,
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693
where we also used (2.3). Since the roles of x and y may be reversed in this inequality, this
implies
k fx − f y k =
sup
khk=kx− yk
| f x (h) − f y (h)|
kx − yk
6 2q Kkx − ykq−1
This proves the (q − 1)-Hölder continuity of the norm k · k on the unit sphere.
We proceed with the proof of (2.2); the (q − 1)-Hölder continuity of ψq as well as the local
(p − 1)-Hölder continuity of ψ p follow from it. For all x, y ∈ E with x 6= 0 and y 6= 0 we have
ψ0p (x) = pkxk p−1 f x .
It is easy to check that f x = f x and k f x k = 1. Following once more the argument of [7, Lemma
kxk
V.3.5], this gives
kψ0p (x) − ψ0p ( y)k = pkxk p−1 f x − k yk p−1 f y 6 pkxk p−1 ( f x − f y ) + p(kxk p−1 − k yk p−1 ) f y kxk
k yk
k yk
x
y q−1
+ pkxk p−1 − k yk p−1 −
6 p2q Kkxk p−1 kxk k yk
q−1
6 p2q Kkxk p−q k yk1−q xk yk − ykxk + pkxk p−1 − k yk p−1 q−1
+ pkxk p−1 − k yk p−1 = p2q Kkxk p−q k yk1−q k yk(x − y) + y(k yk − kxk)
6 p2q Kkxk p−q k yk1−q (2k ykkx − yk)q−1 + pkxk p−1 − k yk p−1 = p22q−1 Kkxk p−q kx − ykq−1 + pkxk p−1 − k yk p−1 .
(2.4)
If q 6 p 6 2, then by the inequality |t r − s r | 6 |t − s| r , valid for 0 < r 6 1 and s, t ∈ [0, ∞), we
have
kxk p−1 − k yk p−1 6 kxk − k yk p−1 6 kx − yk p−1 6 (kxk + k yk) p−q kx − ykq−1 .
If p > 2, by applying the mean value theorem, for some θ ∈ [0, 1] we have
kxk p−1 − k yk p−1 = (p − 1)
kθ x + (1 − θ ) yk p−2 fθ x+(1−θ ) y (x − y)
6 (p − 1)(kxk + k yk) p−2 kx − yk
6 (p − 1)(kxk + k yk) p−2 (kxk + k yk)2−q kx − ykq−1
= (p − 1)(kxk + k yk) p−q kx − ykq−1 .
Also, since ψ0p (0) = 0, for y 6= 0 we have
y p−1
y q−1
kψ0p (0) − ψ0p ( y)k = pk yk p−1 = pk yk p−1 6 pk yk p−1 = pk yk p−q k ykq−1 .
k yk
k yk
The above lemma will be combined with the next one, which gives a first order Taylor formula
with a remainder term involving the first derivative only.
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Lemma 2.2. Let E and F be Banach spaces, let 0 < α 6 1, and let ψ : E → F be a Fréchet
differentiable function whose Fréchet derivative ψ0 : E → L (E, F ) is locally α-Hölder continuous.
Then for all x, y ∈ E we have
ψ( y) = ψ(x) + ψ0 (x)( y − x) + R(x, y),
where
R(x, y) =
Z
1
(ψ0 (x + r( y − x))( y − x) − ψ0 (x)( y − x)) dr.
(2.5)
0
Proof. Pick w ∈ E such that kwk 6 1 and consider the function f : R → F by
f (θ ) := ψ(x + θ w).
For all x ∗ ∈ F ∗ , ⟨ f 0 , x ∗ ⟩ is locally α-Hölder continuous. To see this, note that for |θ1 |, |θ2 | 6 R and
kxk 6 R we have kx + θ1 wk, kx + θ2 wk 6 2R, so by assumption there exists a constant C2R such
that
|⟨ f 0 (θ1 ) − f 0 (θ2 ), x ∗ ⟩| = |⟨ψ0 (x + θ1 w)w, x ∗ ⟩ − ⟨ψ0 (x + θ2 w)w, x ∗ ⟩|
6 kψ0 (x + θ1 w) − ψ0 (x + θ2 w)k kx ∗ k 6 C2R |θ1 − θ2 |α kx ∗ k.
Applying Taylor’s formula and [1, Lemma 1, Theorem 3] to the function ⟨ f , x ∗ ⟩ we obtain
⟨ f (t) − f (0), x ∗ ⟩ = t⟨ f 0 (0), x ∗ ⟩ + ⟨R f (0, t), x ∗ ⟩,
where R f (0, t) =
w=
y−x
.
k y−xk
R1
0
t( f 0 (r t) − f 0 (0)) dr. Now let x, y ∈ E be given and set t = k y − xk and
With these choices we obtain
⟨ψ( y), x ∗ ⟩ − ⟨ψ(x), x ∗ ⟩ − ⟨ψ0 (x)( y − x), x ∗ ⟩ = ⟨ψ(x + t w), x ∗ ⟩ − ⟨ψ(x), x ∗ ⟩ − t⟨ψ(0 x)w, x ∗ ⟩
= ⟨ f (t) − f (0) − t f 0 (0), x ∗ ⟩
Z1
t⟨ f 0 (r t) − f 0 (0), x ∗ ⟩ dr
=
0
1
=
Z
⟨ψ0 (x + r( y − x))( y − x) − ψ0 (x)( y − x), x ∗ ⟩ dr.
0
∗
∗
Since x ∈ F was arbitrary, this proves the lemma.
3
An Itô formula for k · k p
From now on we shall always assume that E is a 2-smooth Banach space. We fix T > 0 and let
(Ω, F , P) be a probability space with a filtration (F t ) t∈[0,T ] . Let H be a real Hilbert space, and
denote by γ(H, E) the Banach space of all γ-radonifying operators from H to E. We denote by
M ([0, T ]; γ(H, E)) the space of all progressively measurable processes ξ : [0, T ] × Ω → γ(H, E)
such that
Z T
0
kξ t k2γ(H,E) dt < ∞ P-almost surely.
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695
The space of all such ξ which satisfy
Z
E
T
kξ t k2γ(H,E) dt
0
p
2
<∞
is denoted by M p ([0, T ]; γ(H, E)), 0 < p < ∞.
On (Ω, F , P), let (Wt ) t∈[0,T ] be an (F t ) t∈[0,T ] -cylindrical Brownian motion in H. For adapted
simple processes ξ ∈ M ([0, T ]; γ(H, E)) of the form
ξt =
n−1
X
1(t i ,t i+1 ] (t) ⊗ Ai ,
i=0
where Π = {0 = t 0 < t 1 < · · · < t n = T } is a partition of the interval [0, T ] and the random
variables Ai are F t i -measurable and take values in the space of all finite rank operators from H to
E, we define the random variable I(ξ) ∈ L 0 (Ω, F T ; E) by
I(ξ) :=
n−1
X
Ai (Wt i+1 − Wt i )
i=0
where (h ⊗ x)Wt := (Wt h) ⊗ x. It is well known that
Z T
EkI(ξ)k2 6 C 2 E
0
kξ t k2γ(H,E) dt,
where C depends on p and E only. It follows that I has a unique extension to a bounded linear
operator M 2 ([0, T ]; γ(H, E)) to L 2 (Ω, F T ; E). By a standard localisation argument, I extends
continuous linear operator from M ([0, T ]; γ(H, E)) to L 0 (Ω, F T ; E). In what follows we write
Z t
ξs dWs := I(1(0,t] ξ),
t ∈ [0, T ].
0
This stochastic integral has the following properties:
Rt
1. For all ξ ∈ M ([0, T ]; γ(H, E)) the process t → 0 ξs dWs is an E-valued continuous local
martingale, which is a martingale if ξ ∈ M 2 ([0, T ]; γ(H, E)).
2. For all ξ ∈ M ([0, T ]; γ(H, E)) and stopping times τ with values in [0, T ],
Zτ
Z T
ξ t dWt =
1[0,τ] (t)ξ t dWt
P-almost surely.
(3.1)
3. For all ξ ∈ M 2 ([0, T ]; γ(H, E)) and 0 6 u < t 6 T ,
2
Z t
Z t
E ξs dWs |Fu 6 C E
kξs k2γ(H,E) ds |Fu .
(3.2)
0
0
u
u
4. (Burkholder’s inequality [2, 6]) For all 0 < p < ∞ there exists a constant C, depending only
on p and E, such that for all ξ ∈ M p ([0, T ]; γ(H, E)) and t ∈ [0, T ],
Z s
p
Z t
p
2
(3.3)
E sup ξu dWu 6 C E
kξs k2γ(H,E) ds .
s∈[0,t]
0
0
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An excellent survey of the theory of stochastic integration in 2-smooth Banach spaces with complete proofs is given in Ondreját’s thesis [16], where also further references to the literature can
be found.
In what follows we fix p > 2 and set ψ(x) := ψ p (x) = kxk p . Since we assume that E is 2-smooth,
this function is Fréchet differentiable. Following the notation of Lemma 2.2 we set
Rψ (x, y) :=
Z
1
(ψ0 (x + r( y − x))( y − x) − ψ0 (x)( y − x)) dr.
0
We have the following version of Itô’s formula.
Theorem 3.1 (Itô formula). Let E be a 2-smooth Banach space and let 2 6 p < ∞. Let (a t ) t∈[0,T ]
be an E-valued progressively measurable process such that
T
Z
E
ka t k dt
p
<∞
0
and let (g t ) t∈[0,T ] be a process in M p ([0, T ]; γ(H, E)). Fix x ∈ E and let (X t ) t∈[0,T ] be given by
Xt = x +
Z
t
as ds +
0
Z
t
gs dWs .
0
The process s 7→ ψ0 (X s )gs is progressively measurable and belongs to M 1 ([0, T ]; H), and for all
t ∈ [0, T ] we have
ψ(X t ) = ψ(x) +
Z
t
0
ψ (X s )(as ) ds +
t
Z
0
0
ψ (X s )(gs ) dWs + lim
n→∞
0
m(n)−1
X
n
Rψ (X t in ∧t , X t i+1
∧t )
(3.4)
i=0
n
with convergence in probability, for any sequence of partitions Πn = {0 = t 0n < t 1n < · · · < t m(n)
= T}
n
n
whose meshes kΠn k := max06i 6m(n)−1 |t i+1 − t i | tend to 0 as n → ∞. Moreover, there exists a
constant C and, for each " > 0, a constant C" , both independent of a and g, such that
E lim inf
n→∞
m(n)−1
X
Z t
p
2
n
|Rψ (X t in ∧t , X t i+1
kgs k2γ(H,E) ds .
∧t )| 6 "C E sup kX s k + C" E
p
s∈[0,t]
i=0
(3.5)
0
1− 2
The proof shows that we may take C" = C 0 (" p + 1) for some constant C 0 independent of a, g,
and ".
Before we start the proof of the theorem we state some lemmas. The first is an immediate consequence of Burkholder’s inequality (3.3).
Lemma 3.2. Under the assumptions of Theorem 3.1 we have
Z
E sup kX t k p 6 C E
06 t 6 T
0
T
kas k ds
p
Z
+ CE
0
T
kgs k2γ(H,E) ds
p
2
.
Lemma 3.3. Under the assumptions of Theorem 3.1, the process t 7→ ψ0 (X t )(g t ) is progressively
measurable and belongs to M 1 ([0, T ]; H).
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697
Proof. By the identity kψ0 (x)k = pkxk p−1 and Hölder’s inequality,
T
Z
E
kψ0 (X t )(g t )k2H dt
1
2
T
Z
6E
0
0
kψ0 (X t )k2 kg t k2γ(H,E) dt
6 E sup kX t k p−1
t∈[0,T ]
Z
0
1
2
T
kg t k2γ(H,E) ds
2
T
p−1 Z
p
p
6 C E sup kX t k
E
t∈[0,T ]
1
0
kg t k2γ(H,E) ds
p 1
2
p
,
and the right-hand side is finite by the previous lemma. The progressively measurability is clear.
This lemma implies that the stochastic integral in (3.4) is well-defined.
Lemma 3.4. Let 0 6 u 6 t 6 T be arbitrary and fixed. Under the assumptions of Theorem 3.1,
the process s 7→ ψ0 (X u )(gs ) is progressively measurable and belongs to M 1 ([0, T ]; H). Moreover,
P-almost surely,
Z t
Z t
ψ0 (X u )
ψ0 (X u )(gs ) dWs .
gs dWs =
u
u
Proof. By similar estimates as in the previous lemma,
Z t
p 1
Z t
1
p−1 2
2
p
p p
0
2
6 C(EkX u k )
E
.
kgs k2γ(H,E) ds
E
kψ (X u )(gs )kH ds
u
u
The progressively measurability is again clear. To prove the identity we first assume that g is a
simple adapted process of the form
gs =
n−1
X
1(t i ,t i+1 ] (s)Ai ,
i=0
where Π = {u = t 0 < t 1 < · · · < t n = t} is a partition of the interval [0, T ] and the random
variables are F t i -measurable and take values in the space of all finite rank operators from H to E.
Then,
0
ψ (X u )
Z
t
n−1
X
gs dWs = ψ0 (X u )
Ai (Wt i+1 − Wt i )
u
i=0
=
n−1
X
i=0
0
ψ (X u )(Ai (Wt i+1 − Wt i )) =
Z
t
ψ0 (X u )(gs ) dWs .
u
For general progressively measurable g ∈ L p (Ω; L 2 ([0, T ]; γ(H, E))), the identity follows by a
routine approximation argument.
Proof of Theorem 3.1. The proof of the theorem proceeds in two steps. All constants occurring in
the proof may depend on E and p, even where this is not indicated explicitly, but not on T . The
numerical value of the constants may change from line to line.
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Electronic Communications in Probability
Step 1 – Applying Lemma 2.2 to the function ψ(x) = kxk p and the process X , we have, for every
t ∈ [0, T ],
ψ(X t ) − ψ(x) =
m(n)−1
X n
ψ(X t i+1
∧t ) − ψ(X t in ∧t )
i=0
=
m(n)−1
X
n
ψ0 (X t in ∧t )(X t i+1
∧t − X t in ∧t ) +
i=0
m(n)−1
X
n
Rψ (X t in ∧t , X t i+1
∧t ).
i=0
We shall prove the identity (3.4) by showing that
lim
m(n)−1
X
n→∞
0
n
ψ (X t in ∧t )(X t i+1
∧t − X t in ) =
t
Z
0
ψ (X s )(as ) ds +
Z
0
i=0
t
ψ0 (X s )(gs ) dWs
0
with convergence in probability. In view of the definition of X t , it is enough to show that
m(n)−1
Z
X
0
n
lim ψ (X t i ∧t )
n→∞
i=0
n
∧t
t i+1
t
Z
as ds −
t in ∧t
0
ψ0 (X s )(as ) ds = 0 P-almost surely
and
lim
n→∞
m(n)−1
X
Z
ψ (X t in ∧t )
n
t i+1
∧t
0
i=0
t
Z
ψ0 (X s )(gs ) dWs = 0 in probability.
gs dWs −
t in ∧t
0
By (2.2), P-almost surely we have
m(n)−1
Z
X
0
lim sup ψ (X t in ∧t )
n→∞
i=0
Z
m(n)−1
X 6 lim sup
n→∞
i=0
6 C sup kX s k
p−2
n
t i+1
∧t
t in ∧t
n
t i+1
∧t
t in ∧t
6 C sup kX s k p−2 × lim sup
n
Z t i+1
∧t
m(n)−1
X
kX t in ∧t − X s k kas k ds
sup
sup
n
06i 6m(n)−1 s∈[t in ∧t,t i+1
∧t]
Z
m(n)−1
X
i=0
t in ∧t
i=0
n→∞
×
0
ψ0 (X s )(as ) ds
(ψ0 (X t in ∧t ) − ψ0 (X s ))(as ) ds
n→∞
s∈[0,T ]
t
Z
as −
× lim sup
s∈[0,T ]
n
t i+1
∧t
kas k ds
kX t in ∧t − X s k
t in ∧t
= 0,
where we used the continuity of the process X in the last line.
Next, by Lemma 3.4 and the inequalities (3.2) and (2.2),
m(n)−1
X
i=0
0
ψ (X
t in ∧t
Z
)
n
t i+1
∧t
Z
gs dWs −
t in ∧t
0
t
ψ0 (X s )(gs ) dWs
(3.6)
Stochastic convolutions in 2-smooth Banach spaces
=
n
Z t i+1
∧t
m(n)−1
X
t in ∧t
i=0
=
t m(n)−1
X
Z
0
699
0
ψ (X t in ∧t )(gs ) dWs −
t
Z
ψ0 (X s )(gs ) dWs
0
0
0
n
1(t in ,t i+1
] (s)(ψ (X t in ∧t ) − ψ (X s ))(g s ) dWs .
i=0
The localized stochastic integral being continuous from M ([0, t]; γ(H, E))) into L 0 (Ω, F t ; E), in
order to prove that the right-hand side converges to 0 in probability it suffices to prove that
m(n)−1
X
0
0
n
1(t in ,t i+1
lim s 7→
] (s)(ψ (X t in ∧t ) − ψ (X s ))(g s )
n→∞
i=0
L 2 ([0,t];H)
= 0 in probability.
For this, in turn, it suffices to observe that P-almost surely
m(n)−1
X
0
0
n
n
lim 1(t in ,t i+1
(s)(ψ
(X
)
−
ψ
(X
))
]
t i ∧t
s n→∞
i=0
= lim
sup
sup
n→∞ 06i 6n−1 s∈[t n ∧t,t n ∧t]
i
i+1
L ∞ ([0,t];E ∗ )
kψ0 (X t in ∧t ) − ψ0 (X s )k = 0
by the path continuity of X .
Step 2 – In this step we prove the estimate (3.5). By (2.2), for all x, y ∈ E and r ∈ [0, 1] we have
|ψ0 (x + r( y − x)) − ψ0 (x)| 6 (kxk p−2 kx − yk + kx − yk p−1 ).
Combining this with (2.5) we obtain
p−2
2
p
n
n
n
|Rψ (X t in ∧t , X t i+1
kX t i+1
∧t − X t in ∧t k .
∧t )| 6 CkX t in ∧t k
∧t − X t in ∧t k + CkX t i+1
We shall estimate the two terms on the right hand of (3.7) side separately.
For the first term, using the inequality |a + b|2 6 2|a|2 + 2|b|2 we obtain
m(n)−1
X
2
n
kX t in ∧t k p−2 kX t i+1
∧t − X t in ∧t k
i=0
62
m(n)−1
X
kX t in ∧t k
i=0
=:
I1n
+
Z
n
t i+1
∧t
p−2 t in ∧t
m(n)−1
2
Z
X
p−2 as ds + 2
kX t in ∧t k i=0
n
t i+1
∧t
t in ∧t
2
gs dWs I2n .
For the first term we have
I1n
6 2C sup kX s k
s∈[0,t]
p−2
Z
× sup i
Z
p−2
6 2C sup kX s k
× sup s∈[0,t]
i
n
t i+1
∧t
t in ∧t
n
t i+1
∧t
t in ∧t
Z
m(n)−1
X as ds ×
i=0
n
t i+1
∧t
t in ∧t
Z t
as ds ×
kas k ds.
0
as ds
(3.7)
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Electronic Communications in Probability
n
By letting n → ∞ we have max06i 6m(n)−1 (t i+1
− t in ) → 0, so
sup
06i 6m(n)−1
Z
n
t i+1
∧t
as ds → 0
t in ∧t
as n → ∞. Therefore,
lim I n
n→∞ 1
= 0, P-almost surely.
To estimate I2 we use (3.2) and Young’s inequality with " > 0 to infer
E
lim inf I2n
n
6 lim inf
n
E I2n
m(n)−1
X
= lim inf E
n
kX
t in ∧t
k
Z
i=0
= lim inf
m(n)−1
X
n
i=0
6 C lim inf
m(n)−1
X
n
n
∧t
t i+1
t in ∧t
Z
E kX t in ∧t k p−2
i=0
t in ∧t
n
t i+1 ∧t
s∈[0,t]
t in ∧t
i=0
= C E sup kX s k p−2
n
t i+1
∧t
t in ∧t
kgs k2γ(H,E) ds
t
Z
s∈[0,t]
kgs k2γ(H,E) dsF t in ∧t
kgs k2γ(H,E) ds
Z
m(n)−1
X
p−2
6 C lim inf E sup kX s k
n
2 gs dWs F t in ∧t
n
∧t
t i+1
Z
E kX t in ∧t k p−2 E
i=0
6 C lim inf
2
gs dWs t in ∧t
Z
E kX t in ∧t k p−2 E m(n)−1
X
n
n
t i+1
∧t
p−2 0
kgs k2γ(H,E) ds
Z t
p
p
2
1− 2
p
6 C" E sup kX s k + C"
E
kgs k2γ(H,E) ds .
s∈[0,t]
0
Next we estimate the second term in (3.7). We have
m(n)−1
X
p
n
kX t i+1
∧t − X t in ∧t k 6 C
Z n
m(n)−1
X t i+1 ∧t
i=0
i=0
=:
I3n
+
t in ∧t
Z
m(n)−1
p
X as ds + C
i=0
n
t i+1
∧t
t in ∧t
p
gs dWs I4n .
A similar consideration as before yields
lim I n
n→∞ 3
6 C lim
sup
n→∞ 06i 6m(n)−1
Z
n
t i+1
∧t
t in ∧t
p−1 Z t
as ds
×
kas k ds = 0.
0
Moreover, by Burkholder’s inequality (3.3),
E lim inf I4n
n
6
lim inf E I4n
n
= C lim inf
n
m(n)−1
X
i=0
Z
E
n
t i+1
∧t
t in ∧t
p
gs dWs Stochastic convolutions in 2-smooth Banach spaces
6 C lim inf
n
701
m(n)−1
X
Z
E
kgs k2γ(H,E) ds
t in ∧t
i=0
Z
m(n)−1
X
6 C lim inf E
n
n
t i+1
∧t
n
t i+1
∧t
kgs k2γ(H,E) ds
t in ∧t
i=0
p
2
p
2
Z t
p
2
= CE
kgs k2γ(H,E) ds .
0
Collecting terms, for any " > 0 we obtain the estimate
E lim inf
n→∞
m(n)−1
X
n
|Rψ (X t in ∧t , X t i+1
∧t )|
i=0
Z t
p
p
2
1− 2
p
+ 1)E
6 C" E sup kX s k + C("
kgs k2γ(H,E) ds .
s∈[0,t]
0
In the proof of Theorem 1.2 we will also need the following simple observation.
Lemma 3.5. P-Almost surely we have
lim inf sup
n→∞ t∈[0,T ]
m(n)−1
X
n
|Rψ (X t in ∧t , X t i+1
∧t )| 6 lim inf
n→∞
i=0
m(n)−1
X
n )|.
|Rψ (X t in , X t i+1
(3.8)
i=0
n
n
Proof. Fix t ∈ (0, T ] and let k(n) be the unique index such that t ∈ (t k(n)
, t k(n)+1
]. Then
m(n)−1
X
n
|Rψ (X t in ∧t , X t i+1
∧t )|
i=0
=
k(n)−1
X
n )| + |R (X n
, X t )|
|Rψ (X t in , X t i+1
ψ
t
k(n)
i=0
m(n)−1
X
6
n )| + |R (X n
|Rψ (X t in , X t i+1
, X t )|
ψ
t
k(n)
i=0
m(n)−1
X
6
n )| + CkX n
|Rψ (X t in , X t i+1
k p−2 kX t − X t n k2 + CkX t − X t n k p
t
k(n)
k(n)
k(n)
i=0
m(n)−1
X
6
p−2
n )| + C sup kX k
|Rψ (X t in , X t i+1
kX t − X t n k2 + CkX t − X t n k p .
s
i=0
s∈[0,T ]
k(n)
k(n)
Now (3.8) follows by taking the limes inferior for n → ∞ and using path continuity.
4
Proof of Theorem 1.2
We proceed in four steps. In Steps 1 and 2 we establish the estimate in the theorem for g ∈
M p ([0, T ]; γ(H, E)) with 2 6 p < ∞. In order to be able to cover exponents 0 < p < 2 in Step 3,
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Electronic Communications in Probability
we need a stopped version of the inequalities proved in Steps 1 and 2. For reasons of economy of
presentations, we therefore build in a stopping time τ from the start. In Step 4 we finally consider
the case where g ∈ M ([0, T ]; γ(H, E)).
We shall apply (a special case of) Lenglart’s inequality [13, Corollaire II] which states that if
(ξ t ) t∈[0,T ] and (a t ) t∈[0,T ] are continuous non-negative adapted processes, the latter non-decreasing,
such that Eξτ 6 Eaτ for all stopping times τ with values in [0, T ], then for all 0 < r < 1 one has
E sup ξ rt 6
06 t 6 T
2−r
1−r
Ea Tr .
(4.1)
Step 1 – Fix p > 2 and suppose first that g ∈ M p ([0, T ]; γ(H, D(A)). As is well known (see [16]),
Rt
under this condition the process X t = 0 e(t−s)A gs dWs is a strong solution to the equation
dX t = AX t dt + g t dWt , t > 0; X 0 = 0.
In other words, X satisfies
Z
t
Xt =
AX s ds +
t
Z
0
gs dWs ∀t ∈ [0, T ] P-almost surely.
0
Hence if τ is a stopping time with values in [0, T ], then by (3.1),
Z t
Z t
X t∧τ =
1[0,τ] (s)AX s ds +
0
1[0,τ] (s)gs dWs ∀t ∈ [0, T ], P-almost surely.
0
Let us check next that a t := 1[0,τ] (t)AX t satisfies the assumptions of Theorem 3.1. Indeed, with
h t := 1[0,τ] (t)Ag t we have, using the contractivity of the semigroup S and Burkholder’s inequality
(3.3),
p
Z T
p
Z T Z t
E
ka t k dt 6 E
e(t−s)Ahs dWs dt
0
0
Z
6 CT
p−1
0
T
E
0
Z
6 CT
p−1
Z t
p
e(t−s)Ahs dWs dt
0
T
Z
t
E
Z
6 CT E
0
T
p
0
0
ke(t−s)Ahs k2γ(H,E) ds
khs k2γ(H,E) ds
p
2
p
2
dt
< ∞.
Hence we may apply Theorem 3.1 and infer that
Z t
1[0,τ] (s)ψ0 (X s )(AX s ) ds
kX t∧τ k p =
0
+
Z
t
1[0,τ] (s)ψ0 (X s )(gs ) dWs + lim
n→∞
0
Z
6
0
m(n)−1
X
t
1[0,τ] (s)ψ0 (X s )(gs ) dWs + lim
n→∞
m(n)−1
X
i=0
n
Rψ (X t in ∧t∧τ , X t i+1
∧t∧τ )
i=0
n
Rψ (X t in ∧t∧τ , X t i+1
∧t∧τ ),
Stochastic convolutions in 2-smooth Banach spaces
703
since ψ0 (x)(Ax) 6 0 for all x ∈ D(A) by the contractivity of e tA (see [3, Lemma 4.2]). Hence, by
Lemma 3.5,
E sup kX t∧τ k p
t∈[0,T ]
t
Z
1[0,τ] (s)ψ0 (X s )(gs ) dWs + E sup lim inf
6 E sup
t∈[0,T ]
t∈[0,T ] n→∞
0
t
Z
1[0,τ] (s)ψ0 (X s )(gs ) dWs + E lim inf
6 E sup
t∈[0,T ]
n→∞
0
n
|Rψ (X t in ∧t∧τ , X t i+1
∧t∧τ )|
i=0
n
|Rψ (X t in ∧τ , X t i+1
∧τ )|
i=0
t
Z
1[0,τ] (s)ψ0 (X s )(gs ) dWs
6 C E sup
t∈[0,T ]
m(n)−1
X
m(n)−1
X
0
Z
+ "C E sup kX s∧τ k + C" E
p
s∈[0,T ]
0
T
1[0,τ] (s)kgs k2γ(H,E) ds
p
2
.
By Burkholder’s inequality (3.3) and the identity kψ0 ( y)k = pk yk p−1 ,
Z t
E sup 1[0,τ] (s)ψ0 (X s )(gs ) dWs t∈[0,T ]
0
T
Z
6 CE
Z
= CE
0
T
0
1
2
2
1[0,τ] (s)ψ0 (X s ) kgs k2γ(H,E) ds
1
2(p−1)
2
1[0,τ] (s)X s kgs k2γ(H,E) ds
T
Z
p−1
6 C E sup kX t∧τ k
t∈[0,T ]
1[0,τ] (s)kgs k2γ(H,E) ds
0
p−1 Z
p
p
p
6 C p E sup kX t∧τ k
E
t∈[0,T ]
0
Z
p
6 C" E sup kX t∧τ k + C" E
t∈[0,T ]
0
1 2
T
1[0,τ] (s)kgs k2γ(H,E) ds
T
1[0,τ] (s)kgs k2γ(H,E) ds
p
2
p 1
2
p
,
where we also used the Hölder’s inequality and Young’s inequality.
Combining these estimates and taking " > 0 small enough, we infer that
T
Z
E sup kX t∧τ k 6 C E
p
t∈[0,T ]
0
1[0,τ] (s)kgs k2γ(H,E) ds
p
2
.
Step 2 – Now let g ∈ M p ([0, T ]; γ(H, E) be arbitrary. Set g n = n(nI − A)−1 g, n > 1. These
processes satisfy the assumptions of Step 1 and we have kg n kγ(H,E) 6 kgkγ(H,E) pointwise. Define
Rt
X tn = 0 e(t−s)A gsn ds. From Step 1 we know that for any stopping time τ in [0, T ] we have
E sup
t∈[0,T ]
n
kX t∧τ
kp
Z
6 CE
0
T
1[0,τ] (s)kgsn k2γ(H,E) ds
p
2
.
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Electronic Communications in Probability
In particular, as n, m → ∞,
E sup kX tn − X tm k p → 0.
t∈[0,T ]
In these circumstances there is a process X̄ such that limn→∞ E sup t∈[0,T ] kX̄ tn − X t k p = 0 and
T
Z
E sup kX̄ t∧τ k 6 C E
1[0,τ] (s)kgs k2γ(H,E) ds
p
t∈[0,T ]
0
p
2
.
(4.2)
Also, notice that for every t ∈ [0, T ], we have
EkX tn
Z t
Z t
Z t
p
p
(t−s)A n
(t−s)A
− X t k = E
e
gs ds −
e
gs ds 6 C E
kgsn − gs k2γ(H,E) ds .
p
0
0
0
Hence X tn → X t in L p (Ω; E). Therefore, X̄ is a modification of X . This concludes the proof for
p > 2.
Step 3 – In this step we extend the result to exponents 0 < p < 2. First consider the case where
g ∈ M 2 ([0, T ]; γ(H, E)). By (4.2), for all stopping times τ in [0, T ] we have
2
Z
τ
EkX τ k 6 C E
0
kgs k2γ(H,E) ds.
It then follows from Lenglart’s inequality (4.1) that for all 0 < p < 2,
Z
E sup kX t k 6 C E
p
t∈[0,T ]
0
T
kgs k2γ(H,E) ds
p
2
.
For g ∈ M p ([0, T ]; γ(H, E)) the result follows by approximation.
Step 4 – Finally, the existence of a continuous version for the process X under the assumption
g ∈ M ([0, T ]; γ(H, E)) follows by a standard localisation argument.
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