Stochastic integration in Banach spaces and applications to parabolic evolution equations

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Stochastic Integration in Banach Spaces

and

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Stochastic Integration in Banach Spaces

and

Applications to Parabolic Evolution Equations

Proefschrift

ter verkrijging van de graad van doctor aan de Technische Universiteit Delft,

op gezag van de Rector Magnificus Prof. dr. ir. J.T. Fokkema, voorzitter van het College voor Promoties,

in het openbaar te verdedigen

op dinsdag 19 december 2006 om 15.00 uur door

Mark Christiaan VERAAR

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Prof. dr. Ph.P.J.E. Cl´ement Prof. dr. J.M.A.M. van Neerven Samenstelling promotiecommissie:

Rector Magnificus voorzitter

Prof. dr. Ph.P.J.E. Cl´ement, Technische Universiteit Delft, promotor Prof. dr. J.M.A.M. van Neerven, Technische Universiteit Delft, promotor Prof. dr. Z. Brze´zniak, University of York

Prof. dr. B. de Pagter, Technische Universiteit Delft, Prof. dr. S.M. Verduyn Lunel, Universiteit Leiden

Prof. dr. L.W. Weis, Universit¨at Karlsruhe Prof. dr. J. Zabczyk, Polish Academy of Sciences

Het onderzoek beschreven in dit proefschrift is mede gefinancierd door de Neder-landse Organisatie voor Wetenschappelijk Onderzoek (NWO), onder projectnummer 639.032.201.

Het Stieltjes Instituut heeft bijgedragen in de drukkosten van het proefschrift.

ISBN-10: 90-9021380-5 ISBN-13: 978-90-9021380-4

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Preface

This Ph.D. thesis was written during my period as a Ph.D. student at the Delft University of Technology. In this preface I would like to express my thanks to all who contributed in some way to the realization of this thesis.

First of all, I would like to thank my advisor Jan van Neerven, for his guidance in the last four years. He introduced me to the subject of this thesis and he taught me a lot of mathematics. It has been great working together and I hope that the collaboration will continue in the future. I wish to thank Philippe Cl´ement for many useful discussions and his interest in my research. I also would like to thank Lutz Weis for his kind hospitality during my stays at the Universit¨at Karlsruhe (Germany) and the University of South-Carolina (USA). The always pleasant discussions have been very important to me, and I hope that the fruitful collaboration will be continued.

I wish to thank Onno van Gaans, Stefan Geiss, Tuomas Hyt¨onen, Roland Schnaubelt and Mario Walther for helpful discussions. I am grateful to Sonja Cox for reading the introduction and propositions of this thesis and for her comments.

I would like to thank all the colleagues of the Analysis group in Delft for the pleasant working atmosphere. In particular I am grateful to my colleagues Guido Sweers, Erik Koelink and Ben de Pagter, and to my fellow Ph.D. students Anna Dall’Acqua, Timofey Gerasimov, Wolter Groenevelt, Jan Maas and Yvette van Norden. Many thanks go to the colleagues in Karlsruhe, Markus Duelli, Bernhard Haak, Cornelia Kaiser, Peer Kunstmann and Jan Zimmerschied, for all their help during my stays. I also would like to thank the colleagues in South-Carolina for their kind hospitality.

I am grateful to the organizers of the Tulka internet seminar in the years 2003-2006 for the stimulating courses. I wish to thank the people of the European Research Training Network “Evolution Equations for Deterministic and Stochastic Systems” (HPRN-CT-2002-00281) for the interesting workshops in Brest, Delft, Jena, Pisa and Vienna.

During my research I was financially supported by the Netherlands Organization for Scientific Research (NWO), under project number 639.032.201, and by the Marie Curie Fellowship Program in Karlsruhe.

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Chapter 1 Introduction

Stochastic partial differential equations (SPDEs) of evolution type are usually modelled as ordinary stochastic differential equations (SDEs) in an infinite-dimensional state space. In many examples such as the stochastic heat and wave equation, this viewpoint may lead to existence and uniqueness results and regularity properties. To model the equation in such a way one needs a stochastic integration theory for processes with values in an infinite-dimensional space. Since real-valued stochastic integration theory extends directly to processes with values in Hilbert spaces, this is the class of spaces in which SPDEs are usually modelled. This approach has been considered by many authors using semigroup methods from the 70th’s up to now.

There are situations where it is more natural to model the SPDE in a function space which is not a Hilbert space but only a Banach space. The main problem for this is to find a “good” stochastic integration theory for processes with values in a Banach space. In the 70th’s and 80th’s several authors found negative results in this direction, and it turned out that the stochastic integration theory for Hilbert spaces does not extend to the Banach space setting.

In this thesis we show that if one reformulates the integration theory for Hilbert spaces, then it does generalize to a certain class of Banach spaces. For this class of spaces we give a complete description of the stochastically integrable processes and we show that two-sided estimates for the stochastic integral hold. We develop a stochastic calculus for the stochastic integral. In particular we show that the Itˆo formula holds. The results lead to new applications in SPDEs.

1.1 Some history on stochastic integration

Let W be a standard Brownian motion on a probability space (Ω, A, P) with filtration F = (Ft)t≥0. The stochastic integral

R1

0 φ(t) dW (t) of a function φ ∈ L

2_{(0, 1) was first}

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The case of adapted and joint measurable processes φ : [0, 1] × Ω → R was first considered by Itˆo in [60]. In this case the Itˆo isometry holds:

E Z 1 0 φ(t) dW (t) 2 = kφk2_L2_((0,1)×Ω) for adapted φ ∈ L2((0, 1) × Ω). (1.1.2)

It is easy to check that (1.1.1) and (1.1.2) can be generalized to the case where φ takes values in a Hilbert space. This is due to the fact that the norm comes from an inner product. If φ takes values in a general Banach space (E, k · k), stochastic integration is more difficult. In [137] for E = lp _{with p ∈ [1, 2), Yor constructed}

a uniformly bounded measurable and adapted process φ : [0, 1] × Ω → E which is not stochastically integrable with respect to W in any reasonable sense. A possible definition of stochastic integrability can be found in Definition 1.3.1. Later on, in [117] Rosi´nski and Suchanecki showed that there even exists a uniformly bounded measurable function φ : [0, 1] → E that is not stochastically integrable with respect to W .

On the other hand, for the class of Banach spaces E that have type 2 (e.g. Lp

-spaces for 2 ≤ p < ∞), Hoffmann-Jørgensen and Pisier showed in [59] that there exists a constant C such that

E Z 1 0 φ(t) dW (t) 2 ≤ C2_kφk2

L2_(0,1;E) for φ ∈ L2(0, 1; E). (1.1.3)

In other words a one-sided version of (1.1.1) holds for Banach spaces with type 2. In the case that E has martingale type 2 or equivalently up to renorming, E is 2-uniformly smooth (e.g. Lp-spaces for 2 ≤ p < ∞), one can even show that there is a constant C such that

E Z 1 0 φ(t) dW (t) 2 ≤ C2_|φk2

L2_{((0,1)×Ω;E)} for adapted φ ∈ L2((0, 1) × Ω; E). (1.1.4)

This has been considered by several authors (cf. [13, 19, 42, 102] and the references therein).

The results (1.1.3) and (1.1.4) are very useful but they do not give two-sided estimates or characterizations of integrability. In [117] Rosi´nski and Suchanecki characterized the functions φ : [0, 1] → E that are stochastically integrable with respect to W in terms of Gaussian characteristic functions. This result has been extended to more general integrators by Rosi´nski in [116].

In [50] Garling showed that for UMD spaces E and all p ∈ (1, ∞) there exist constants c, C such that for all adapted step processes φ : [0, 1] × Ω → E, one has

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1.2. γ-Spaces 3 the form (1.1.5) are referred to as decoupling inequalities. Garling also showed that the validity of (1.1.5) for all adapted step processes φ implies the UMD property.

The UMD property is defined in terms of unconditionality of martingales difference sequences. Examples of UMD Banach spaces are the Lp_{-spaces for p ∈ (1, ∞). UMD}

spaces are important for vector-valued harmonic analysis. This is due to the fact that the Hilbert transform is bounded on Lp_{(R; E) for all p ∈ (1, ∞) if E is a UMD space.} This result has been proved by Burkholder in [27]. In [50] Garling has given a short proof based on (1.1.5). The converse also holds, i.e. if for some p ∈ (1, ∞) the Hilbert transform is bounded on Lp_{(R; E), then E is a UMD space. This has been} showed by Bourgain in [17].

UMD spaces also are the “right” spaces for studying maximal Lp_{-regularity for}

evolution equations. For this class of spaces Weis [133] has characterized maximal Lp -regularity in terms of R-boundedness of certain resolvents. For the proof he extended the Mikhlin multiplier theorem to a UMD space setting. The randomized notion R-boundedness will play an important rˆole in this thesis as well.

In [88] McConnell has generalized (1.1.5) to so-called tangent martingale dif-ference sequences. This result has been proved independently in [57] by Hitczenko. Their proofs are based on the existence of certain biconcave functions related to the UMD property and constructed by Burkholder in [28]. Using the inequalities for tangent martingale difference sequences, McConnell has given sufficient conditions for measurable and adapted processes φ : [0, 1] × Ω → E to be stochastically integrable with respect to W . He showed that if almost all paths of φ are stochastically integrable with respect to an independent Brownian motion, then φ is stochastically integrable with respect to W .

In the above results no two-sided estimates for the stochastic integral are given. In [23] by Brze´zniak and van Neerven and in [100] by van Neerven and Weis, the authors have studied stochastic integrability of operator-valued functions Φ : [0, 1] → B(H, E) with respect to WH. Here H is a separable Hilbert space and WH is a

cylindrical Brownian motion. They have extended the results of [117] to the operator-valued situation and obtained an isometry in terms of γ-spaces. Before we explain this in more detail we will discuss γ-spaces.

1.2 γ-Spaces

Let E be a Banach space and let H be a separable Hilbert space. Let (γn)n≥1 be a

Gaussian sequence, i.e. a sequence of independent standard Gaussian random variables on a probability space (Ω, A, P).

A bounded operator R ∈ B(H, E) is said to be γ-radonifying if there exists an orthonormal basis (hn)n≥1 of H such that

P

n≥1γnRhn converges in L2(Ω; E). For

such an operator we define

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This quantity does not depend on the sequence (γn)n≥1 and the basis (hn)n≥1, and

defines a norm on the space γ(H, E) of all γ-radonifying operators from H into E. Endowed with this norm, γ(H, E) is a Banach space and it defines an operator ideal. The space γ(H, E) is a closed linear subspace of the space of almost summing operators from H into E which was first introduced by Linde and Pietsch in [79].

Let (S, Σ, µ) be a separable measure space. We say that φ : S → E belongs to L2(S) scalarly if for all x∗ ∈ E∗_{, hφ, x}∗_{i ∈ L}2_{(S). For a strongly measurable function}

φ : S → E which is scalarly in L2_{(S), consider the integral operator I}

φ : L2(S) → E given by Iφf = Z S φ(s)f (s) dµ(s), f ∈ L2(S).

This integral is well-defined as a Pettis integral. We define the γ-norm of φ as kφkγ(S;E) = kIφkγ(L2_(S),E).

This randomized norm or γ-norm has been introduced by Kalton and Weis in [67]. For Hilbert spaces E one has kφkγ(S,E) = kφkL2_(S;E) and for certain spaces E (for

example Lp_{-spaces) it is equivalent with a square-function norm. For this reason the}

γ-norm is sometimes referred to as a generalized square-function. The space γ(S; E) is not complete in general. However, it is a subspace of the Banach space γ(L2(S), E) in a canonical way.

Many classical results from Hilbert space theory do not generalize to Banach spaces in a direct way. In some cases however, a reformulation with γ-norms allows such a generalization. An illustrative example of this is the Fourier-Plancherel formula. For a function φ ∈ L1_{(R; E) ∩ L}2_{(R; E) it is not true in general that the Fourier transform}

b

φ satisfies the isometry k bφkL2_(R;E) = kφk_L2_(R;E). More precisely, if this holds for all

functions φ, then E is isomorphic to a Hilbert space. However, for general Banach spaces E one still has

kbφk_γ(R;E) = kφk_γ(R;E). We refer to Chapter 3 for more details.

The γ-norms or generalized square-functions have turned out to be useful in various fields of analysis. In [49, 67, 66] γ-norms have been used to study the H∞-calculus. In [55] applications to control have been obtained and in [62] the γ-spaces have been applied for wavelet decompositions. In [48] the randomized norms have been used in the study of local Banach space theory.

At the end of the previous section we have already mentioned that γ-norms have been used in [23] and [100] to study stochastic integrability. We can now formulate this result more precisely. For notational convenience we do not consider the operator-valued situation. The authors show that a function φ : [0, 1] → E is stochastically integrable if and only if φ ∈ γ(0, 1; E). In this case the following isometry holds:

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1.3. Stochastic integration in UMD spaces 5 Since (1.2.1) is an isometry, this seems to be the right generalization of (1.1.1) to the Banach space setting. The good structure of γ(0, 1; E) makes it possible to study the stochastic integral in great detail.

It is not difficult to see that one has a continuous embedding

L2(0, 1; E) ,→ γ(0, 1; E) (1.2.2)

if and only if E has type 2. As a consequence of this and (1.2.1) one also obtains (1.1.3). We will extend this embedding result to spaces with type p where p ∈ [1, 2]. Recall that for p < ∞, Lp-spaces have type min{p, 2} and every Banach space has type 1.

For s ∈ R, 1 ≤ p, q ≤ ∞ let Bs

p,q(Rd; E) denote the vector-valued Besov space.

Theorem 1.2.1. Let E be a Banach space and let p ∈ [1, 2]. Then the space E has type p if and only if B

d p−

d 2

p,p (Rd; E) ,→ γ(Rd; E).

Theorem 1.2.1 will be proved in Section 3.3. A similar result holds for domains S ⊂ Rd_{. As a consequence we obtain sufficient conditions for stochastic integrability.}

In particular if E has type p, then for all α > 1_p−1

2, every function φ ∈ C

α_{([0, 1]; E) is}

stochastically integrable. Here Cα_{([0, 1]; E) is the space of E-valued α-H¨}_older

contin-uous functions. Conversely, one can show that if α ∈ (0,1₂) and every φ ∈ Cα([0, 1]; E) is stochastically integrable, then E has type p for all p ∈ (1, 2) with α < 1

p − 1 2. In

particular for E = lp _{with p ∈ [1, 2), for all α <} 1 p −

1

2, there exists an α-H¨older

continuous function φ : [0, 1] → E which is not stochastically integrable with respect to W . This extends examples in [117, 137].

1.3 Stochastic integration in UMD spaces

One of the aims of this thesis is to extend the results in [23] and [100] to operator-valued processes. Again for notational convenience we will explain the results in the case of E-valued processes. Before we state the result let us give a precise definition of the stochastic integral. It is obvious how to define the stochastic integral for adapted step processes, and we will extend this to more general processes below.

Recall that for a Banach space E, L0(Ω; E) stands for the space of strongly mea-surable functions from Ω into E, with the topology induced by convergence in proba-bility. For a strongly measurable process φ : [0, 1] × Ω → E we say that φ belongs to L0(Ω; L2(0, 1)) scalarly if for all x∗ ∈ E∗_{, hφ, x}∗_{i ∈ L}0_{(Ω; L}2_{(0, 1)).}

Definition 1.3.1. For a strongly measurable and adapted process φ : [0, 1] × Ω → E that is scalarly in L0_{(Ω; L}2_{(0, 1)) we say that φ is stochastically integrable with respect}

to W if there exists a sequence (φn)n≥1 of adapted step processes such that:

(i) for all x∗ ∈ E∗ _{we have lim} n→∞hφn, x

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(ii) there exists a process ζ ∈ L0_{(Ω; C([0, 1]; E)) such that} ζ = lim n→∞ Z · 0 φn(t) dW (t) in L0(Ω; C([0, 1]; E)).

The process ζ is called the stochastic integral of φ with respect to W , notation ζ =

Z ·

0

φ(t) dW (t).

In this way R₀·φ(t) dW (t) is uniquely defined up to indistinguishability. Moreover, it is a continuous local martingale starting at zero.

We will give a very short proof of the decoupling inequality (1.1.5) and using this and (1.2.1) we will prove the following result.

Theorem 1.3.2. Let E be a UMD space. For a strongly measurable and adapted process φ : [0, 1] × Ω → E which is scalarly in L0(Ω; L2(0, 1)) the following assertions are equivalent:

(1) φ is stochastically integrable.

(2) There exists a process ζ ∈ L0_{(Ω; C([0, 1]; E)) such that for all x}∗ _{∈ E}∗_{, we have}

hζ, x∗i = Z ·

0

hφ(t), x∗i dW (t) in L0_{(Ω; C([0, 1])).}

(3) φ ∈ γ(0, 1; E) almost surely.

(4) For almost all ω ∈ Ω, φ(·, ω) is stochastically integrable with respect to ˜W . In this case, for all p ∈ (1, ∞),

E sup t∈[0,1] Z t 0 φ(s) dW (s) p

hp,E Ekφkp_γ(0,1;E). (1.3.1)

The implication (4) ⇒ (1) is the mentioned result of McConnell from [88] which he proved with different methods.

The theorem does not contain the isometry (1.2.1) for processes. However, (1.3.1) is a good replacement since it is a two-sided estimate. The estimate (1.3.1) can be seen as a stochastic integral version of the Burkholder-Davis-Gundy inequalities. It will allow us to prove an E-valued version of the Brownian martingale representation theorem. The equivalent condition in (2) is very useful and rather surprising. It shows essentially that a “Bochner type” integrability property is equivalent to a “Pettis type” integrability property.

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1.3. Stochastic integration in UMD spaces 7 We will consider several other criteria for stochastic integrability. In Banach func-tion spaces, Lp-moments of the stochastic integral are estimated in terms of Lp-norms of square-functions. We also show that local Lp-martingales are stochastically in-tegrable. In particular the stochastic integral process R₀·φ(t) dW (t) is stochastically integrable with respect to W .

Theorem 1.3.2 will also be proved in the case that the integrator W is replaced by an arbitrary continuous local martingale M . It seems that the equivalence (4) does not extend to this situation. At some level it is surprising that integration of a process with respect to a general continuous local martingale can be checked using convergence of Gaussian sums in (3). However, seeing the γ-norm as a generalized square-function, this result is very natural. The proof of the result for arbitrary continuous local martingales is based on Theorem 1.3.2 and the result of Dambis, Dubins and Schwartz which says that every continuous local martingale is a time transformed Brownian motion

Although we have been able to give a short proof of (1.1.5), the proof in [50] is still very interesting. This comes from the fact that Garling proves the one-sided estimates E Z 1 0 φ(t) dW (t) p ≤ C E˜E Z 1 0 φ(t) d ˜W (t) p (1.3.2) and c E˜E Z 1 0 φ(t) d ˜W (t) p ≤ E Z 1 0 φ(t) dW (t) p (1.3.3)

from randomized versions of the UMD property. These randomized properties were introduced in [51] by Garling under the abbreviations LERMT (Lower Estimates for Random Martingale Transforms) and UERMT (Upper Estimates for Random Mar-tingale Transforms). We will prefer to use the notation UMD− and UMD+_{, because}

it emphasizes the relation with UMD. Here the superscript − stands for Lower and the superscript + stands for Upper. In [50], (1.3.2) has been proved for UMD− spaces and (1.3.3) has been proved for UMD+ _{spaces. Therefore, the characterizations in}

Theorem 1.3.2 have one-sided versions for UMD− and UMD+ space. For example the implication (3) ⇒ (1) remains valid for UMD− spaces. This is interesting since L1 is a UMD− space.

It is natural to ask for which spaces E, (1.3.2) and (1.3.3) hold. We already noted that in [50] it has been shown that if both inequalities hold, then E is a UMD space. We will extend this argument. However, we will only show that (1.3.2) implies the UMD− property for the special class of Paley-Walsh martingales. A similar result holds for UMD+_{. It is unknown whether this implies the properties for arbitrary}

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1.4 Itˆ

o’s formula

One of the main tools of stochastic calculus is the Itˆo formula. We will also obtain an Itˆo formula for our stochastic integral. The result is formulated below for stochastic integrals of the formR₀·, Φ(t) dWH(t), where H is a separable Hilbert space, Φ : [0, 1] ×

Ω → B(H, E), and WH is a cylindrical Brownian motion.

Although, we have only defined the stochastic integral for E-valued processes in the last section, we will briefly comment on the case of operator-valued processes. First of all, the process Φ is assumed to be H-strongly measurable and adapted, i.e. for all h ∈ H, Φh is strongly measurable and adapted. The cylindrical Brownian motion can be interpreted as WH(t) =

P

n≥1hn⊗ Wn, where (hn)n≥1 is an orthonormal basis for

H and (Wn)n≥1 is a sequence of independent standard Brownian motions. Now, one

can think of the stochastic integral of the process Φ as Z · 0 Φ(t) dWH(t) = X n≥1 Z · 0 Φ(t)hndWn(t),

where the series converses in L0_{(Ω; C([0, 1]; E)).} _{The integrable processes Φ can}

be characterized as in Theorem 1.3.2, where one has to replace γ(L2(0, 1), E) by γ(L2_{(0, 1; H), E) and γ(0, 1; E) by γ(0, 1; H, E).}

After this brief explanation on the stochastic integral for operator-valued processes, we return to the Itˆo formula. Let E and F be Banach spaces and let R ∈ γ(H, E) and T ∈ B(E, B(E, F )) be given. Let (hn)n≥1 be an arbitrary orthonormal basis for

H and define the trace of T with respect to R as TrRT :=

X

n≥1

(T Rhn)(Rhn).

This sum converges unconditionally and it does not depend on the choice of the or-thonormal basis.

We can now state the Itˆo formula which will be proved in Section 4.8.

Theorem 1.4.1. Let E and F be UMD spaces. Assume that f : R+× E → F is of

class C1,2. Let Φ : [0, 1] × Ω → B(H, E) be an H-strongly measurable and adapted process which is stochastically integrable with respect to WH and assume that the paths

of Φ belong to L2_{(0, 1; γ(H, E)) a.s. Let ψ : [0, 1] × Ω → E be strongly measurable}

and adapted with paths in L1(0, 1; E) a.s. Let ξ : Ω → E be strongly F0-measurable.

Define ζ : [0, 1] × Ω → E as ζ = ξ + Z · 0 ψ(s) ds + Z · 0 Φ(s) dWH(s).

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1.5. Some history on stochastic evolution equations 9 all t ∈ [0, 1], f (t, ζ(t)) − f (0, ξ) = Z t 0 D1f (s, ζ(s)) ds + Z t 0 D2f (s, ζ(s))ψ(s) ds + Z t 0 D2f (s, ζ(s))Φ(s) dWH(s) +1 2 Z t 0 TrΦ(s) D22f (s, ζ(s)) ds. (1.4.1)

Here the deterministic integrals are defined as Bochner integrals. The stochastic integral is defined as in Definition 1.3.1. There is also a version of the Itˆo formula for UMD− spaces.

The proof of (1.4.1) is based on Theorem 1.3.2 (2) and standard approximation arguments. For martingale type 2 spaces E and F the version of the above result can be found in [102] (also see [21]). Notice that Theorem 1.4.1 can also be applied to the situation where f : E × E∗ _{→ R is given by f(x, x}∗) = hx, x∗i. We do not know how such a result could be obtained with the Itˆo formula from [21, 102], unless E is isomorphic to a Hilbert space.

The careful reader has noticed the assumption that Φ belong to L2_{(0, 1; γ(H, E))}

a.s. in Theorem 1.4.1. This L2-assumption is unnatural in the sense that for the existence of the stochastic integral in the definition of ζ this is not needed. The assumption is needed in the approximation argument in the proof of the Itˆo formula. Although the condition is not very strong, we will show that at least in the special case where f : E × E∗ _{→ R is as above, the L}2_{-assumption on Φ can be omitted.}

1.5 Some history on stochastic evolution equations

In the theory of deterministic partial differential equations of evolution type it is common to formulate partial differential equations as ordinary differential equations in a function space. This approach is also used for stochastic partial differential equations. Therefore, it is useful to study stochastic equations in an abstract formulation. The abstract equation we will be considering is

(

dU (t) = A(t, U (t)) dt + B(t, U (t)) dWH(t), t ∈ [0, T ],

U (0) = u0.

(1.5.1)

Here (U (t))t∈[0,T ] is the unknown and A and B are certain unbounded and non-linear

operators. The noise term is modelled with the cylindrical Brownian motion WH and

u0 is the initial value which could be random.

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space. It has been applied in the stochastic case by Pardoux in [107] and Krylov and Rozovski˘ı in [71].

Semigroup methods are another way to study (1.5.1). Usually this method is applied in Hilbert spaces, because real-valued stochastic integration theory generalizes to such spaces. Many authors have studied (1.5.1) with semigroup methods. This started with Curtain and Falb [33] and Dawson [38] and has been continued by Da Prato and collaborators (cf. [37] and the references therein). One of the important tools that has been developed, is the factorization method from [35] by Da Prato, Kwapie´n and Zabczyk. This method allows one to prove space-time regularity results for stochastic convolutions. For a detailed discussion we refer to the monograph of Da Prato and Zabczyk [37] and the references therein.

In the papers [19] and [20] Brze´zniak has studied particular cases of (1.5.1) in a Banach space E. In [19] maximal regularity results have been considered and in [20] semi-linear equations have been studied using fixed point methods. The stochastic integral he uses is the one for martingale type 2 spaces E (see (1.1.4)). We will return to this topic in Section 1.6.

In the linear case with additive noise, i.e. (

dU (t) = A(t)U (t) dt + B(t) dWH(t), t ∈ [0, T ],

U (0) = u0,

(1.5.2)

it is possible to study (1.5.2) in an arbitrary Banach space. This is because (1.2.1) holds in every Banach space. The case where A is the generator of a strongly con-tinuous semigroup and B is constant, has been studied in [23] by Brze´zniak and van Neerven and in [100] by van Neerven and Weis. For analytic semigroup generators A, regularity of the solution of (1.5.2) has been studied by Dettweiler, van Neerven and Weis [43]. The situation where A depends on time and generates a strongly continuous evolution family has been considered in [131]. There the factor-ization method of [35] has been extended to the case where (A(t))t∈[0,T ] satisfies the

Acquistapace-Terreni conditions which have been introduced by Acquistapace and Terreni in [4] (see Section 10.2).

In the definition of a solution of a stochastic equation one can either fix the prob-ability space on which the processes live in advance or make this variable as well. The former is usually called a strong solution, whereas the latter is a martingale or weak solution. Martingale solutions are important since it allows one to get existence results for equations with less regular coefficients. However, in accordance with most references so far, we will only consider strong solutions.

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1.6. Stochastic evolution equations in UMD spaces 11 This has been generalized to the setting of UMD− _{spaces by Zimmerschied in [140].} His methods are based on the integration theory developed in this thesis (see Section 1.3) and a representation result of Ondrej´at [105].

Another possible approach to study (1.5.1) is Lyons’s rough path analysis. This has been taken up in the Banach space setting for bounded operators in [77] by Ledoux, Lyons and Qian.

1.6 Stochastic evolution equations in UMD spaces

We will consider the following class of semi-linear equations on a UMD Banach space E.

(

dU (t) = (A(t)U (t) + F (t, U (t))) dt + B(t, U (t)) dWH(t), t ∈ [0, T ],

U (0) = u0.

(1.6.1)

Here (A(t))t∈[0,T ] is a family of densely defined closed operators on E that generates an

evolution family (P (t, s))0≤s≤t≤T. The processes F and B are defined on certain spaces

in between E and the domains of (A(t))t∈[0,T ]. We will define strong, variational, mild

and weak solutions for (1.6.1). Here, strong and weak solution are not the probabilistic notions we discussed in Section 1.5, but analytic solution concepts. The formula of a mild solution is given by

U (t) = P (t, 0)u0+ Z t 0 P (t, s)F (s, U (s)) ds + Z t 0 P (t, s)B(s, U (s)) dWH(s).

We prove several equivalences between the solution types. Most of these equivalences are part of mathematical folklore. In case of additive noise some of the concepts were already considered by Chojnowska-Michalik in [30] (also see [37] and the references therein).

For Hilbert spaces E in the case that A generates a strong continuous semigroup, (1.6.1) has been considered by Da Prato and Zabczyk in [37, Chapter 7]. Existence and uniqueness results have been obtained using fixed point arguments. The case where A depends on time has been considered by Seidler in [122]. For this purpose, he extended the factorization method from [35] to the non-autonomous setting under restrictive assumptions on the fractional domain spaces. For UMD spaces E with type 2 in the case that A generates an analytic semigroup, (1.6.1) has been studied by Brze´zniak. Again, this is based on fixed point arguments and the factorization method.

As a first step we consider (1.6.1) in a UMD space E with type 2. We assume that (A(t))t∈[0,T ] satisfies the Tanabe conditions in [124, Section 5.2]. These condition are:

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on the fractional domain spaces and we prove the following result in Section 8.5. For a ∈ [0, 1) let V_a∞,loc(0, T ; E) be the space of processes φ : [0, T ] × Ω → (E, D)a,1which

are adapted and have continuous paths. The space (E, D)a,1 is the real interpolation

space between E and D. If a = 0, we let (E, D)a,1 = E.

Theorem 1.6.1. Let E be a UMD space with type 2. Assume that (A(t))t∈[0,T ] satisfies

the Tanabe conditions and that F and B are locally Lipschitz and of linear growth as in Section 8.5 (A2)0 and (A3)0 and that u0 ∈ (E, D)1

2,1. Then there exists a unique

mild solution U of (1.6.1) in V_a∞,loc(0, T ; E) and it has a version such that for almost all ω ∈ Ω,

t 7→ U (t, ω) ∈ Cλ([0, T ]; (E, D)a+δ,1),

where λ > 0 and δ ≥ 0 satisfy λ + δ < min{1 − (a + θF),1₂ − (a + θB)}.

The fixed point space we used is of the form

Lr(Ω; Lp(0, T ; (E, D)a,1)). (1.6.2)

Theorem 1.6.1 is applied to a second order equation with colored noise similar to the equation considered in [119] by Sanz-Sol´e and Vuillermot. The equation is modelled in Lp with p ∈ [2, ∞). We also prove a maximal regularity result for this equation. Secondly, we apply Theorem 1.6.1 to a general elliptic equation on bounded domains with space-time white noise. This will be explained in more detail in Section 1.7.1.

We will consider (1.6.1) on a general UMD− space as well. This will be difficult enough already for the case that A generates an analytic semigroup. We will extend the factorization method to UMD− spaces using the γ-norm. The problem (1.6.1) will be studied using fixed point methods. Since we have to consider deterministic and stochastic convolutions we need Bochner and γ-norms. Therefore, our fixed point space will be the intersection of a Bochner and a γ-space. The space has to be small enough in order to obtain existence and regularity properties for the stochastic convolution.

To be able to apply a fixed point argument we need to estimate both the determinis-tic and stochasdeterminis-tic convolution in the Bochner and weighted γ norm. The determinisdeterminis-tic convolution can be estimated in the γ-norm thanks to a combination of Theorem 1.2.1 and results for convolutions in Besov spaces. It will however give restrictions on F related to the type of the space E. The γ-norm of the stochastic convolution can be estimated likewise or with Pisier’s property (α). The Lp_{-spaces with p ∈ [1, ∞)}

satisfy this property. The notion of a Lipschitz function also has to be generalized to γ-norms. This leads to a randomized Lipschitz condition, which we call L2

γ-Lipschitz.

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1.6. Stochastic evolution equations in UMD spaces 13 Lipschitz as in Section 9.7 (A2)0 and of linear growth as in (9.5.1), B is locally L2

γ

-Lipschitz as in Section 9.7 (A3)0 and of linear growth as in (9.5.2) and u0 ∈ (E, D)1 2,1.

Then for every α ∈ (0,1₂) and p ∈ (2, ∞) such that α ∈ (a + θB,1₂ − 1_p) and a + θF < 3

2− 1 q−

1

p, there exists a unique mild solution U of (1.6.1) in V p,loc

α,a (0, T ; E) and it has

a version such that for almost all ω ∈ Ω,

t 7→ U (t, ω) ∈ Cλ([0, T ]; (E, D)a+δ,1),

where λ > 0 and δ ≥ 0 satisfy λ + δ < min{1₂ − (a + θB), 1 − (a + θF)}.

There is also a version of this result for spaces without property (α). This will give restrictions on the function B related to the type of E.

As an application of Theorem 1.6.2 we will consider a perturbed heat equation in Lp_{-spaces with p ∈ [1, ∞). We will also extend the above examples for L}p_{-spaces with}

p ∈ [2, ∞) to all p ∈ (1, ∞).

The last class of equations we will consider is of the form: dU (t) = A(t)U (t)dt + N X n=1 BnU (t) dWn(t), t ∈ [0, T ], U (0, x) = u0. (1.6.3)

Here the linear operators (A(t))t∈[0,T ] are closed and densely defined on a Banach

space E, the operators (Bn)Nn=1 are generators of strongly continuous groups on E,

and (Wn)Nn=1 are real-valued independent F -adapted Brownian motions. The main

problem here is that the (Bn)Nn=1 are unbounded with D(A(t)) ⊂

TN

n=1D(B 2 n) and

that we want the solution to be a strong solution. Typically (A(t))t∈[0,T ] is a family

of second order differential operators and (Bn)n≥1 are first order operators. For the

precise definition of a strong solution we refer to Section 7.5.

The problem (1.6.3) has been considered by Da Prato, Iannelli and Tubaro [34] and Da Prato and Zabczyk [37, Section 6.5] in the autonomous case and in Hilbert spaces. They showed that (1.6.3) is equivalent to a deterministic problem, which they solve pathwise. We will extend their results to the non-autonomous setting and to UMD Banach spaces. This will be done with the Itˆo formula from Section 1.4. In particular we need this formula for f : E×E∗ _{→ R given by f(x, x}∗) = hx, x∗i. To solve the deterministic problem we will assume that (A(t))t∈[0,T ] satisfies the

Acquistapace-Terreni conditions (AT1) and (AT2) introduced by Acquistapace and Acquistapace-Terreni in [4] (see Section 10.2). In Section 10.5 we will obtain the following result. The hypotheses (H1), (H2), (H3) and (H4) can be found in Section 10.3 and condition (K) is explained in Section 10.5.

Theorem 1.6.3. Let E be a UMD Banach space. Assume that Hypotheses (H1), (H2), (H3) and (H4) are fulfilled and that (AT1), (AT2) and (K) are satisfied for A(t) −1₂PN

n=1B 2

n− µ for all µ ∈ R large enough. If u0 ∈ D(A(0)) almost surely, then

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We apply Theorem 1.6.3 to a second order equation with time varying boundary conditions and to Zakai’s equation. Both equations are modelled in Lp-spaces with p ∈ (1, ∞). The Zakai equation will be explained in more detail in Section 1.7.2.

Before Acquistapace and Terreni wrote their paper [4], they also studied (1.6.3) in Hilbert spaces in [3]. They have assumed that (A(t))t∈[0,T ] satisfies the Kato-Tanabe

conditions [124, Section 5.3]. This gives technical problems for the deterministic prob-lem.

In [19], in the setting of martingale type 2 spaces E, Brze´zniak has studied a class of equations which is more general than (10.1.1). For E = Lp_(Rd) with 2 ≤ p < ∞, the existence of solutions with paths in L2_{(0, T ; W}2,p_(Rd_{)) has been obtained. We do}

not know whether the techniques of [34] can be extended to the setting of martingale type 2 spaces E, since this would require an extension of the Itˆo formula for the duality mapping.

Another approach was taken by Krylov [70], who developed an Lp_{-theory for a}

general class of time-dependent parabolic stochastic partial differential equations on Rd by analytic methods. The equations are non-linear and many coefficients are only assumed to be measurable. In this high generality he has been able to obtain a solution with paths in Lp(0, T ; W2,p_(Rd)) with p ∈ [2, ∞).

1.7 Applications

In this section we will give two applications of the theory developed so far. The first application will be concerned with a partial differential equation perturbed with a space-time white noise in higher dimensions. We will obtain existence and uniqueness conditions and discuss several regularity results. In the second application we consider similar questions for a special case of the Zakai equation.

1.7.1 Space-time white noise in higher dimensions

Let S ⊂ Rd _{be a bounded C}∞_{-domain and consider}

∂u

∂t(t, s) = A(t, s, D)u(t, s) + f (t, s, u(t, s)) +g(t, s, u(t, s))∂w

∂t(t, s), s ∈ S, t ∈ (0, T ],

Bj(s, D)u(t, s) = 0, s ∈ ∂S, t ∈ (0, T ], j = 1, . . . , m (1.7.1)

u(0, s) = u0(s), s ∈ S.

Here A is of the form

A(t, s, D) = X

|α|≤2m

aα(t, s)Dα,

where D = (∂1, . . . , ∂d), and for j = 1, . . . , m,

Bj(s, D) =

X

|β|≤mj

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1.7. Applications 15 where 1 ≤ mj < 2m is an integer. For the principal part of A, Aπ(t, s, D) =

P

|α|=2maα(t, s)Dα of A we assume that there is a κ > 0 such that

(−1)m+1 X

|α|=2m

aα(t, s)ξα ≥ κ|ξ|2m, t ∈ [0, T ], s ∈ S, ξ ∈ Rd.

For |α| ≤ 2m the coefficients aα are in Cµ([0, T ]; C(S)). For the coefficients of the

boundary value operator assume that for j = 1, . . . , m and |β| ≤ mj, bjβ ∈ C2m−mj(S).

and (Bj)mj=1 is a normal system of Dirichlet type, i.e. 0 ≤ mj < m (cf. [124, Section

3.7]).

The functions f, g : [0, T ] × Ω × S × R → R are jointly measurable, and adapted in the sense that for each t ∈ [0, T ], f (t, ·), g(t, ·) is Ft⊗ BS⊗ BR-measurable. Finally, w

is a spatio-temporal white noise and u0 : S × Ω → R is a BS ⊗ F0-measurable initial

value condition.

Consider the following condition:

(C) The functions f and g are locally Lipschitz in the fourth variable uniformly in [0, T ] × Ω × S, i.e. for each R > 0 the exists constants LR

f and LRg such that

|f (t, ω, s, x)−f (t, ω, s, y)| ≤ LR_f|x−y|, t ∈ [0, T ], ω ∈ Ω, s ∈ S and |x|, |y| < R, |g(t, ω, s, x)−g(t, ω, s, y)| ≤ LR_g|x−y|, t ∈ [0, T ], ω ∈ Ω, s ∈ S and |x|, |y| < R. The functions f and g are of linear growth in the fourth variable uniformly in [0, T ] × Ω × S, i.e. the exists constants Cf and Cg such that

|f (t, ω, s, x)| ≤ Cf(1 + |x|), t ∈ [0, T ], ω ∈ Ω, s ∈ S, x ∈ R,

|g(t, ω, s, x)| ≤ Cg(1 + |x|), t ∈ [0, T ], ω ∈ Ω, s ∈ S, x ∈ R.

The main theorem of this section will be formulated in the terms of the spaces H_{Bs,p

j}(S), B

s

p,1,{Bj}(S) and C

s

{Bj}(S). The definitions of these spaces can be found in

Section 8.7.

The problem (1.7.1) will be modelled as a stochastic evolution equation of the form (1.6.1) in the space E = Lp(S) with p ∈ [2, ∞). We say that (1.7.1) has a mild solution if the corresponding functional analytic model (1.6.1) has a mild solution. The next theorem will be proved in Section 8.7 and is a consequence of Theorem 1.6.1.

Theorem 1.7.1. Assume _md < 2. Assume (C) and let p ∈ [2, ∞) be such that _2mpd <

1 2 − d 4m. If a ∈ [ d 2mp, 1 2 − d

4m) is such that (8.7.2) holds for (a, p), and if for almost

all ω ∈ Ω we have u0(·, ω) ∈ B_p,1,{B2ma _j}(S), then there exists a unique mild solution

u : [0, T ] × Ω → R of (1.7.1) such that almost surely t 7→ u(t, ·) is continuous as an B_p,1,{B2ma

j}(S)-valued process.

Moreover, if for almost all ω ∈ Ω, u0(·, ω) ∈ B m−d₂

p,1,{Bj}(S), then u has paths in

Cλ_{([0, T ]; B}2mδ

p,1,{Bj}(S)) for all δ >

d

2mp and λ > 0 that satisfy δ + λ < 1 2 −

d 4m and

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By Sobolev embedding one obtains H¨older continuous solutions in time and space. For instance assume that u0 ∈ C

m−d 2

{Bj}(S). Then it follows in combination with [126,

Theorem 4.6.1] that the solution u has paths in Cλ([0, T ]; C_{B2mδ

j}(S)) for all δ, λ > 0 that satisfy δ + λ < 1 2 − d 4m.

1.7.2 Zakai’s equation

Consider the following instance of the Zakai equation. ∂u

∂t(t, x) = A(t, x, D)u(t, x) + B(x, D)u(t, x) dW (t) dt , t ∈ [0, T ], x ∈ R d u(0, x) = u0(x), x ∈ Rd. (1.7.2) Here A(t, x, D) = d X i,j=1 aij(t, x)DiDj + d X i=1 qi(t, x)Di+ r(t, x), B(x, D) = d X i=1 bi(x)Di+ c(x).

All coefficients are real-valued and we take aij, qi, r ∈ B([0, T ]; Cb1(Rd)), where B

stands for bounded. The coefficients aij, qi and r are µ-H¨older continuous in time for

some µ ∈ (0, 1], uniformly in Rd_{. Furthermore we assume that the matrices (a}

ij(t, x))i,j

are symmetric, and there exists a constant ν > 0 such that for all t ∈ [0, T ], we have

d X i,j=1 aij(t, x) − 1 2bi(x)bj(x) λiλj ≥ ν d X i=1 λ2_i_{, x ∈ R}d_{, λ ∈ R}d. (1.7.3)

For the coefficients of B we assume bi, c ∈ Cb2(Rd). Finally, W is a standard Brownian

motion.

The problem (1.7.2) will be modelled as a stochastic evolution equation of the form (1.6.3) in the space E = Lp(S) with p ∈ (1, ∞). We will say that (1.7.2) has a strong solution if the corresponding functional analytic model (1.6.3) has a strong solution. The following theorem will be proved in Section 10.8 and is a consequence of Theorem 1.6.3.

Theorem 1.7.2. Let p ∈ (1, ∞) and assume the above conditions. If u0 ∈ W2,p(Rd)

almost surely, then there exists a unique strong solution u of (1.7.2) on [0, T ] with almost all paths in Cα_{([0, T ]; E) ∩ C([0, T ]; W}2,p_(Rd_{)) for all α ∈ (0,}1

2).

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1.8. Overview 17

1.8 Overview

This thesis consists of two parts. Part I consists of Chapters 3-6 and is devoted to stochastic integration theory. Chapters 7-10 represent Part II which is concerned with stochastic evolution equations.

In Chapter 2 we will explain some preliminary results on measurability, Gaussian random variables, UMD spaces, (co)type, R-boundedness and Besov spaces. In Chap-ter 3 we will consider γ-spaces and prove Theorem 1.2.1 which is the main result in [65]. We also give a construction of the stochastic integral for operator-valued func-tions and prove several preliminary results on γ-spaces. The stochastic integration theory for operator-valued processes will be presented in Chapter 4. This is based on [96, 95]. We also prove the Itˆo formula which is taken from [97]. In Chapter 5 we consider stochastic integration and randomized UMD spaces. This chapter is based on [129]. Integration with respect to arbitrary continuous local martingales will be studied in Chapter 6 and is based on [130].

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Chapter 2 Preliminaries

2.1 Introduction

In this preliminary chapter we will recall some definitions and results that we will frequently use. In Section 2.2 we discuss several measurability concepts of functions taking values in a metric space. Section 2.3 is concerned with Gaussian random vari-ables taking values in a Banach space. Many of the results in this thesis are formulated for Banach spaces with the UMD property. This is the subject of Section 2.4. We also prove a decoupling inequality for random sums taking values in a UMD space. This inequality is fundamental in Chapter 4. In Section 2.5 we recall the Banach space properties (martingale) type and cotype. Moreover, two inequalities for vector-valued random sums are discussed. In Section 2.6 we explain the notion of R-boundedness. This randomized boundedness condition will be useful at several places. In the final Section 2.7 we define the vector-valued Besov spaces.

For some unexplained notations we refer to the List of symbols at the end of the thesis. Here the reader may also find the Index.

The scalar field of the vector spaces in this thesis will be the real numbers R, unless stated otherwise.

2.2 Measurability

Let (S, Σ) be a measurable space and let (E, d) be a metric space.

A function φ : S → E is called (Borel) measurable if for all B ∈ BE, we have

φ−1(B) ∈ Σ.

A function φ : S → E is called strongly measurable if it is the pointwise limit of a sequence of simple functions. We will often use the following measurability theorem. Proposition 2.2.1 (Pettis’s measurability theorem). Let E be a complete metric space. Let Γ be a set of real-valued continuous functions defined on E that separates points of E. For a function φ : S → E, the following assertions are equivalent:

(1) φ is strongly measurable.

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(2) φ is separably valued and measurable.

(3) φ is separably valued and for all f ∈ Γ, f ◦ φ is measurable.

(4) There exists a sequence of countably valued measurable functions (φn)n≥1 such

that φ = limn→∞φn uniformly in S.

As a consequence the pointwise limit of a sequence of strongly measurable functions is again strongly measurable.

Now assume that E is a Banach space. The Pettis measurability theorem is often applied with Γ = E∗. If φ : S → E is such that hφ, x∗i is measurable for all x∗ _{∈ E}∗_,

we say that φ is weakly measurable. In this way a function φ : S → E is strongly measurable if and only if φ is separably valued and weakly measurable.

Let E1 and E2 be Banach spaces. An operator-valued function Φ : S → B(E1, E2)

will be called E1-strongly measurable if for all x ∈ E1, the E2-valued function Φx is

strongly measurable. Observe that if Φ : S → B(E1, E2) is E1-strongly measurable

and f : S → E1 is strongly measurable, then Φf is strongly measurable.

If (S, Σ, µ) is a measure space and φ : S → E is defined as an equivalence class of functions, then we say that φ is strongly measurable if there is a version of φ which is strongly measurable.

If (S, Σ, µ) is a finite measure space, we introduce the space L0_{(S; E) of all strongly}

measurable functions φ : S → E, identifying functions which are almost everywhere equal. This space is a complete metric space under the metric

dS(φ1, φ2) =

Z

S

d(φ1(s), φ2(s)) ∧ 1 dµ(s).

For functions φ, φ1, φ2, . . . ∈ L0(S; E) we have φ = limn→∞φn if and only if φ =

limn→∞φn in measure.

For a detailed study on vector-valued functions we refer to [46, Chapter II]. In par-ticular, we will use the Bochner and Pettis integral. For the definition and properties of the conditional expectations and martingale theory we refer to [46, Chapter V]. We only comment on the following simple extension of the Doob regularization theorem for martingales to the vector-valued situation. In particular it gives a condition under which every martingale has a jointly measurable version.

Proposition 2.2.2. Let E be a Banach space and let (Ω, A, P) be a complete probability space with a filtration F = (Ft)t∈R+ that satisfies the usual conditions. If M : R+×Ω →

E is an F -martingale, then M has a cadlag version.

Proof. It suffices to consider the case that M is a martingale which is constant after some time T .

We can find a sequence of FT-simple functions (fn)n≥1 such that MT = limn→∞fn

in L1_{(Ω; E). Let (M}n₎

n≥1 be the sequence of martingales defined by Mtn = E(fn|Ft).

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2.3. Gaussian random variables 21 7.27]) that each Mn _{has a cadlag version ˜}_Mn_{. Furthermore, by [63, Proposition 7.15]}

and the contractiveness of the conditional expectation, for arbitrary δ > 0, we have δ P sup

t∈[0,T ]

k ˜M_tn− ˜M_tmk > δ_{≤ EkM}n

T − MTmk → 0 if n, m → ∞.

Hence ( ˜Mn₎

n≥1 is a Cauchy sequence in probability in the space (CL([0, T ]; E), k · k∞)

of cadlag function on [0, T ]. Since CL([0, T ]; E) is complete, ˜Mn _{is convergent to some}

˜

M in CL([0, T ]; E) in probability. For all t ∈ [0, T ] and all δ > 0, we have δP(k ˜Mt− Mtk > δ) ≤ δP k ˜Mt− ˜Mtnk > δ 2 + δPkMn t − Mtk > δ 2 ≤ δPk ˜Mt− ˜Mtnk > δ 2 + 2EkMtn− Mtk ≤ δPk ˜Mt− ˜Mtnk > δ 2 + 2Ekfn− MTk.

Since the latter converges to 0 as n tends to infinity and δ > 0 was arbitrary, it follows that ˜Mt= Mt almost surely. This proves that ˜M is the required version of M .

2.3 Gaussian random variables

Let E be a Banach space and let (Ω, A, P) be a probability space.

A mapping η : Ω → E is called a random variable if it is strongly measurable. A random variable is called Gaussian if for all x∗ ∈ E∗_{, hη, x}∗_{i is a real-valued Gaussian}

random variable.

If η is an E-valued Gaussian random variable, then a theorem of Fernique says that there exists an ε > 0 such that Eeεkηk2 < ∞ (cf. [76, Section 2.7]). In particular all p-th moments of ξ are finite.

For an E-valued Gaussian random variable ξ we define its mean m ∈ E and covariance operator Q ∈ B(E∗, E) as

m = Eξ, Qx∗ = Ehξ − m, x∗i(ξ − m).

Clearly, Q is a positive and symmetric operator. In this case we say that ξ has distribution N (m, Q). Conversely, an operator Q ∈ B(E∗, E) is called a Gaussian covariance operator if there exists an E-valued Gaussian random variable ξ such that Q is its covariance operator.

Let (γn)n≥1 be a Gaussian sequence, i.e. an i.i.d. sequence of standard real-valued

Gaussian random variables. For a sequence (xn)n≥1 in E, the a.s. convergence,

con-vergence in probability and Lp_{-convergence for some (all) p ∈ [1, ∞) of}

X

n≥1

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are equivalent and the limit is again an E-valued Gaussian random variable (cf. [76, Theorems 2.1.1 and 2.2.1]).

Let (rn)n≥1 be a Rademacher sequence, i.e. an i.i.d. sequence with

P(r1 = 1) = P(r1 = −1) =

1 2.

For N ≥ 1 and x1, . . . , xN ∈ E, recall the Kahane-Khinchine inequalities (cf. [45,

Section 11.1] or [76, Proposition 3.4.1]): for all p, q ∈ [1, ∞), we have

(E N X n=1 rnxn p )1p hp,q (E N X n=1 rnxn q )1q_.

Also for a centered E-valued Gaussian random variable ξ the Kahane-Khinchine in-equalities (cf. [76, Corollary 3.4.1]) hold: for all p, q ∈ [1, ∞), we have

(Ekξkp)1p

hp,q (Ekξkq)

1 q.

As a consequence one obtains the following convergence result for E-valued Gaussian random variables (see [117]). Let (ξn)n≥1 be a sequence of E-valued Gaussian random

variables and ξ : Ω → E is a random variable. If limn→∞ξn = ξ in probability, then ξ

is a Gaussian random variable and for all p ∈ [1, ∞), limn→∞ξn= ξ in Lp(Ω; E).

Let I be an index set. A mapping ζ : I × Ω → E is called a Gaussian process if for all n ≥ 1 and all i1, i2, . . . , in ∈ I,

(ζi1, ζi2, . . . , ζin)

is an En_{-valued Gaussian random variable. Clearly, ζ is a Gaussian process if and}

only if for all x∗ ∈ E∗_{, hζ, x}∗_{i is a Gaussian process.}

A strongly measurable process W : R+× Ω → E is called a Brownian motion if

for all x∗ ∈ E∗_{, hW, x}∗_{i is a Brownian motion starting at 0. Let Q be the covariance}

of W (1). For the process W we have (1) W (0) = 0,

(2) W has a version with continuous paths, (3) W has independent increments,

(4) For all 0 ≤ s < t < ∞, W (t) − W (s) has distribution N (0, (t − s)Q).

In this situation we say that W is a Brownian motion with covariance Q. Notice that every process W that satisfies (3) and (4) has a path-wise continuous version (cf. [63, Theorem 3.23]).

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2.4. UMD spaces and decoupling inequalities 23 the reproducing kernel Hilbert space or Cameron-Martin space (cf. [16, 37]) associated with Q and let i : HQ → E be the inclusion operator. Then ii∗ = Q and i ∈ γ(H, E).

Let (hn)n≥1 be an orthonormal basis for HQ and let (Wn)n≥1 be independent

real-valued Brownian motions. Then W (t) = P

n≥1Wn(t)ihn converges in L

2_{(Ω; E) and}

satisfies (1), (3) and (4) and there is a version which satisfies (1)-(4).

2.4 UMD spaces and decoupling inequalities

Let (Ω, A, P) be a probability space. Let (rn)n≥1 be a Rademacher sequence on Ω. Let

G0 = {∅, Ω} and Gn = σ(rk, k = 1, . . . , n). Recall that a martingale difference sequence

(dn)Nn=1 is a Paley-Walsh martingale difference sequence if it is a martingale difference

sequence with respect to the filtration (Gn)Nn=0.

A Banach space E is a UMD(p) space for p ∈ (1, ∞) if for every N ≥ 1, every martingale difference sequence (dn)Nn=1 in Lp(Ω, E) and every {−1, 1}-valued sequence

(εn)Nn=1, we have E N X n=1 εndn p1_p .E,p E N X n=1 dn p1_p . (2.4.1)

The smallest constant such that (2.4.1) holds is called the UMD(p) constant of E and is denoted by βp(E). Similarly, we say that E is a UMDPW(p)-space if one only

considers Paley-Walsh martingale difference sequences in the definition of UMD(p). The corresponding constant is denoted by βPW

p (E). In [85] it has been shown that

UMDPW(p) already implies UMD(p) and βp(E) = βpPW(E). It was shown in [26]

that if E is UMD(p) space for some p ∈ (1, ∞), then E is a UMD(p) space for all p ∈ (1, ∞). Spaces with this property will be referred to as UMD spaces. UMD stands for unconditional martingale differences. Some properties of UMD spaces are:

• If E is UMD, then every closed subspace of F ⊂ E is UMD with βp(F ) ≤ βp(E)

for all p ∈ (1, ∞).

• E is UMD if and only if E∗ _{is UMD with β}

q(E∗) ≤ βp(E) for all p, q ∈ (1, ∞)

with 1_p +1_q = 1.

• If E is UMD and F ⊂ E is a closed subspace, then the quotient space E/F is UMD with βp(E/F ) ≤ βp(E) for all p ∈ (1, ∞).

• UMD spaces are super-reflexive.

• UMD spaces have non-trivial type (see Section 2.5).

Most “classical” reflexive spaces are examples of UMD space. For instance the reflexive ranges of the Lp_{-spaces, the Hardy spaces H}p _{and the Schatten classes C}p _are

UMD spaces.

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This random variable will again be denoted by ξ. A similar extension will be used for random variables on ˜Ω. An independent copy of a random variable (or process) ξ on Ω is defined as ˜ξ(ω, ˜ω) = ξ(˜ω). Sometimes ˜ξ will be used as a random variable on ˜Ω. Expectations with respect to Ω will be denoted by E and expectation with respect to

˜

Ω will be denoted by ˜_{E. Expectations with respect to Ω ×}_{Ω will be denoted by E˜}˜ _E. We prove a general decoupling inequality. Let F = (Fn)n≥0 be a filtration on Ω.

Let (ηn)n≥1 be a real-valued sequence of centered F -adapted random variables such

that for each n ≥ 1, ηn is independent of Fn−1. Let (˜ηn)n≥1 be an independent copy of

(ηn)n≥1 and let ˜F on ˜Ω be a copy of F . Let (vn)n≥1 be an E-valued F ⊗ ˜F -predictable

sequence, i.e. for each n ≥ 1, vn is Fn−1⊗ ˜Fn−1-measurable.

Theorem 2.4.1. Let E be a UMD space and let p ∈ (1, ∞). Then for all N ≥ 1,

EE˜ N X n=1 ηnvn p hp,E EE˜ N X n=1 ˜ ηnvn p . (2.4.2)

Proof. For n = 1, . . . , N define

d2n−1 := 1₂(ηnvn+ ˜ηnvn) and d2n := 1₂(ηnvn− ˜ηnvn).

Then (dj)2Nj=1 is a martingale difference sequence with respect to the filtration (Gj)2Nj=1,

where for n ≥ 1,

G2n = σ(Fn, ˜Fn),

G2n−1 = σ(Fn−1, ˜Fn−1, ηn+ ˜ηn).

Indeed, (dn)2Nn=1 is (Gn)2Nn=1-adapted. For n = 1, . . . , N ,

EE(d˜ 2n+1|G2n) = 1₂vn+1(Eηn+1+ ˜E˜ηn+1) = 0,

since ηn+1 and ˜ηn+1 are independent of G2n and centered. For n = 1, . . . , N ,

EE(d˜ 2n|G2n−1) = 1 2vnE ˜ E(ηn− ˜ηn|G2n−1) (i) = 1 2vnE ˜ E(ηn− ˜ηn|ηn+ ˜ηn) (ii) = 0.

Here (i) follows from the independence of σ(ηn, ˜ηn) and σ(Fn−1, ˜Fn−1) (cf. [115, Section

II.41]) and in (ii) we used that ηn and ˜ηn are i.i.d. Since N X n=1 ηnvn = 2N X j=1 dj and N X n=1 ˜ ηnvn = 2N X j=1 (−1)j+1dj,

the result follows from the UMD property applied to (dj)2Nj=1 and ((−1)j+1dj)2Nj=1.

Remark 2.4.2. In the above setting one can also take (ηn)n≥1 and (˜ηn)n≥1 to be

E-valued and (vn)n≥1 to be B(E, F )-valued, where E is an arbitrary Banach space and F

is a UMD space. If one assumes that each vnis an E-strongly Fn−1⊗ ˜Fn−1-measurable,

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2.5. Type and cotype 25

2.5 Type and cotype

Let p ∈ [1, 2] and q ∈ [2, ∞]. A Banach space E is said to have type p if there exists a constant C ≥ 0 such that for all finite subsets {x1, . . . , xN} of E we have

E N X n=1 rnxn 21₂ ≤ C N X n=1 kxnkp _p1 .

The least possible constant C is called the type p constant of E and is denoted by Tp(E). A Banach space E is said to have cotype q if there exists a constant C ≥ 0

such that for all finite subsets {x1, . . . , xN} of E we have

_XN n=1 kxnkq 1_q ≤ C_E N X n=1 rnxn 21₂ ,

with the obvious modification in the case q = ∞. The least possible constant C is called the cotype q constant of E and is denoted by Cp(E).

Every Banach space has type 1 and cotype ∞ with constant 1. Therefore, we say that E has non-trivial type (non-trivial cotype) if E has type p for some p ∈ (1, 2] (cotype q for some 2 ≤ q < ∞). If a Banach space E has non-trivial type, it has non-trivial cotype. Hilbert spaces have type 2 and cotype 2 with constants 1. For p ∈ [1, ∞) the Lp_{-spaces have type min{p, 2} and cotype max{p, 2}.}

The following two results on random sums will be very useful.

Lemma 2.5.1. Let E be a Banach space. Let (rk)Nn=1 be a Rademacher sequence and

let (ξn)Nn=1 be a sequence of independent symmetric real-valued random variables with

α := min

n≤NE|ξn| ∈ (0, ∞). Then the following estimate holds for all p ∈ [1, ∞) and all

choices of (xn)Nn=1 in E: E N X n=1 rnxn p1/p ≤ 1 α E N X n=1 ξnxn p1/p . For a real-valued random variable ξ and q ∈ [1, ∞) we define

kξkq,1 =

Z

R+

P(|ξ| > t)1/qdt. Note that if s > q, kξkq,1 ≤ _s−qq (E|ξ|s)1/s.

Lemma 2.5.2. Let E be a Banach space with finite cotype q0, let p ∈ [1, ∞) and let

q = max{q0, p}. Let (rn)Nn=1 be a Rademacher sequence and let (ξn)Nn=1 be an i.i.d.

sequence of symmetric random variables. Then there exists a constant C > 0 such that for all choices of (xn)Nn=1 in E:

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In the definitions of type and cotype the rˆole of the Rademacher variables may be replaced by Gaussian variables without altering the class of spaces under consideration. For type this follows from directly from Lemma 2.5.1, but for cotype this is a delicate problem (see [87]).

The least constants arising from these equivalent definitions are called the Gaussian type p constant and the Gaussian cotype q constant of E respectively, notation T_pγ(E) and Cγ

q(E).

Every Hilbert space has both type 2 and cotype 2, and a famous result of Kwapie´n asserts that up to isomorphism this property characterizes the class of Hilbert spaces (see [74]).

Let (S, Σ, µ) be a measure space that is non-trivial in the sense that it contains at least one set of finite positive measure. If 1 < p ≤ 2 and p ≤ r < ∞, then a Banach space E has type p if and only if Lr_{(S; E) has type p (cf. [45, Theorem}

11.12]). Similarly, if 2 ≤ q < ∞ and 1 ≤ r ≤ q, then E has cotype q if and only if Lr(S; E) has cotype q. Moreover,

Tp(L2(S; E)) = Tp(E), Cq(L2(S; E)) = Cq(E). (2.5.1)

Let p ∈ [1, 2] and q ∈ [2, ∞). A Banach space E is said to have martingale type p if there exists a constant C ≥ 0 such that for every N ≥ 1 and every martingale difference sequence (dn)n≥1 in Lp(Ω; E) we have

E N X n=1 dn p ≤ Cp N X n=1 Ekdnkp.

The least possible constant C is called the martingale type p constant of E and is denoted by M Tp(E). A Banach space E is said to have martingale cotype q if there

exists a constant C ≥ 0 such that for every N ≥ 1 and every martingale difference sequence (dn)n≥1 in Lp(Ω; E) we have N X n=1 Ekdnkq ≤ CqE N X n=1 dn q .

The least possible constant C is called the martingale cotype q constant of E and is denoted by M Cq(E). Usually, the notions martingale type and cotype are introduced

for martingales not necessarily staring at 0. By an easy randomization argument as in [26, Remark 1.1] one can see this gives the same definition.

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2.6. R-Boundedness 27 In [111] it has been proved that a space E has martingale type p (martingale cotype q) if and only if there is an equivalent norm on E which is p-uniformly smooth (q-uniformly convex).

2.6 R-Boundedness

Let (rn)n≥1be a Rademacher sequence on a probability space (Ω, A, P) and let E1 and

E2 be Banach spaces. A collection T ⊂ B(E1, E2) is said to be R-bounded if there

exists a constant M ≥ 0 such that E N X n=1 rnTnxn 2 E2 1₂ ≤ M_E N X n=1 rnxn 2 E1 1₂ ,

for all N ≥ 1 and all sequences (Tn)Nn=1 in T and (xn)Nn=1 in E1. The least constant

M for which this estimate holds is called the R-bound of T , notation R(T ). By the Kahane-Khinchine inequalities, the rôle of the exponent 2 may be replaced by any exponent 1 ≤ p < ∞ (at the expense of a possibly different constant). Replacing the rôle of the Rademacher sequence by a Gaussian sequence, we obtain the related notion of γ-boundedness. Again by the Kahane-Khinchine inequalities, the rôle of the exponent 2 may be replaced by any exponent 1 ≤ p < ∞. By an easy randomization argument, every R-bounded family is γ-bounded and we have γ(T ) ≤ R(T ). If E1

has finite cotype, the notions of R-boundedness and γ-boundedness are equivalent (cf. Lemmas 2.5.1 and 2.5.2) and we have R(T ) hE1 γ(T ). If E1 and E2 are Hilbert spaces,

both notions reduce to uniform boundedness and we have γ(T ) = R(T ) = sup_{T ∈T} kT k.

2.7 Besov spaces

In this section we recall the definition of Besov spaces using the so-called Littlewood-Paley decomposition. The Fourier transform bf of a function f ∈ L1_(Rd_{; E) will be}

normalized as b f (ξ) = 1 (2π)d/2 Z Rd f (x)e−ix·ξdx, _{ξ ∈ R}d.

Let φ ∈ S (Rd_{) be a fixed Schwartz function whose Fourier transform b}_{φ is}

nonneg-ative and has support in {ξ ∈ Rd _: 1

2 ≤ |ξ| ≤ 2} and which satisfies

X

k∈Z

b

φ(2−kξ) = 1 _{for ξ ∈ R}d_{\ {0}.}

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Similar as in the real case one can define S (Rd_{; E) as the usual Schwartz space}

of rapidly decreasing E-valued smooth functions on Rd. As in the real case this is a Fr´echet space. Let the space of E-valued tempered distributions S0_(Rd; E) be defined as the continuous linear operators from S (Rd_{) into E.}

For 1 ≤ p, q ≤ ∞ and s ∈ R the Besov space Bp,qs (Rd; E) is defined as the space of

all E-valued tempered distributions f ∈S0_(Rd; E) for which kf kBs p,q(Rd;E) := 2 ks ϕk∗ f k≥0 lq_(Lp_(Rd_;E))

is finite. Endowed with this norm, B_p,qs _(Rd; E) is a Banach space, and up to an equivalent norm this space is independent of the choice of the initial function φ. The sequence (ϕk∗ f )k≥0 is called the Littlewood-Paley decomposition of f associated with

the function φ.

The following continuous inclusions hold: B_p,qs ₁_(Rd; E) ,→ B_p,qs ₂_(Rd; E), Bs1 p,q(R d ; E) ,→ Bs2 p,q(R d ; E) for all s, s1, s2 ∈ R, p, q, q1, q2 ∈ [1, ∞] with q1 ≤ q2, s2 ≤ s1. Also note that

B_p,10 _(Rd; E) ,→ Lp_(Rd; E) ,→ B_p,∞0 _(Rd; E). If 1 ≤ p, q < ∞, then Bs

p,q(Rd; E) contains the Schwartz space S (Rd; E) as a dense

subspace.

Besov spaces on domains S ⊂ Rd_{can be defined by taking restrictions (cf. [11, 126]).}

2.8 Notes and comments

The Pettis measurability theorem (Proposition 2.2.1) can be found in many books. The version we state has been proved in [128, Propositions I.1.9 and I.1.10]. The vector-valued extension of Doob’s regularization theorem in Proposition 2.2.2 might be well-known. The case of Hilbert spaces E has been considered in [72] by Kunita. His proof uses basis expansions.

For more details on vector-valued random sums and Gaussian random variables we refer to the monographs [16, 76].

The class of UMD Banach spaces has been studied by Maurey [85], Pisier [112], Aldous [8] and Burkholder [26] and many others. In [26] a geometric character-ization of UMD spaces has been obtained. UMD spaces play an important rˆole in harmonic analysis for vector-valued functions. This is due to the fact that the Hilbert transform on Lp_{(R; E) is bounded for some (for all) p ∈ (1, ∞) if and only if E is a} UMD space. The “if” part has been proved by Burkholder in [27] and the “only if” part is due to Bourgain in [17]. Details on UMD spaces can be found in [26, 29] and references given therein.

Stochastic integration in Banach spaces and applications to parabolic evolution equations

Stochastic Integration in Banach Spaces

and

Stochastic Integration in Banach Spaces

and

Applications to Parabolic Evolution Equations

Mark Christiaan VERAAR

Preface

Contents

I

Stochastic integration

31

II

Stochastic equations

121

Chapter 1

Introduction

1.1

Some history on stochastic integration

1.2

γ-Spaces

1.3

Stochastic integration in UMD spaces

1.4

Itˆ

o’s formula

1.5

Some history on stochastic evolution equations

1.6

Stochastic evolution equations in UMD spaces

1.7

Applications

1.7.1

Space-time white noise in higher dimensions

1.7.2

Zakai’s equation

1.8

Overview

Chapter 2

Preliminaries

2.1

Introduction

2.2

Measurability

2.3

Gaussian random variables

2.4

UMD spaces and decoupling inequalities

2.5

Type and cotype

2.6

R-Boundedness

2.7

Besov spaces

2.8

Notes and comments