MARTINGALE SOLUTIONS FOR STOCHASTIC EQUATION OF

arXiv:1010.5933v1 [math.PR] 28 Oct 2010

MARTINGALE SOLUTIONS FOR STOCHASTIC EQUATION OF REACTION DIFFUSION TYPE DRIVEN ´ BY LEVY NOISE OR POISSON RANDOM MEASURE ´ ZDZISLAW BRZEZNIAK AND ERIKA HAUSENBLAS Abstract. In this paper we are interested in a reaction diffusion equation driven by Poissonian noise respective Lévy noise. For this aim we first show existence of a martingale solution for an SPDE of parabolic type driven by a Poisson random measure with only continuous and bounded coefficients. This result is transferred to an parabolic SPDE driven by Lévy noise. In a second step, we show existence of a martingale solution of reaction diffusion type, also driven by Poissonian noise respective Lévy noise. Our results answer positively a long standing open question about existence of martingale solutions driven by genuine Lévy processes.

Keywords and phrases: Stochastic integral of jump type, stochastic partial differential equations, Poisson random measures, Lévy processes Reaction Diffusion Equation. AMS subject classification (2002): Primary 60H15; Secondary 60G57. 1. Introduction The main subject of our article is a stochastic reaction diffusion equation driven by a Lévy noise (or a Poissonian) noise. For instance, let L = {L(t) : t ≥ 0} be a one dimensional Lévy process whose characteristic measure ν has finite p-moment for some p ∈ (1, 2], O ⊂ Rd be a bounded domain with smooth boundary and ∆ be the Laplace operator with Dirichlet boundary conditions. A typical example of the problem we study is a the following stochastic partial differential equation (1.1)   du(t, ξ) = ∆u(t, ξ) dt + u(t, ξ) − u(t, ξ)3 dt      1  )1R\{0} (u(t, ξ)) dL(t), + sin(u(t, ξ)) sin( u(t,ξ)  u(t, ξ) = 0, ξ ∈ ∂O,       u(0, ξ) = u0 (ξ), ξ ∈ O .

Date: October 29, 2010. This work was supported by the FWF-Project P17273-N12. 1

t > 0,

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ZB AND EH, OCTOBER 29, 2010

One consequence of the main result in this paper, i.e. Theorem 5.2, ag process is that for any u0 ∈ C0 (O) there exists an C0 (O)-valued cádl´ u = {u(t) : t ≥ 0} which is a martingale solution to problem (1.1). Our main result allows treatment of equations with more general coefficients than in problem (1.1), for instance the diffusion coefficient, i.e. the function g(u) = sin(u) sin( u1 )1R\{0} (u) is just an example of a bounded and continuous function, the drift, i.e. the function f (u) = u − u3 is an example of an dissipative function f : R → R of polynomial growth and the Laplace operator ∆ is a special case of a second order, with variable coefficients, uniformly elliptic dissipative operator. Moreover, our results are applicable to equations with infinite dimensional Lévy processes as well as to more general classes of the initial data, for instance the Lq (O) spaces with q ≥ p, as well as the Sobolev spaces H0γ,q (O). The above and other examples are carefully presented in Sections 6 and 7. The approach we use in this paper follows basically a recent work [11] by one of the authours, in which a similar problem but with Wiener process was treated. The current work is a major improvement of the work [11]. Indeed, the Itô integral in martingale type 2 Banach spaces with respect to a cylindrical Wiener process on which [11] is so heavily relied, is replaced by the Itô integral in martingale type p Banach spaces with respect to Poisson random measures, see [14]. The compactness argument used in [11] depends on the Hölder continuity of the trajectories of the stochastic convolutions driven by a Wiener process. However, since the trajectories of the stochastic convolutions driven by a Lévy process are not continuous, the counterpart of the Hölder continuity, i.e the cádl´ ag property of the trajectories, seems natural to be used. Unfortunately, as many known counterexamples show, see for instance a recent monograph [57], as well as even more recent papers [17] and [12], the trajectories of the stochastic convolutions driven by a Lévy process may not even be c´ adl´ ag in the space the Lévy process lives and hence this issue has to be handled with special care. Finally, the third but not the least difference with respect to [11] is the martingale representation theorem. In [11] a known result by Dettweiler [19] was used. In the Lévy process case we have not found, to our great surprise, an appropriate martingale representation Theorem. In the current paper we proved a generalization of the result from Dettweiler [19]. Finally, instead of using stopping times as in in [11], we used interpolation methods in order to control certain norms of the solution, see Theorem C.1.

Stochastic Reaction Diffusion Equation of Poisson Type

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As we indicated in the Abstract, to the best of our knowledge, our paper answers positively a long standing open problem on the existence of solutions to stochastic reaction diffusion equations with multiplicative Lévy (or jump) noise. Parabolic SPDEs driven by additive Lévy noise were introduced by Walsh [67] and Gyöngy in [35]. Walsh, whose motivation came from neurophysiology, studied a particular example of the cable equation which describes the behaviour of voltage potentials of spatially extended neurons, see also Tuckwell [66]. Gyöngy considered stochastic equations with general Hilbert space valued semimartingales replacing the Wiener process and generalized the existence and uniqueness theorem of Krylov and B. L. Rozovskii [46]. The question we are studying in our paper are of similar type to those studied by Walsh and Tuckwell but more general as we allow more irregular and non-additive noise. However, it is of different type than studied by Gyöngy: the difference is of the similar order as between [11] and [46] in the Wiener process case. Since the early eighties many works have been written on the topic. In particular, Albeverio, Wu, Zhang [2], Applebaum and Wu [6], Bié [8], Hausenblas [38, 39], Kallianpur and Xiong [43, 44], and Knoche [45]. Some of these papers use the framework Poisson random measures while others of Lévy noise. However, all these papers typically deal with Lipschitz coefficients and/or Hilbert spaces and hence none of them is applicable to the stochastic reaction diffusion equations. There are not so many works about SPDEs driven by Lévy processes in Banach spaces. Here we can only mention the papers [38, 39] by the second authour, a very recent paper [14] by both authours and [49] by Mandrekar and R¨ udiger (who actually study ordinary stochastic differential equations in martingale type 2 Banch spaces). On the other hand, Peszat and Zabczyk in [57] formulate there results in the framework of Hilbert spaces, Zhang and R¨ ockner [59] generalized the Gyöngy [35] and Pardoux [54] results by firstly, studying evolution equations driven by both Wiener and Lévy processes and well considered globally Lipschitz (and not only linear) coefficients, and secondly studying the large deviation principle and exponential integrability of the solutions. Mytnik in [53] established existence of a weak solution for the stochastic heat equation driven by stable noise with the noise coefficients of polynomial type and without any drift term f . Our results are hence not only incomparable but also the methods of proof are different. In [52] Mueller, Mytnik

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and Stan investigated the heat equation with one-sided time independent stable noise. Martingale solution of SPDEs driven by Levy processes in Hilbert spaces are not treated quite often in the literature. Mytnik [53] constructed a weak solution to SPDEs with non Lipschitz coefficients driven by space time stable Lévy noise. Mueller [51] studied non Lipschitz SPDEs driven by nonnegative Lévy noise of index α ∈ (0, 1). Concerning nonlinear stochastic equation with Lévy processes not many paper exists. The most recent paper [21] by Dong considers Burgers equation with compound Poisson noise and thus in fact deals with deterministic Burgers equation on random intervals. Some discussion of stochastic Burgers equation with additive Lévy noise is contained in [17], where it is shown how integrability properties of trajectories of the corresponding OrnsteinUhlenbeck process play an important rˆ ole. The paper is organised as follows: In sections 2 and 3 we introduce the definitions necessary to formulate the main results. Then the main results are presented in Section 4 and Section 5. First we considered an SPDE with continuous and bounded coefficients. Here, the main Theorem is formulated int terms of Poisson random measures, i.e. Theorem 4.5, and in terms of Lévy processes, i.e. Theorem 4.7. Then, in Section 5 we consider an SPDE of Reaction Diffusion type and list the exact conditions under which we were able to show existence of a martingale solution. Two examples illustrating the applicability of our results are then present in Section 6 and Section 7. To be more precise, in Section 6 we present an SPDE of reaction diffusion type driven by a Lévy noise of spectral type, in Section 6 we present an SPDE of reaction diffusion type driven by a space time Lévy white noise. The remaining Sections and the Appendix are devoted to the proofs of our results. Notation 1.1. By N we denote the set of natural numbers, i.e. N = ¯ we denote the set N ∪ {+∞}. Whenever we speak {0, 1, 2, · · · } and by N ¯ about N (or N)-valued measurable functions we implicitly assume that that ¯ set is equipped with the trivial σ-field 2N (or 2N ). By R+ we will denote the interval [0, ∞) and by R∗ the set R \ {0}. If X is a topological space, then by B(X) we will denote the Borel σ-field on X. By λd we will denote the Lebesgue measure on (Rd , B(Rd )), by λ the Lebesgue measure on (R, B(R)). If (S, S) is a measurable space then by M (S) we denote the set of all real valued measures on (S, S), and by M(S) the σ-field on M (S) generated by


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functions iB : M (S) ∋ µ 7→ µ(B) ∈ R, B ∈ S. By M+ (S) we denote the set of all non negative measures on S, and by M(S) the σ-field on M+ (S) generated by functions iB : M+ (S) ∋ µ 7→ µ(B) ∈ R+ , B ∈ S. Finally, by MI (S) we denote the family of all N-valued measures on (S, S), and by MI (S) the σ-field on MI (S) generated by functions iB : M (S) ∋ µ 7→ ¯ B ∈ S. If (S, S) is a measurable space then we will denote by µ(B) ∈ N, S ⊗ B(R+ ) the product σ-field on S × R+ and by ν ⊗ λ the product measure of ν and the Lebesgue measure λ. 2. Stochastic Preliminaries In this paper we use the terminus of Poisson random measure to deal with Equations of type (1.1). Therefore, we introduce in the first part of this Section Poisson random measures. In the second Part of this Section we point out the relationship between Poisson random measures and Lévy processes. Let 1 < p ≤ 2 be fixed throughout the whole paper. Moreover, throughout the whole paper, we assume that (Ω, F, F, P) is a complete filtered probability space. Here, for simplicity, we denoted the filtration {Ft }t≥0 by F. Definition 2.1. (see Pisier [58]) A Banach space E is of martingale type p, iff there exists a constant C = C(E, p) such that for any E-valued discrete martingale (M0 , M1 , M2 , . . .) with M0 = 0 the following inequality holds X sup E|Mn |p ≤ C E|Mn − Mn−1 |p . n≥1

n≥1

The following definitions are presented here for the sake of completeness because the notion of time homogeneous random measure is introduced in many, not always equivalent ways. Definition 2.2. (see [40], Def. I.8.1) Let (Z, Z) be a measurable space. A Poisson random measure η on (Z, Z) over (Ω, F, F, P) is a measurable function η : (Ω, F) → (MI (Z × R+ ), MI (Z × R+ )), such that ¯ is a Poisson random (i) for each B ∈ Z ⊗ B(R+ ), η(B) := iB ◦ η : Ω → N variable with parameter1 Eη(B); (ii) η is independently scattered, i.e. if the sets Bj ∈ Z ⊗B(R+ ), j = 1, · · · , n, are disjoint, then the random variables η(Bj ), j = 1, · · · , n, are independent; ¯ (iii) for each U ∈ Z, the N-valued process (N (t, U ))t≥0 defined by N (t, U ) := η(U × (0, t]), t ≥ 0 1If Eη(B) = ∞, then obviously η(B) = ∞ a.s..

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is F-adapted and its increments are independent of the past, i.e. if t > s ≥ 0, then N (t, U ) − N (s, U ) = η(U × (s, t]) is independent of Fs . Remark 2.3. In the framework of Definition 2.2 the assignment ν : Z ∋ A 7→ E η(A × (0, 1)) defines a uniquely determined measure.

Given a complete filtered probability space (Ω, F, F, P), where F = {Ft }t≥0 denotes the filtration, the predictable random field P on Ω × R+ is the σ– field generated by all continuous F–adapted processes (see e.g. Kallenberg [42, Chapter 25]). A real valued stochastic process {x(t) : t ≥ 0}, defined on a filtered probability space (Ω, F, F, P) is called predictable, if the mapping x : Ω × R+ → R is P/B(R)-measurable. A random measure γ on Z × B(R+ ) over (Ω; F, F, P) is called predictable, iff for each U ∈ Z, the R–valued process R+ ∋ t 7→ γ(U × (0, t]) is predictable. Definition 2.4. Assume that (Z, Z) is a measurable space and ν is a nonnegative measure on (Z, Z). Assume that η is a time homogeneous Poisson random measure with intensity measure ν on (Z, Z) over (Ω, F, F, P). The compensator of η is the unique predictable random measure, denoted by γ, on Z × B(R+ ) over (Ω, F, F, P) such that for each T < ∞ and A ∈ Z with ˜ (t, A)}t∈(0,T ] defined by Eη(A × (0, T ]) < ∞, the R-valued processes {N ˜ (t, A) := η(A × (0, t]) − γ(A × (0, t]), N

0 < t ≤ T,

is a martingale on (Ω, F, F, P). Remark 2.5. Assume that η is a time homogeneous Poisson random measure with intensity ν on (S, S) over (Ω, F, F, P). It turns out that the compensator γ of η is uniquely determined and moreover γ : Z × B(R+ ) ∋ (A, I) 7→ ν(A) × λ(I). The difference between a time homogeneous Poisson random measure η and its compensator γ, i.e. η˜ = η − γ, is called a compensated Poisson random measure. The classical Itô stochastic integral has been generalised in several directions, for example in Banach spaces of martingale type p. Since it would exceed the scope of the paper, we have decided not to present a detailed introduction on this topic and restrict ourselves only to the necessary definitions. A short summary of stochastic integration in Banach spaces of


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martingale type p is given in Brze´zniak [9], Hausenblas [38], or Brze´zniak and Hausenblas [14]. We finish with the following version of the Stochastic Fubini Theorem (see [68]). Theorem 2.6. Assume that E is a Banach space of martingale type p and ξ : [a, b] × R+ × Z → E is a progressively measurable process. Then, for each T ∈ [a, b], a.s. Z b Z TZ Z TZ Z b ξ(s, r; z) η˜(dz; dr) ds = ξ(s, r; z) ds η˜(dz; dr) a

0

0

Z

Z

a

2.1. L´ evy processes and Poisson random measures. Given a Lévy process, one can construct a corresponding Poisson random measure. Vice versa, given a Poisson random measure, one gets easily a corresponding Lévy process. To illustrate this fact, let us recall firstly the definition of a Lévy process. Definition 2.7. Let E be a Banach space. A stochastic process L = {L(t) : t ≥ 0} over a probability space (Ω, F, P) is an E-valued Lévy process if the following conditions are satisfied. (i) for any choice n ∈ N and 0 ≤ t0 < t1 < · · · tn , the random variables L(t0 ), L(t1 ) − L(t0 ), . . ., X(tn ) − X(tn−1 ) are independent; (ii) L0 = 0 a.s.; (iii) For all 0 ≤ s < t, the law of L(t + s) − L(s) does not depend on s; (iv) L is stochastically continuous; (v) the trajectories of L are a.s. c´ adl´ ag on E. Let F = {Ft }t≥0 be a filtration on F. We say that L is a Lévy process over (Ω, F, F, P), if L is an E–valued and F-adapted Lévy process. The characteristic function of a Lévy process is uniquely defined and is given by the Lévy-Khinchin formula. In particular, if E is a Banach space of type p and L = {L(t) : t ≥ 0} is an E-valued Lévy process, there exist a positive operator Q : E ′ → E, a non negative measure ν concentrated on R E \ {0} such that E 1 ∧ |z|p ν(dz) < ∞, and an element m ∈ E such that (we refer e.g. to [3, 4, 23, 1]) 1 EeihL(1),xi = exp ihm, xi − hQx, xi Z 2 1 − eihy,xi + 1(−1,1) (|y|)ihy, xi ν(dy) , x ∈ E ′ . + E

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The measure ν is called characteristic measure of the Lévy process L. A Lévy process is of pure jump type iff Q = 0. Moreover, the triplet (Q, m, ν) uniquely determines the law of the Lévy process. Now, starting with an E–valued Lévy process over a filtered probability space (Ω, F, F, P) one can construct an integer valued random measure as follows. For each (B, I) ∈ B(R) × B(R+ ) let ηL (B × I) := #{s ∈ I | ∆s L ∈ B} ∈ N0 ∪ {∞}2. If E = Rd , it can be shown that ηL defined above is a time homogeneous Poisson random measure (see Theorem 19.2 [62, Chapter 4]). Vice versa, let η be a a time homogeneous Poisson random measure on a Banach space E of type p, p ∈ (0, 2]. Then the integral (Dettweiler [29]) Z tZ z η˜(dz, ds) t 7→ 0

Z

is well defined, if the intensity measure is a Lévy measure, whose definition is given below. Definition 2.8. (see Linde [23, Chapter 5.4]) Let E be a separable Banach space and let E ′ be its dual. A symmetric3 σ-finite Borel measure λ on E is called a Lévy measure if and only if (i) λ({0}) = 0, and (ii) the function4 E ′ ∋ a 7→ exp

Z

E

(coshx, ai − 1) λ(dx)

is a characteristic function of a Radon measure on E. An arbitrary σ-finite Borel measure λ on E is called a Lévy measure provided its symmetric part 21 (λ + λ− ), where λ− (A) := λ(−A), A ∈ B(E), is a Lévy measure. The class of all Lévy measures on (E, B(E)) will be denoted by L(E). Remark 2.9. (see e.g. [29]) If E is a Banach space of type p, a measure R ν ∈ M+ (E) is a Lévy measure iff ν({0}) = 0 and E |z|p ν(dz) < ∞.

2The jump process ∆X = {∆ X, 0 ≤ t < ∞} of a process X is defined by ∆ X(t) := t t

X(t) − X(t−) = X(t) − lims↑0 X(s), t > 0 and ∆0 X := 0. 3i.e. λ(A) = λ(−A) for all A ∈ B(E), 4 R As remarked in Linde [23, Chapter 5.4] we do not need to suppose that the integral (coshx, ai−1) λ(dx) is finite. However, see Corollary 5.4.2 in ibidem, if λ is a symmetric E Lévy measure, then, for each a ∈ E ′ , the integral in question is finite.


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Suppose E is a Banach space of martingale type p. Therefore, for a time homogeneous Poisson random measure η on E with intensity measure ν ∈ L(E), the process Z tZ z η˜(dz, ds), t ≥ 0, L(t) := 0

E

is an E–valued Lévy process with triplet (0, m, νˆ), where Z z ν(dz). νˆ = ν {x∈E,|x|≤1} and m = {x∈E,|x|≥1}

For more details about the connection of Banach spaces of type p and stochastic integration we refer to Dettweiler [29]. 3. Analytic Assumptions and Hypotheses Let us begin with a list of assumptions which will be frequently used throughout this and later sections. Whenever we use any of them this will be specifically written. (H1) B is an Banach space of martingale type p. (H2-a) A is a positive operator in B, i.e. a densely defined and closed operator for which there exists M > 0 such that for λ ≥ 0 k(A + λ)−1 k ≤

M , λ > 0. 1+λ

Moreover, (A + λ)−1 : B → B is assumed to be a compact operator. (H2-b) −A is a generator of an analytic semigroup e−tA t≥0 on B. (H3) There exist positive constants K and ϑ satisfying π (3.1) ϑ < 2 such that (3.2)

kAis k ≤ Keϑ|s| , s ∈ R.

Let us now formulate some consequences of the last assumption. We begin with recalling a result from [24] Lemma 3.1. Suppose that a linear operator A in a Banach space E satisfies the conditions (H2-a), (H2-b) and (H3). Then, if µ ≥ 0, the operator A + µI satisfies those conditions as well. The condition (H2-a) is satisfied ˜ such that with the same constant M and moreover there exists a constant K for each µ ≥ 0, (3.3)

˜ ϑ|s| , s ∈ R. k(µ + A)is k ≤ Ke

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Theorem 3.2. Suppose that a linear operator A in a Banach space E satisfies the conditions (H2-a), (H2-b) and (H3). Then there exists a constant ˜ such that for all µ ≥ 0 and α ∈ [0, 1], K (3.4) (3.5) (3.6)

˜ 2 (µM + 1)α , k(µ + A)α A−α k ≤ K µM α α −α 2 ˜ 2 (1 + M )α , ˜ ≤K kA (µ + A) k ≤ K 1 + 1+µ α M −α k(µ + A) k ≤ . 1+µ

Proof. Let us fix µ ≥ 0 and α ∈ [0, 1]. Consider an L(E, E)-valued function f (z), 0 ≤ Re z ≤ 1, defined by f (z) = (µ + A)z A−z . This function is continuous in the closed strip 0 ≤ Re z ≤ 1 and analytic in its interior. From the assumptions we infer that (3.7)

˜ 2 e2ϑ|s| , s ∈ R; kf (is)k ≤ k(µ + A)is kkA−is k ≤ K kf (1 + is)k ≤ k(µ + A)is kkA−is kk(µ + A)A−1 k

(3.8)

˜ 2 (µM + 1)e2ϑ|s| , s ∈ R. ≤ K

Therefore, the inequality (3.4) follows by applying the Hadamamard’s three line theorem, see e.g. Appendix to IX.4 in [60]. The proof of the other two inequalities is analogous. Lemma 3.3. Assume that −A is an infinitesimal generator of an analytic semigroup {e−tA }t≥0 on a Banach space E. Assume that for an ω ∈ (0, π2 ), M > 0 and r > 0, compare with Assumption II.6.1 in [56], (3.9)

Σ− := {z ∈ C : |Argz| ≤ π − ω} ∪ BC (0, r) ⊂ ρ(−A),

where BC (0, r) is the open ball in C centred at 0 of radius r and ρ(−A) is the resolvent set of the operator −A, and M k(A + z)−1 k ≤ (3.10) , z ∈ Σ− . 1 + |z| Then, there exists a constant δ > 0 and constants Mα > 0 for α ≥ 0, such that Mα −δt kAα e−tA k ≤ (3.11) e , t > 0; tα (3.12) kA−α (e−tA − I)k ≤ Mα tα e−δt , t > 0. Proof. For the first part see Theorem 6.13(c) in [56, Pazy]. If α ≤ 1, for the second part see Theorem 6.13(c) therein. The general case follows by induction in [α], the integer part of α.


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4. Martingale Solutions Consider the following stochastic evolution equation. (4.1)     

du(t) + Au(t) dt = F (t, u(t)) dt + u(0) = u0 .

R

Z

G(t, u(t); z) η˜(dz, dt),

t≥0

We suppose that B is a separable Banach space and −A is a generator of an analytic semigroup {e−tA }t≥0 on B. More detailed conditions on B and A are listed in assumptions (H1)-(H4) in Section 3. We shall define now a mild and, later on, a martingale solution to Problem (4.1).

Definition 4.1. Assume that p ∈ (1, 2] and B is a separable Banach space of martingale type p such that the Conditions (H1)–(H2) and (H3) from Section 3 are satisfied. Assume that (Z, Z) is a measurable space and η˜ is a compensated time homogeneous Poisson random measure on (Z, Z) over (Ω, F, F, P) with intensity measure ν. Suppose that F is a densely defined function from [0, ∞) × B to B and G is a densely defined function from R [0, ∞) × B to Lp (Z, ν, B), such that Z 1{0} (G(t, x; z))ν(dz) = 0 for all t ∈ R+ . Assume that u0 ∈ B. Assume that u is a B-valued, progressively measurable and c´ adl´ ag process such that (F (t, u(t))t≥0 and (G(t, u(t))t≥0 are well defined B-valued, resp. Lp (Z, ν, B)-valued and progressively measurable processes. Then u is called a mild solution on B to the Problem (4.1) iff for any t > 0, (4.2) Z t Z tZ p −(t−r)A |F (r, u(r))| B dr + G(r, u(r); z) ν(dz) dr < ∞ E e 0

0

Z

B

and, P-a.s., (4.3)

Z t e−(t−r)A F (r, u(r)) dr u(t) = e−tA u0 + 0 Z tZ e−(t−r)A G(r, u(r); z) η(dz, ˜ dr). + 0

Z

Definition 4.2. Assume that p ∈ (1, 2] and B is a separable Banach space of martingale type p such that the Conditions (H1)–(H2) and (H3) from Section 3 are satisfied. Let (Z, Z) be a measurable space and ν a σ–finite measure on (Z, Z). Suppose that F is a densely defined function from [0, ∞)×B in B and G is a densely defined function from [0, ∞) × B in Lp (Z, ν, B),

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R such that Z 1{0} (G(t, x; z))ν(dz) = 0 for all t ∈ R+ . Let u0 ∈ B. A martingale solution on B to the Problem (4.1) is a system (4.4)

(Ω, F, P, F, {η(t)}t≥0 , {u(t)}t≥0 )

such that (i) (Ω, F, F, P) is a complete filtered probability space with filtration F = {Ft }t≥0 , (ii) {η(t)}t≥0 is a time homogeneous Poisson random measure on (Z, B(Z)) over (Ω, F, F, P) with intensity measure ν, (iii) u = {u(t)}t≥0 is a mild solution on B to the Problem (4.1). We say that the martingale solution (4.4) to (4.1) is unique iff given another martingale solution to (4.1) Ω′ , F ′ , P′ , F′ , {η ′ (z; t)}t≥0,z∈Z , {u′ (t)}t≥0 , the laws of the processes u and u′ on the space D(R+ ; B) are equal.

In the next paragraph we will formulate assumptions for our first main result. Firstly, we begin with introducing an auxiliary Banach space E on which the functions F and G will be defined. Assumption 4.1. We assume that E is separable Banach space of martingale type p such that (i) E ֒→ B continuously and densely, (ii) E is invariant under the semigroup {e−tA }t≥0 , and the restriction of {e−tA }t≥0 to E is an analytic semigroup on E. Remark 4.3. Later on we will take E to be a real interpolation space of B (δ, p)5. Since by [9, Theorem A.7] the space D B (δ, p) is also a the form DA A B (δ, p) Banach space of martingale type p and since {e−tA }t≥0 restricted to DA B (δ, p), then the is also an analytic semigroup, it follows that, if E = DA Assumption 4.1 is satisfied. Assumption 4.2. There exists δF ∈ [0, 1) such that the map A−δF F : [0, ∞) × E → E is measurable, bounded and locally continuous with respect to the second variable, uniformly with respect to the first one. Let us denote n o RF := sup A−δF F (t, x) : (t, x) ∈ R+ × E . E

5For 0 < δ < 1, 1 ≤ p ≤ ∞, DB (δ, p) := {x ∈ B : t1−δ |Ae−tA x| ∈ Lp (0, 1)} (see e.g. ∗ A

Bergh and L¨ ofstr¨ om [7, Chapter 6.7]), where δ ∈ (0, 1] is a certain constant. If δ ∈ (1, 2], B B then DA (δ, p) = DA 2 (δ/2, p) (see Lunardi [48, Proposition 3.1.8, p. 63] ).


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Assumption 4.3. Let (Z, Z) be a measurable space and ν be a σ-finite measure on it. There exists a δG ∈ [0, p1 ) such that the function A−δG G : R+ × E → Lp (Z, ν; E) is measurable, bounded and locally continuous with respect to the second variable, uniformly with respect to the first one. Moreover, we assume that R (i) Z 1{0} (G(t, x; z))ν(dz) = 0 for all x ∈ E and t ∈ R+ ; (ii) there exists a function h : Z → B such that |g(s, x, z)|B ≤ h(z) for all R (s, x, z) ∈ R+ × B × Z and Z |h(z)|p ν(dz) < ∞; R (iii) RG := Z |A−δG G(t, x; z)|pE ν(dz) : (t, x) ∈ R+ × E is finite; (iv) for any u0 ∈ E and each ε > 0 there exists a δ > 0 such that Z |A−δG [G(t, u; z) − G(t, u0 ; z)] |pE ν(dz) ≤ ε Z

for all t ∈ R+ and all u ∈ E satisfying |u − u0 |E ≤ δ.

Remark 4.4. If the maps A−δF F (t, ·) : E → E and A−δG G(t, ·) : E → Lp (Z, ν; E) are Lipschitz continuous, uniformly with respect to t ∈ R+ , i.e. there exists a K > 0 such that for all t ∈ R+ and all u1 , u2 ∈ E, |A−δF F (t, u2 ) − F (t, u1 ) |E ≤ K|u2 − u1 |E , Z

Z

|A−δG

and p G(t, u2 ; z) − G(t, u1 ; z) |E ν(dz) ≤ K|u2 − u1 |pE ,

then the SPDEs (4.1) has a unique strong solution. In our work we are interested in the case when both these conditions are relaxed. Finally, we will impose the following assumption on the initial condition. Assumption 4.4. There exists δI
max(δG + p1 , δf − 1 + p1 , δI ). Suppose, that a function F satisfies the Assumption 4.2 and a function G satisfeis B (δ, p). the Assumption 4.3 with E = DA Then, for any u0 satisfying Assumption 4.4, there exists a martingale solution (Ω, F, P, F, {η(t)}t≥0 , {u(t)}t≥0 )

14


to Equation (4.1) on B such that Z ∞ (4.5) e−λt E|u(t)|pE dt < ∞. 0

Remark 4.6. The Assumption (H4) is not needed, since it can be compensated by taking A + λI instead of A. As we pointed out in paragraph 2.1, one can construct from a Lévy process a time homogeneous Poisson random measure on a Banach space and vice versa. Hence, Theorem 4.5 can be written in terms of a Lévy process. Let Z be a Banach space, L = {L(t) : t ≥ 0} be a Z–valued Lévy process, F and G certain mappings specified in the next paragraph. Thus we are interested in the following stochastic equation.    du(t) + Au(t) dt = F (t, u(t)) dt + G(t, u(t)) dL(t), (4.6)   u(0) = u0 . As before we need an underlying Banach space B and an auxiliary space E. Therefore, again, we assume that B be a separable Banach space and −A is a generator of an analytic semigroup {e−tA }t≥0 on B satisfying the assumptions (H1)-(H4). Furthermore, E satisfies the assumption 4.1. Now, by Theorem 4.5 one can give conditions under which a solutions exists. But before doing this, Assumption 4.3 have to be rewritten.

Assumption 4.5. Let Z be a Banach space. There exists δG ∈ [0, 1p ) such that the function A−δG G : R+ × E → L(Z, E) is measurable, bounded and locally continuous with respect to the second variable, uniformly with respect to the first one. In particular, we assume that (i) RG := |A−δG G(t, x)|L(Z,E) : (t, x) ∈ R+ × E is finite; (ii) for any u0 ∈ E and each ε > 0 there exists a δ > 0 such that |A−δG [G(t, u) − G(t, u0 )] |L(Z,E) ≤ ε for all t ∈ R+ and all u ∈ E satisfying |u − u0 |E ≤ δ. Theorem 4.7. Assume that p ∈ (1, 2] and B is a separable Banach space of martingale type p such that the Conditions (H1)–(H2) and (H4) from Section 3 are satisfied. Let Z be a Banach space of type p and L = {L(t) : 0 ≤ t < ∞} be a Z–valued Lévy process of pure jump type over a complete filtered probability space (Ω, F, F, P) with characteristic ν ∈ M(Z).


15

Let δ be a constant such that δ > max(δG + 1p , δf − 1 + 1p , δI ). Suppose, that for a function F the Assumption 4.2 and a function G the Assumption 4.5 B (δ, p) are satisfied. with E = DA Then, for any u0 satisfying Assumption 4.4, there exists a system (Ω, F, P, F, {L(t)}t≥0 , {u(t)}t≥0 ) such that (i) (Ω, F, F, P) is a complete filtered probability space with filtration F = {Ft }t≥0 ; (ii) {L(t)}t≥0 is a Z-valued Lévy process with characteristic measure ν over (Ω, F, F, P); (iii) u = {u(t)}t≥0 is a B-valued, adapted and c´ adl´ ag process such that (F (t, u(t))t≥0 and (G(t, u(t))t≥0 are well defined B-valued, resp. L(Z, B) valued and progressively measurable processes with P-a.s. Z t Z t −(t−r)A −tA e−(t−r)A G(r, u(r)) dL(t). e F (r, u(r)) dr + u(t) = e u0 + 0

0

Moreover, there exists a λ ∈ R such that it satisfies Z ∞ (4.7) e−λt E|u(t)|pE dt < ∞. 0

5. Stochastic reaction diffusion equations The next assumption uses a notion of a subdifferential of [26]. Given x, y ∈ X the map ϕ : R ∋ s 7→ |x + sy| ∈ X therefore it is right and left differentiable. Define D± |x|y right/left derivative of ϕ at 0. Then the subdifferential ∂|x| is defined by

a norm ϕ, see is convex and to be the the of |x|, x ∈ X,

∂|x| := {x∗ ∈ X ∗ : D− |x|y ≤ hy, x∗ i ≤ D+ |x|y, ∀y ∈ X} , where X ∗ is the dual space to X. One can show that not only ∂|x| is a nonempty, closed and convex set, but also ∂|x| = {x∗ ∈ X ∗ : hx, x∗ i = |x| and |x∗ | ≤ 1}. In particular, ∂|0| is the unit ball in X ∗ . Assumption 5.1. The mapping F : [0, ∞) × X → X is separately continuous with respect to both variables. Moreover, there exist a constant k ∈ R

16


and an increasing function a : R+ → R+ with limt→∞ a(t) = ∞ such that for all x ∈ D(A), y ∈ X and t ≥ 0 (5.1)

< −Ax + F (t, x + y), z >≤ a(|y|X ) − k|x|X ,

for any z ∈ x∗ = ∂|x|. Remark 5.1. Later on we will take for X an intermediate space between B and E. To show our next result, i.e. Theorem 5.2, we approximate the possible unbounded mapping F by a bounded mapping and apply Theorem 4.5. Therefore, we will introduce the following Assumption. Studying our examples you will note that this Assumption is satisfied in many interesting applications. Assumption 5.2. There exists a sequence of bounded mappings Fn : [0, ∞)× X → X such that • Fn converges pointwise to F in B • Fn satisfies the Assumption 5.1 uniformly in n. Later on we shall also need some of the following conditions. Assumption 5.3. There exist numbers k0 ≥ 0 and q ≥ p such that the Assumption (5.1) is satisfied with a function a : R+ → R+ defined by a(r) = k0 (1 + r q ),

r ≥ 0.

Assumption 5.4. (i) There exists constants M, a > 0 such that |e−tA |L(X,X) ≤ M e−at ,

t ≥ 0.

(i) The constant k from Assumption 5.1 is positive, i.e. k > 0. In the main result of this section we replace the boundedness assumption of F by the dissipativity of the drift −A + F . Thus, under the weaker hypotheses on the perturbation, we can proof the following Theorem. Theorem 5.2. Assume that p ∈ (1, 2] and B is a separable Banach space of martingale type p such that the Conditions (H1)–(H2) and (H4) from Section 3 are satisfied. Let Z be a measurable space and η˜ be a compensated time homogeneous Poisson random measure on B(Z) over a probability space (Ω, F, F, P) with intensity ν ∈ M(Z). Let δ > max(0, δF − 1 + 1p , δG + 1p , δI ), γ < 1 − p1 , and θ = 1 − pq be fixed. B (δ, p), X = [E, B] , and X := D B (γ, p). Suppose, that for a Put E = DA 0 1 θ A


17

function G the Assumption 4.3 is satisfied. If X is a separable Banach space such that X0 ֒→ X ֒→ X1 continuously, then for any function F : [0, ∞) × X → X satisfing the Assumptions 5.1–5.4 and any u0 satisfying Assumption 4.4 there exists a martingale solution (Ω, F, P, F, {η(t)}t≥0 , {u(t)}t≥0 ) on B of Problem (4.1). Moreover, this solution is q integrable in X, i.e. there exists a real number λ < ∞ such that Z ∞ e−λt E|u(t)|qX dt < ∞. 0

Remark 5.3. Similarly to Theorem 4.7, which was reformulated in terms of Lévy processes by Theorem 4.7, we can reformulate Theorem 5.2 in terms of Lévy processes. However, because it would exceed the scope of the paper we omit it. 6. The Reaction-Diffusion Equation with L´ evy Noise of Spectral Type Through the whole Section, let O be a bounded open domain in Rd , d ≥ 1, with C ∞ boundary, α > 0 and let p ∈ (1, 2] be a fixed number. Let {Li : i ∈ N} be a family of i.i.d. real valued Lévy processes with characteristic νR , where νR is a Lévy measure over R. Our aim in the section is to specify the conditions under which Theorem 5.2 covers an equation of the following type (6.1)  du(t, ξ) = ∆u(t, ξ) dt − |u(t, ξ)|q sgn(u(t, ξ)) + u(t, ξ) dt    P  −α e (ξ) dL (t),  + sin(u(t, ξ)) ∞ t > 0, ξ ∈ O,  i i i=1 i u(t, ξ) = 0, ξ ∈ ∂O, t > 0,       u(0, ξ) = u0 (ξ), ξ ∈ O,

where ∆ is the Laplacian with the Dirichlet boundary conditions, {ei : i ∈ N} are the eigenfunctions of ∆ and α is choosen, e.g. , according to d d 1 1 3 3 < α− + − + . p 2p 2 p p q First, we will describe how to reformulate the sum of Lévy processes in terms of a Poisson random measure. Then we define a more general deterministic setting.

18


6.1. The stochastic setting. Let {ei : i ∈ N} be an orthonormal basis in L2 (O) and let νR be a Lévy measure on R such that Z |z|p νR (dz) < ∞. Cν := z∈R

We define a Lévy measure ν by

ν : P(N) × B(R) ∋ (I × B) 7→

∞ X

1I (n) νR (B).

n=1

Let η be a time homogeneous Poisson random measure on N × R over a complete filtered probability space (Ω, F, F, P) with intensity ν. Let λ = ˜ = {L(t) ˜ (λi )i∈N ∈ l1 (R) be a positive sequence and define a process L :t≥ 0} by ∞ Z tZ X ˜ λi ei z η˜({i}, dz, ds) t 7→ L(t) := i=1

If λi =

i−α ,

0

R

i ∈ N, then

∞ X ˜ i−α ei Li (t) , Law L(t) = Law i=1

on

L2 (O).

6.2. The deterministic setting. Let O be a bounded open domain in Rd , d ≥ 1, with C ∞ boundary. Let d X ∂ ∂ A := (6.2) aij (x) + a0 (x) ∂xi ∂xj i,j=1

be a second order uniformly elliptic differential operator, where ai,j = aj,i for i, j = 1, . . . , d. Assume that the functions aij and a0 are of C ∞ class on ¯ In particular, suppose that there exists a C > 0 such that for all x ∈ O O. and λ ∈ Rd (6.3)

d X

aij (x)λi λj ≥ C|λ|2 ,

i,j=1

Then, hypothesis (H1), (H2) and (H3) are satisfied for all B = H0γ,p (O), γ ∈ R (see e.g. [56, Chapter 5.2]) 6.3. The equation. Let A2 be the realisation of A in L2 (O), i.e. the operator A from (6.2). Assume, for simplicity, that A2 is selfadjoint. Then, since O is bounded, A−1 2 exists and is compact. Hence, an ONB {ei , i ∈ N} of 2 L (O) consisting of the eigenvectors of the operator A2 with real eigenvalues d {ρi , i ∈ N} exists. It is known that ρi ∼ i 2 as i → ∞.


19

Let A be the operator defined by D(A) := {u ∈ H0γ+2,p (O) | Au ∈ H γ,p (O)}, (6.4) A := A. Now, we are interested in an SPDEs of the following type (6.5)  du(t) = Au(t) dt + F (u(t)) dt +    RR sin(u(t)) (i,z)∈N×R ei z η˜({i} × dz, dt),    u(t) = u0 ,

t > 0,

where F = Fβ,q : Cb0 (O) → Cb0 (O) is defined by

Fβ,q (u)(ξ) = −|u(ξ)|q sgn(u(ξ)) + βu(ξ) . Remark 6.1. Observe, Equation (6.1) is the same type as Equation (6.5) , only written in terms of Lévy processes. The following Theorem can be proved by verifying the assumptions of Theorem 5.2. Theorem 6.2. Under the conditions described above let us assume that λn = O(n−α ) for a real number α > 0 and that there exist numbers r ∈ [p, ∞) and δ ∈ R such that the inequality d 2 d 1 1 3 3 1 1 d < and δ + < α− + − + α− + r 2p 2 r p q p 2p 2 p is satisfied. Then there exists a martingale solution to Problem (6.5) in B = H0δ,r (O). Corollary 6.3. Let O ⊂ Rd be a bounded domain with smooth boundary and let ∆ be the Laplace operator. Let L = {L(t) : t ≥ 0} be a one dimensional Lévy process with characteristic measure ν such that for a p ∈ (1, 2] we have R p R |x| ν(dx) < ∞. We consider the following stochastic partial differential equation   du(t, ξ) = ∆u(t, ξ) dt − u(t, ξ)3 dt + u(t, ξ) dt       + sin(u(t, ξ)) dL(t), t > 0, (6.6)  u(t, ξ) = 0, ξ ∈ ∂O,       u(0, ξ) = u0 (ξ), ξ ∈ O,

where u0 ∈ Lq (O), q ≥ p. It follows from our main result, i.e. Theorem 5.2, that there exists an Lq (O)-valued c´ adl´ ag process u = {u(t) : 0 ≤ t < ∞} which is a martingale solution to problem (6.6). If u0 ∈ H0γ,q (O) with γ > dq ,

20


then there exists a H0γ,q (O)-valued c´ adl´ ag process u = {u(t) : 0 ≤ t < ∞} which is a martingale solution to problem (6.6). Proof of Corollary 6.3. Since the Lévy process is a one dimensional Lévy process and the function R ∋ ξ 7→ sin(ξ) is infinitely often differentiable, the α in Theorem 6.2 can be taken arbitrary large. Thus, the only restriction is given by the initial condition. Since u0 is supposed to be in Lq (O) and Lq (O) is of martingale type p, the trajectories of the solution u to Problem (6.6) are c´ adl´ ag in Lq (O). The second claim can be shown similarly. Proof of Theorem 6.2. We will show, that Theorem 5.2 is applicable. We will first analyze the stochastic term, then, secondly, the nonlinearity, and finally, as third step, we will choose the right Banach space to which the solution will belong. Part I: The stochastic perturbation: The first step is to identify the Banach space, in which the stochastic perturbation takes values. Claim 6.1. Fix r ≥ 2. If λi = i−α , i ∈ N, for γ 1 1 + − (6.7) α >1+p d 2 r then the function G : N × R ∋ (i, x) 7→ x ei ∈ Lp (Z, ν; H0γ,r (O)) satisfies ZZ ∞ Z X |G(i, z)|pH γ,r (O) λi νR (dz) < ∞. |G(i, z)|pH γ,r (O) dν(i, dz) = N×R

0

0

R

i=1

Proof of Claim 6.1. This can be proved by straightforward calculations. By interpolation, for all γ ∈ (0, 2) γ

1− γ

|u|H γ,r ≤ |u|H2 2,r |u|Lr 2 ,

u ∈ H 2,r (O).

Secondly, by a special case of the Gagliardo Nirenberg inequality, for all j ∈ N0 and r ≥ 1 with 1r = dj + 12 − m d α, there exists a constant C such that j D u r ≤ C |D m u|α2 |u|1−α u ∈ H 2,2 (O). L L2 , L 1 Hence, for any γ ∈ (0, 2), r ≥ 2 and m ≥ 2, m ∈ N, with γ ≤ d 1r + m d − 2 there exists a constant C such that γ

d +( 1 − 1r ) m

m 2 |u|H γ,r ≤ C |u|H m,2

1−

γ

|u|L2 m

d −( 21 − r1 ) m

,

u ∈ H 2,2 (O).

Since A is strongly elliptic and of second order, we know by [56, Chapter 4.10.1, Chapter 5.6.2] that for the i-th eigenfunction ei , i ∈ N, |ei |H 2,2 =


21

2

O(i− d ). Therefore, ∞ ∞ Z X X p( γ2 +( 12 − r1 ) d2 ) p λi |ei |H 2,2 λi |x ei |H γ,r νR (dx) ≤ C i=1

R

i=1

≤ C

(6.8)

∞ X

γ

1

1

λi ip( d +( 2 − r )) .

i=1

Now, since (6.7) the RHS of (6.8) is finite.

Part II: The nonlinearity: Let us consider a function f by (6.9)

f : [0, ∞) × O × R ∋ (t, ξ, u) 7→ f (t, ξ, u) ∈ R.

Obviously, f is a separately continuous real valued function and satisfies the following condition (6.10) −K(1+|u|q 1{u≥0} ) ≤ f (t, ξ, u) ≤ K(1+|u|q 1{u≤0} ), t ≥ 0, ξ ∈ O, u ∈ R, where K > 0. It follows, that if f satisfies the condition (6.10) then f (t, ξ, v+ z) sgn v ≤ K(1 + |z|q ) for all v, z ∈ R and t ≥ 0, ξ ∈ O. Therefore, a map F : X → X, where F (u), u ∈ X, is defined by (6.11)

F (u)(t, ξ) := f (t, ξ, u(ξ)),

satisfies Assumption 5.1 and 5.2 with X = sequence (fn )n∈N , where   f (t, ξ, u), fn (t, x, u) := f (t, ξ, n),   f (t, ξ, −n),

ξ ∈ O,

C0 (R)6. Approximating f by a if u ∈ [−n, n] if u ≥ n, if u ≤ −n,

we obtain a sequence (Fn )n∈N defined by

Fn : R+ × X ∋ (t, u) 7→ {O ∋ ξ 7→ fn (t, ξ, u(ξ))} ∈ X, which satisfies Assumption 5.2. Part III: The triplet of Banach spaces: To verify that there exists a solution to equation (6.5), one has to specify the underlying Banach spaces B, E and X0 . For this purpose, we choose γ0 < γ1 < γ2 and let us define auxiliarly spaces ¯ E = H0γ2 ,r (O),

¯ X0 = H0γ1 ,r (O),

6C (O) ¯ := {u ∈ C(O) ¯ : u|∂O = 0}. 0

¯ B = H0γ0 ,r (O).

22


Then, we fix r ∈ [p, ∞) and γ such that d γ < (α − 1) − d p

1 1 − 2 r

,

and put X1 = H γ,r (O). Let us note that due to the assumptions of the Theorem 5.2 the embedding X1 ֒→ B has to be continuous. Therefore, γ0 < γ has to be satisfied. Moreover, Claim 6.1 implies that there exists a δG < p1 , such that A−δG X1 ⊂ E. Hence, G satisfies the Assumption 4.3. Now, we choose the entities γ0 , γ1 and γ2 according to the following constrains. (i)

γ2 < γ + 2p ,

(ii)

(γ2 − γ0 )(1 − pq ) < 2, ,

(iii)

γ2 − γ0 > 2(δG + p1 ),

(iv)

γ1 > dr ,

(v)

γ1 = γ2 − (γ2 − γ0 )(1 − pq ).

For the time being, we assume that conditions (i) to (iv) are valid. Consequently, E, X, X0 , X1 and B satisfy the assumption of Theorem 5.2. Indeed, if 2 4 d δG + p1 . From (ii) follows that X0 ֒→ X contiuously. From (iv) follows that θ > 1 − pq . 7. The Reaction-Diffusion Equation of an arbitrary Order with Space Time L´ evy Noise In this chapter we want to apply our main result, i.e. Theorem 5.2, to a SPDEs of reaction diffusion type driven by so called space time Poissonian noise or impulsive white noise. This kind of noise is a generalisation of the space time white noise and is treated quite often in the literature, e.g. in Peszat and Zabczyk [57, Definition 7.24] or St Lubert Bié [8].


23

First, we will introduce the space time Poissonian noise, and, secondly, the deterministic part of the equation. Finally, we present our result, i.e. the exact condition under which a martingale solution of such an reaction diffusion type equation with space time Poissonian noise exists. dz, 7.1. The space time Poissonian white noise. Similar to the space time Gaussian white noise one can construct a space time Lévy white noise or space time Poissonian white noise. But before doing this, let us recall the definition of a Gaussian white noise (see e.g. Dalang [25]). Definition 7.1. Let (Ω, F, P) be a complete probability space and (S, S, σ) a measure space. A Gaussian white noise on (S, S, σ) is an F M(S) measurable mapping W : Ω → M (S) such that (i) for every A ∈ S such that σ(A) < ∞, W (A) := iA ◦ W is a real valued, Gaussian random variable with mean 0 and variance σ(A); (ii) if the sets A1 , A2 ∈ S are disjoint, then the random variables W (A1 ) and W (A2 ) are independent and W (A1 ∪ A2 ) = W (A1 ) + W (A2 ). Now having defined the Gaussian white noise, the space time Gaussian white noise can be defined as follows. Let O ⊂ Rd be a bounded domain with smooth boundary. Put S = O × [0, ∞), S = B(O) ⊗ B([0, ∞)) and σ the Lebesgue measure on E. Then, by definition, the space time Gaussian white noise is the measure valued process process {W ( · × [0, t)); t ≥ 0}. If (Ω, F, F, P) is a filtered probability space, then we say W is a space time Gaussian white noise over (Ω, F, F, P), if the measure valued process t 7→ W ( · × [0, t)) is F–adapted. Moreover, one can show that the measure valued process t 7→ W (· ×[0, t)) ˆ t )t≥0 , see generates, in a unique way, an L2 (O)-cylindrical Wiener process (W [13, Definition 4.1]. In particular, for any A ∈ O such that λ(A) < ∞, and ˆ t (1A ) = W (A × [0, t)). any t ≥ 0, W In a similar way, one can define a Lévy white noise and a space time Lévy white noise. Definition 7.2. Let (Ω, F, P) be a complete probability space, let (S, S, σ) be a measurable space and let ν ∈ L(R) (see Definition 2.8). Then a L´ evy white noise on (S, S, σ) with intensity jump size measure ν is an

24


F M(S)-measurable mapping

L : Ω → M (S)

such that (i) for all A ∈ S such that σ(A) < ∞, L(A) := iA ◦ L is a infinite divisible R-valued random variables satisfying Z iθx iθL(A) 1 − e − i sin(θx) ν(dx) ; e = exp σ(A) R

(ii) if the sets A1 , A2 ∈ S are disjoint, then the random variables L(A1 ) and L(A2 ) are independent and L(A1 ∪ A2 ) = L(A1 ) + L(A2 ). Definition 7.3. Let (Ω, F, P) be a complete probability space. Suppose O ⊂ Rd be a bounded domain with smooth boundary and put S = O × [0, ∞), S = B(O) × B([0, ∞)) and σ the Lebesgue measure. Let ν ∈ L(R). If L : Ω → M (S) is a Lévy white noise on (S, S, σ) with intensity jump size measure ν, then the measure valued process process {L(· × [0, t)); t ≥ 0}, is called a space time L´ evy noise on O with jump size characteristic ν. If (Ω, F, F, P) is a filtered probability space, then we say L is a space time L´ evy white noise over (Ω, F, F, P), if the measure valued process {L( · × [0, t)); t ≥ 0}, is F–adapted. Remark 7.4. Let L be a space time Lévy white noise. The measure valued process t 7→ L( · × [0, t)) is a weakly cylindrical process on L2 (O) (see Definition 3.2 [5]). Proposition 7.5. If L is a space time white noise, then the process t 7→ L( · × [0, t)) is an impulsive cylindrical process on L2 (O) with jump size intensity ν. The distribution of the time derivative ∂L/∂t is an impulsive white noise with the jumps size intensity ν. Proof. For all φ ∈ Cc (O)7 let us define Z φ(ξ)Z(t, dξ), Z(t, φ) :=

t ≥ 0.

O

The operator Cc (O) ∋ φ 7→ Z(t, φ) ∈ L2 (Ω, F, P; R) can be uniquely extended to L2 (O). Thus, by Definition 7.23 [57] an impulsive cylindrical process on L2 (O) with jump size intensity ν is given by the unique L2 (O)valued process such that for all φ ∈ L2 (O) Z(t, φ) = (φ, Z(t)) ,

t ≥ 0.

7C (O) denotes the set of all compactly supported continuous function. c


25

Since the set of simple functions are dense in L2 (O), it is sufficient to prove for all A ∈ B(O)

(7.1)

A) lEe−iθa Z(t,1

Z iθax 1−e − i sin(θax) ν(dx) , = exp tλ(A)

t ≥ 0.

R

But the identity (7.1) follows from the fact that Z(t, 1A ) = L(A×[0, t)) for all t ≥ 0 and from part (i) of Definition 7.2. Now, approximating an arbitrary φ ∈ L2 (O) by a sequence of simple functions, part (ii) of Definition 7.2 gives equivalence of the two Definitions. Going back, again, in the very same way, one can first define a Poissonian white noise, and then, a space time Poissonian white noise. Definition 7.6. Let (Ω, F, P) be a complete probability space, let (S, S, σ) be a measurable space and let ν ∈ L(R). Then the Poissonian white noise on (S, S, σ) with intensity jump size measure ν is a F M(MI (S × R)) measurable mapping η : Ω → M (MI (S × R)) such that (i) for all A × B ∈ S × B(R), η(A × B) := iA×B ◦ η is a Poisson distributed random variable with parameter σ(A) ν(B), provided σ(A) ν(B) < ∞; (ii) if the sets A1 × B1 ∈ S × B(R) and A2 × B2 ∈ S × B(R) are disjoint, then the random variables η(A1 × B1 ) and η(A2 × B2 ) are independent and η ((A1 × B1 ) ∪ (A2 × B2 )) = η(A1 × B1 ) + η(A2 × B2 ). Again, let O ⊂ Rd be a bounded domain with smooth boundary. Put S = O × [0, ∞), S = B(O) ⊗ B([0, ∞)) and σ the Lebesgue measure. Then, by definition, (homogeneous) the space time Poissonian white noise is the measure valued process process [0, ∞) ∋ t 7→ Π(t) ∈ MI (O × [0, ∞)) where {Π(t), t ≥ 0} is defined by (7.2)

Π(t) : B(O) × B(R) ∋ (A, B) 7→ η(A × B × [0, t)) ∈ N.

Again, to fix our notation we will call ν the jump size intensity measure of η. If (Ω, F, F, P) is a filtered probability space, then we say η is a (homogeneous) space time Poissonian white noise over (Ω, F, F, P), if the measure valued process t 7→ Π(t) defined in (7.2) is F–adapted. Compare also [57, Definition 7.2].

26


Remark 7.7. The compensator γ of a homogeneous space time Poisson (white) noise with jump size intensity ν is a measure on O × R × [0, ∞) defined by B(O) × B(R) × B([0, ∞)) ∋ (A, B, I) 7→ γ(A × B × I) = λ(A) ν(B) λ(I). Our main result is formulated in terms of Banach spaces. Therefore, in order to apply our main results to the space time Poissonian white noise, we have to construct a time homogeneous Poisson random measure taking values in a Banach space. Therefore, we will finish this section with the following Proposition. Proposition 7.8. Let O ⊂ Rd be a bounded domain with smooth boundary and ν ∈ L(R). Let η be a space time Poissonian white noise on O with jump d

−d

p ¯ Then the random measure ηZ : Ω → (O). size intensity ν. Let Z = Bp,∞ M (Z × [0, ∞)) defined by

(7.3)

B(Z) × B([0, ∞)) ∋ (B, I) 7→ Z ZZ 1B (ζ δξ ) η(dξ, dζ, dt). ηZ (B × I) := I

R×O

d

−d p ¯ with (O) is a time homogeneous Poisson random measure on Z = Bp,∞ intensity νˆ given by

B(Z) ∋ B 7→ νˆ(B) :=

Z Z R

d

−d

d− d

O

1B (ζδξ ) ν(dζ) dξ. d− d

p Proof. Since (Bp,∞ (O))′ = Bp,1 p (O), see Section 2.1.5 [61], and Bp,1 p (O) ֒→ ¯ see Section 2.2.4 [61, p. 32], the mapping R × O ∋ (ζ, ξ) 7→ ζδξ ∈ Z C0 (O), is well defined and measurable. Therefore, ηZ is P-a.s. a measure on Z = d −d p ¯ That ηZ is a time homogeneous Poisson random measure follows (O). Bp,∞

from the fact that η is a time homogeneous Poisson random measure and 1B does not dependent on the time. Therefore, straightforward calculations show that ηZ is actually a time homogeneous Poisson random measure on Z with intensity νˆ.

7.2. The deterministic setting. Let O be a bounded open domain in Rd with boundary ∂O of C ∞ class. Let A be an second order partial differential


27

operator. We assume that it is given in the following divergence form " d X ∂ ∂u(ξ) aij (ξ) + (Au)(ξ) = − ∂xi ∂ξj i,j=1 # d X ∂u(ξ) di (ξ) (7.4) + e(ξ)u(ξ) , x ∈ O , ∂ξi i=1

¯ of a bounded domain C ∞ with all coefficients of C ∞ class on the closure O domain O and the matrix [aij (x)] not necessary symmetric. We define an operator A = Aq , where q ∈ (1, ∞), in E = Lq (O) by D(A) = H02k,q (O), Au = (−1)k−1 Ak u, if u ∈ D(A). 7.3. The Equation. Fix p ∈ (1, 2] and q ≥ p. We keep the notation introduced in Section 7.2. Let L be a space time Lévy white noise with intensity jump size measure ν ∈ L(R), see Definition 7.2. Suppose that g : R × O → Lp (R, ν; R) is a bounded and continuous function. Now, the SPDEs which we are interested can be heuristically written in the following form  ∂ q   ∂t u(t, ξ) = Au(t, ξ) − |u(t, ξ)| sgn(u(t, ξ)) + b u(t, ξ)    ˙ t)] , ξ ∈ O, t > 0, + g(u(t, ξ); ξ, ·)[L(ξ, (7.5)   u(0, ξ) = u0 (ξ), ξ ∈ O,    u(t, ξ) = 0, for ξ ∈ ∂O, t > 0, where roughly speaking L˙ denotes the Radon-Nikodym derivative of the space time Lévy white noise L, i.e. ˙ t) := ∂L(ξ, t) L(ξ, ∂t ∂ξ We define the solution to the problem (7.5) in the weak sense. Definition 7.9. A weak martingale solution to equation (7.5) is a system (7.6)

(Ω, F, P, F, {L(t, ξ)}t≥0,ξ∈O , {u(t)}t≥0 )

where (i) (Ω, F, F, P) is a complete filtered probability space with filtration F = {Ft }t≥0 , (ii) {L(t, ξ)}t≥0,ξ∈O is a space time Lévy process on O with jump intensity measure ν over (Ω, F, F, P),

28


¯ (iii) u is an F–adapted stochastic process such that for any φ ∈ C ∞ (R+ ×O) the real valued process {hu, φi : t ∈ R+ } satisfy the following equation P–a.s. (7.7) Z

0

∞Z

Z

Z

∞Z

u(s, ξ)A∗ φ(s, ξ) dξ ds u0 (ξ)φ(t, ξ) dξ − u(s, ξ)φ (s, ξ) dξ ds = 0 O O Z ∞Z Z ∞Z q b u(s, ξ)φ(s, ξ) dξ ds |u(s, ξ)| sgn(u(s, ξ))φ(s, ξ) dξ ds + − 0 O 0 O Z ∞ ZZ + g(u(s, ξ); ξ, ζ) φ(ξ) L(dξ, dζ, ds). ′

O

0

R×O

The following result will be proved by applying the Theorem 5.2. Theorem 7.10. Under the conditions described in the subsection 7.3, for − d− d ,p

p ¯ there exists a martingale solution to (7.5) in B = any u0 ∈ H0 (O) −γ,r H0 (O) provided the inequalities 1 1 d 2k +4 − and γ > d − d< p q p p

are satisfied.

The same Theorem can be reformulated in terms of space time Poisson noise. But, since such a result would not differ significantly from the last one, we omit it. Proof. of Theorem 7.10: We will show that the Theorem 5.2 is applicable. As in the proof of Theorem 6.2, we have to find an appropriate quadruple of Banach spaces satisfying the assumption of Theorem 5.2. We will first analyse the stochastic term. Then, secondly, we will analyse the nonlinearity. Finally, we will choose the right Banach spaces which are used in Theorem 5.2. Part I: The stochastic term: First, note that we can replace L by a space time homogeneous Poissonian white noise η with intensity jump size measure ν. This follows from the introduction of the Section 7.1. Now, we construct a Poisson random measure on a Banach space which describes the impulsive white noise with jump size intensity ν from Definition 7.6. For d −d p ¯ and introduce a time homogeneous Poisson (O) this aim we set Z = Bp,∞ random measure ηZ on Z by defining for B ∈ B(Z) and I ∈ B(R+ ) Z ZZ 1B (x δξ ) η(dx, dξ, dt). ηZ (B × I) := I

R×O


29

By Proposition 7.8 it follows that ηZ is a time homogeneous νˆ given by Z Z 1B (ζδξ ) ν(dζ) dξ. B(Z) ∋ B 7→ νˆ(B) := R

Z

As a next point, we have to identify the operator G of Equation 4.1 acting on Lp (Z, νˆ, B). In the first step, we define a map d

−d p ¯ ¯ ×R×O ¯ ∋ (u, ζ, ξ) 7→ g(u(·), ·, ζ) δξ ∈ Bp,∞ (O). G0 : C(O)

Now, by Proposition D.1 the operator G0 can be extended to a continuous ¯ ¯ In fact, if g is bounded and continuous, operator G acting on Lp (O)×R× O. then d

(7.8)

−d p ¯ ×R×O ¯ → Bp,∞ ¯ G : Lp (O) (O)

satisfy the Assumption 4.3. Part II: The nonlinear term: Let us define a function f by setting f : [0, ∞) × O × R → R : (t, x, u) 7→ −|u|q sgn(u) + b u. It is obvious that f is separately continuous. Moreover, see Manthey and Maslowski [50], there exists a constant K > 0 such that (7.9) −K(1 + |u|q 1{u≥0} ) ≤ f (t, x, u) ≤ K(1 + |u|q 1{u≤0} ), t ≥ 0, x ∈ O, u ∈ R. It follows, that if f satisfies the condition (7.10) then f (t, x, v + z) sgn v ≤ K(1 + |z|q ) for all v, z ∈ R and t ≥ 0, x ∈ O. Therefore, as in the previous example 6, put X = C0 (R)8. Now, the map F : X → X, where F (u), u ∈ X, is defined by (compare 6.11) F (u)(t, ξ) := f (t, ξ, u(ξ)),

ξ ∈ O,

satisfies Assumption 5.1 and 5.2 on X. Part III: The quadtruple of Banach spaces: We will specify the underlying Banach spaces B, E, X0 and X1 . For this purpose, let us choose fixed γ0 < γ1 < γ2 and let us define the auxiliar spaces E = H0γ2 ,p (O),

X0 = H0γ1 ,p (O) B = H0γ0 ,p (O).

First, we fix γ such that

d γ 2(δG + p1 ),

(v)

γ1 = γ2 − (γ2 − γ0 )(1 − pq ).

(ii)

(γ2 − γ0 )(1 − pq ) < 2,

(iv)

γ1 > dr ,

For the time being, we assume that if (i) to (v) are valid. Let δI < 1 and put X1 = (I − A)−δI E. Consequently, E, X, X0 , X1 , and B satisfy the assumption of Theorem 5.2. Thus, if d
δG + p1 . From (ii) follows that X0 ֒→ X contiuously. From (iv) follows that θ > 1 − pq . 8. Some Auxiliary Results The purpose of this section is twofold. First we will summarize some results concerning the deterministic convolution process. Here we will use results already shown in [27] and [10]. Secondly, we will state several results concerning the stochastic convolution process. We begin with introducing some notation. Remark 8.1. Without loss of a semigroup of contractions on restriction, since if A satisfies such that A − νI generates a appearing in the estimate (4.5)

generality we can assume that A generates the underlying Banach space B. This is no (H2-b), then there exists a number ν > 0 semigroup of contractions. In this case λ has to be chosen at least larger than ν.

Notation 8.1. For any Banach space Y and numbers q ∈ [1, ∞), λ ∈ R, we denote by ILqλ (R+ ; Y ) the space of (equivalence classes of ) measurable


31

R∞ functions u : [0, ∞) → Y such that 0 e−λs |u(s)|qY ds < ∞. Equipped with the norm 1 Z ∞ q q −λs , u ∈ ILqλ (R+ ; Y ). e |u(s)|Y ds kukILq (R+ ;Y ) := λ

0

ILqλ (R+ ; Y

) is a Banach space, see e.g. [20]. The compact sets of ILqλ (R+ ; Y ) can be characterized by the Sobolev space Wα,p λ (R+ ; Y ) which consists of all p u ∈ ILλ (R+ ; Y ) such that Z ∞Z ∞ |u(t) − u(s)|pY e−λ(t+s) ds dt < ∞. |t − s|1+αp 0 0

The space Wα,p λ (R+ ; Y ) equipped with the norm (see also Appendix B) Z ∞ Z ∞ p1 u(t) − u(s)|pY −λ(t+s) , e kukWα,p (R+ ;Y ) := ds dt λ |t − s|1+αp 0 0

is a (separable, if Y is separable) Banach space. Furthermore, we will denote by C(R+ ; Y ) the space of all continuous functions. We equipp C(R+ ; Y ) with the topology induced on compact intervals. In particular, xn → x in C(R+ ; Y ), iff for all T > 0, 1[0,T ] xn → 1[0,T ] x in C([0, T ]; Y ). We will denote by Cbβ (R+ ; Y ) the space of all continuous and bounded functions u ∈ C(R+ ; Y ) such that for any T > 0 (8.1)

kukC β (0,T ;Y ) :=

sup |u(t)|Y +

0≤t≤T

sup 0≤s 0, 1[0,T ] xn → 1[0,T ] x in D([0, T ]; Y ). Remark 8.2. A known fact is that the Skorohod topology is weaker than the uniform topology (see e.g. [55, Lemma 6.8, p. 248]), i.e. the embedding C(R+ ; Y ) ֒→ D(R+ ; Y ) is continuous. In the part dealing with the stochastic convolution process we will need also the following notation. Notation 8.2. For any Banach space Y and numbers q ∈ [1, ∞), λ ∈ R, by N (R+ ; Y ) we denote the space of (equivalence classes) of progressivelymeasurable processes ξ : R+ × Ω → Y and by Nλq (R+ ; Y ) we denote the subspace of N (R+ ; Y ) consisting of those processes ξ for which, P-a.s. Z ∞ e−λt |ξ(t)|q dt < ∞. 0

32


By Mqλ (R+ ; Y ) we denote the Banach space consisting of those ξ ∈ N (R+ ; Y ) for which Z ∞ e−λt |ξ(t)|q dt < ∞ E 0

Lp (Ω; Y

By ) we will denote the Banach space of (equivalence classes) of measurable functions ξ : (Ω, F) → X such that E|ξ|pY < ∞}, see [20]. If X is a metric space by L0 (Ω; X) we denote the set of measurable functions from (Ω, F) to X.

8.1. The operator Λ. Similarly to Brze´zniak and G¸atarek [11] we define a linear operator A by the formula D(A) = u ∈ ILpλ (R+ ; E) : Au ∈ ILpλ (R+ ; B) , (8.2)

Au := {R+ ∋ t 7→ A(u(t)) ∈ E} ,

u ∈ D(A).

It is known, see [30], that if A + νI satisfies the conditions (H2)–(H3) then A + νI satisfies them as well. For fixed q ∈ (1, ∞) and Banach space B, Hλ1,q (R+ , E) is the Banach spaces of (classes of) functions u ∈ Lqλ (R+ , E) 1,q whose weak derivative u′ belongs to Lqλ (R+ ; E) as well. By H0,λ (R+ ; E) we

will denote the closure in Hλ1,q (R+ , E) of the space {u ∈ C ∞ (R+ ; E) : u(0) = 1,q 0}. It is known that H0,λ (R+ ; E) equals to the subspace of Hλ1,q (R+ , E) consisting of such u with u(0) = 0. With q and E as above we set (8.3)

Bu = u′ ,

u ∈ D(B),

1,q D(B) = H0,λ (R+ ; E).

Define next an operator Λ by (8.4)

Λ := B + A,

D(Λ) := D(B) ∩ D(A).

For more details we refer to [11]. If B is a UMD Banach space and A + νI, for some ν ≥ 0, satisfies the conditions (H2)–(H3) then, since Λ = B − νI + A + νI, by [30] and [34], Λ is a positive operator. In particular, Λ has a bounded inverse. The domain D(Λ) of Λ endowed with a ‘graph’ norm 1 Z ∞ Z ∞ p −λt p −λt ′ p e |Au(s)| ds e |u (s)| ds + (8.5) kuk = 0

0

is a Banach space. Before continuing, we present two results on the fractional powers of the operator ΛT , see [10] for the proof. Proposition 8.3. Assume that the conditions (H1-a), (H2) are satisfied. Assume also that for some ν ≥ 0, A + νI satisfies the condition (H3).


33

Then, for any α ∈ (0, 1], the operator Λ−α is a bounded linear operator in ILpλ (R+ ; B), and, for 0 < α < 1, (8.6) Λ−α f (t) = Z t 1 (t − s)α−1 e−(t−s)A f (s) ds, t ∈ R+ , f ∈ ILpλ (R+ ; B). Γ(α) 0 Lemma 8.4. Assume a Banach space B and a linear operator A satisfy the condition (H2). Suppose that the positive numbers α, β, δ and q > 1 satisfy 1 (8.7) 0 < β < α − + γ − δ. q If T ∈ (0, ∞), then (8.8)

Λ−α : ILq ([0, T ]; D(Aγ )) → C β (0, T ; D(Aδ )) boundedly.

If T = ∞ and the semigroup {e−tA }t≥0 is exponentially bounded on B, i.e. for some a > 0, C > 0 (8.9)

|e−tA |L(B,B) ≤ Ce−at , t ≥ 0,

then (8.10) Λ−α : ILq0 (R+ ; D(Aγ )) → Cbβ (R+ ; D(Aδ )) bounded. Finally, we present slight modifications of the Proposition 2.2 and Theorem 2.6 from [10]. Theorem 8.5. Assume that E is an UMD Banach space and a operator A satisfying condition (H2) is such that A+νI, for a ν ≥ 0, satisfies condition (H3) as well. Assume that α ∈ (0, 1] and γ, δ ≥ 0 are such that δ ≥ γ and δ − γ + 1q < 1 − α + 1r . Then the operator Aδ Λ−α A−γ : ILqλ (R+ ; E) → ILrλ (R+ ; E) is bounded. Moreover, if the operator (A + νI)−1 : E → E is compact, then the operator Aδ Λ−α A−γ : ILqλ (R+ ; E) → ILrλ (R+ ; E) is compact. Remark 8.6. In view of Theorem 8.5 Assumption 4.2 implies that for any p ∈ [1, ∞), Λ−1 : Lp (R+ ; E) → Lp (R+ ; E) is a well defined bounded linear operator. Corollary 8.7. Assume the first set of assumptions Theorem 8.5 are satisfied. Assume that three nonnegative numbers α, β, δ satisfy the following condition 1 (8.11) 0 ≤ β + δ < α − . q

34


q β δ Assume also that γ > λq . Then the operator Λ−α T : ILλ (R+ ; E) → Cγ (R+ ; D(A )) is bounded. Moreover, if the operator (A + νI)−1 : E → E is compact, then q β δ the operator Λ−α T : ILλ (R+ ; E) → Cγ (R+ ; D(A )) is also compact. In particular, if α > 1q and the operator (A + νI)−1 : E → E is compact, the map Λ−α : ILqλ (R+ ; E) → Cγ (R+ ; E) is compact. - compare with the beginning of section 8

8.2. The stochastic convolution. In this section, we will investigate the properties of the stochastic convolution operator defined, for an appropriate process ξ ∈ Nλ (R+ ; Lp (Z, ν, B)), by the formula, Z tZ −(t−s)A e ξ(s; z) η˜(dz, ds) . (8.12) S(ξ) = R+ ∋ t 7→ 0

Z

Throughout the whole subsection we assume that the underlying Banach spaces are fixed. We assume that B is a Banach space of martingale type B (δ, p)9. p. Furthermore, we assume that δ > δG + p1 and we put E = DA Let us recall Remark 4.3. First, the space E is invariant under the action B (δ, p) is also of {e−tA }t≥0 . Secondly, by [9, Theorem A.7] we know that DA a Banach space of martingale type p. Therefore, {e−tA }t≥0 restricted to B (δ, p) will also be an analytic semigroup. E = DA We begin with the following preparatory Propositions, the next follows from Theorem 2.1 [14]. Proposition 8.8. Assume that the conditions (H1) and (H2) are satisfied. Assume also that there exists a ν ≥ 0 such that A + νI satisfies the condition (H3). Assume that 0 ≤ ρ < p1 − δG . Then there exists a constant C > 0 such that for any process ξ with A−δG ξ ∈ Mpλ (R+ ; Lp (Z, ν; E)) and for any λ > 0, Z ∞ e−λt |Aρ S(ξ)(t)|p dt = kAρ SξkpMp (R ;E) E + λ 0

p

(8.13) . ≤ Cλ(δG +ρ)p−1 A−δG ξ p p Mλ (R+ ;L (Z,ν;E))

Proposition 8.9. Assume that the conditions (H1) and (H2) are satisfied. Assume also that for some ν ≥ 0, A+νI satisfies the condition (H3). Then, for any α ∈ (0, p1 − δG ), θ ∈ [0, 1) and any R > 0 the stochastic convolution operator maps any set of the form Z p p −δG p −θλt |A ξ(t; z)|E ν(dz) ≤ R ξ ∈ Mλ (R+ ; L (Z, ν; B)); sup e E 0≤t 0 and ξ ∈ Mpλ (R+ ; Lp (Z, ν; B)) 2

kSξkLp (Ω;Wα,p (R+ ;E)) ≤ C ((1 − θ) λ)(δG +α)− p λ 1 Z p p −δG −θλt ξ(t; z)|E ν(dz) |A sup e E .

(8.14)

0≤t s and ξ ∈ N (R+ ; Lp (Z, ν; B)). Similar calculations as in the Gaussian case lead to the following identity Z tZ e−(t−r)A ξ(r; z) η˜(dz; dr) (Sξ)(t) − (Sξ)(s) = 0 Z Z tZ Z sZ −(t−r)A e−(t−r)A ξ(r; z) η˜(dz; dr) e ξ(r; z) η˜(dz; dr) = − s Z 0 Z Z sZ + e−(t−s)A − I e−(t−r)A ξ(r; z) η˜(dz; dr) 0 Z Z tZ e−(t−r)A ξ(r; z) η˜(dz; dr) + e−(t−s)A − I (Sξ)(s). = s

Z

It is enough to show that each term on the RHS of the above equality satisfies the inequality (8.14). Indeed, by employing the Fubini Theorem 2.6 we infer that Z ∞Z ∞ |Sξ(t) − Sξu(s)|pE ds dt ≤ e−λ(t+s) E |t − s|1+αp 0 0 Z ∞Z t p p −λ(t+s) E |(Sξ)(t) − (Sξ)(s)|E = 2 e ds dt |t − s|1+αp 0 0 p R R t Z ∞Z t E s Z e−(t−r)A ξ(r; z) η˜(dz; dr) p −λ(t+s) E ≤ 2 e ds dt 1+αp |t − s| 0 0 −γ −(t−s)A p Z ∞Z t A e − I L(E) E |Aγ (Sξ)(s)|pE ds dt + 2p e−λ(t+s) |t − s|1+αp 0 0 (8.15) =: 2p (S1 + S2 ) .

By [14, Theorem 2.1] we get the following estimate of the term S1 . p R t R −(t−r)A Z ∞Z t e ν(dz) dr ξ(r; z) E s Z E ds dt S1 ≤ e−λ(t+s) |t − s|1+αp 0 0 Z |A−δG ξ(t; z)|pE ν(dz) ≤ sup e−θλt E 0≤t 0. A L(E)

Then, the Young inequality for convolutions implies for ρ > 0 with α < ρ < 1 p − δG −ρ −(t−s)A p Z ∞Z ∞ A e − I L(E,E) E |Aρ (Sξ)(s)|pE S2 ≤ e−λt ds dt |t − s|1+αp 0 0 Z ∞ Z ∞ −λt (ρ−α)p−1 e−λt E |Aρ (Sξ)(t)|pE dt e |t| dt ≤ C 0 0 Z ∞ e−λt E |Aρ (Sξ)(t)|pE dt. ≤ C1 λ(α−ρ)p × 0

Thus, by Proposition 8.8 we infer that

p

S2 ≤ C λ(δG +ρ)p−1 A−δG ξ

Mpλ (R+ ;Lp (Z,ν;E))

Now, by

−δG p ξ

A

−1 −1

≤ (1 − θ)

Mpλ (R+ ;Lp (Z,ν;E))

−θλt

λ

sup e

E

0≤t δG + 1p , then there exists a constant C > 0 such that Z t Z −(t−s)A −λt −δ ≤ (t2 − t1 )γ Cξ (t1 , t2 ), e ξ(s; z)˜ η (ds, dz) E sup e A t1 ≤t≤t2

Z

t1

E

where

(8.17) Cξ (t1 , t2 ) := C

Z

t2

t1

Z

Z

p1 p −δG −λs e ξ(s; z) ν(dz) ds E A . E

Adding up Proposition 8.8 to Proposition 8.10 gives the following Lemma.


37

Lemma 8.11. Assume that the conditions (H1) and (H2) are satisfied. Assume also that there exists a ν ≥ 0 such that A+ νI satisfies the condition (H3). Assume that q ∈ [p, ∞) and θ = 1 − pq and λ > 0. Put X = [B, E]θ (complex interpolation space). Then, there exists a constant C < ∞ such that for all λ0 ∈ (0, λ) for all ξ ∈ Mpλ0 (R+ ; Lp (Z, ν; B)) one has (δ +α)− 2

p kSξkLr (Ω;Lq (R+ ;X)) ≤ C (λ0 ) G λ 1 Z p p −δG −λ0 t ξ(t; z)|E ν(dz) |A sup e E .

0≤t 0 and 0 < λ0 < λ there exists a number r > 0 such that the stochastic convolution operator S maps subsets of Mpλ0 (R+ ; Lp (Z, ν; B)) of the following form Z p p p −λ0 t −δG e E|A ξ ∈ Mλ0 (R+ ; L (Z, ν; B)) : sup ξ(t; z)|E ν(dz) ≤ R 0≤t λ0 > 0. First, let us recall that due to Remark C.2 following inequality holds. kxkILq (R+ ;(E,B)[θ] ) ≤ kxk1−θ kxkθIL∞ (R+ ;B) , ILp (R+ ;E) λ

λ

x ∈ ILpλ (R+ ; E)∩IL∞ (R+ ; B).

Hence, if rp′ (1 − θ) = p, rp′′ θ = 1 and p1′ + p1′′ = 1, by the Hölder and Young inequality, for all v ∈ Nλq (R+ ; E) ∩ N ∞ (R+ ; B) 1′′ 1′ ′′ θ p p rp′ (1−θ) E kvkrILq (R+ ;(E,B) ) ≤ × E |v|rp E |v|ILp (R ;E) ∞ IL (R+ ;B) [θ]

λ

λ

≤

(8.18)

+

′′ θ 1 1 rp′ (1−θ) E |v|ILp (R ;E) + ′′ E |v|rp ∞ (R ;B) . I L ′ + + λ p p

By Proposition 8.8 and by Proposition 8.10 there exists a C < ∞ such that 1 p E |S(ξ)|pILp (R ;E) + E |S(ξ)|pIL∞ (R+ ;B) ≤ + λ 1 Z ∞ Z p p −λ0 s −δG , ξ ∈ Mpλ0 (R+ ; Lp (Z, ν; B)) ξ(s; z) ν(dz) ds e E A 0

Z

Hence, by (8.18), if r is as above, then for all ξ ∈ Mpλ0 (R+ ; Lp (Z, ν; B)) 1 r E |S(ξ)|rILq (R+ ;(E,B) ) ≤ [θ] λ 1 Z ∞ Z p p −λ0 s −δG . ξ(s; z) ν(dz) ds e E A 0

Z

38


This concludes the proof of the first point of the Lemma. Before proofing the tightness of the laws of the set {S(ξ); ξ ∈ A}. Let us recall that by Theorem B.2 for any E0 ֒→ E compactly and λ > 0 q X := Lpλ,ρ (R+ ; E0 ) ∩ Wα,p λ (R+ ; E) ֒→ ILλ (R+ ; E)

compactly. By Corollary C.5 it follows that [X, IL∞ (R+ ; B)]θ ֒→ ILqλ (R+ ; E) , IL∞ (R+ ; B) θ

compactly. Therefore, by applying Propositions 8.8, 8.9 and 8.10 we infer that there exists a C < ∞ such that 1 r ≤ E |Sξ|r[Wα,p ∞ γ (R+ ;E),IL (R+ ;B)] θ

1 Z p p −δG −λ0 t , C sup e ξ(t; z) ν(dz) A 0≤t 0 such that for any process ξ ∈ Mpλ (R+ ; Lp (Z, ν; B)) with A−δG ξ ∈ Mpλ (R+ ; Lp (Z, ν; E)) one has

p

p

. sup e−λt E A−δG S(ξ)(t) ≤ C A−δG ξ p p E

0≤t 0. As in the previous proof, by the Burkholder inequality the following sequence of calculation are verified. Z t Z p −δG −(t−s)A E e ξ(s; z) η˜(ds; ds) ≤ A 0 Z E Z tZ p −δG −(t−s)A e ξ(s; z) ν(dz) ds E A ≤C E 0 Z Z tZ p ≤ C eλt E A−δG e−λs ξ(s; z) ν(dz) ds. 0

E

Z

In this part of the section, we will investigate the properties of the stochastic convolution term with respect to G. For this purpose let us put Z tZ (t−s)A e G(s; u(s); z) η˜(dz, ds) . (8.19) G(u) = R+ ∋ t 7→ 0

Z


39

Note that G(u) = S(ξ), where and ξ(s; z) = G(s; u(s); z), (s, z) ∈ R+ × Z, and S(ξ) is defined in formula (8.12) on page 34. Thus, in a non-rigorous way, the map G is a composition of the map G with the stochastic convolution operator S. In the next lines we will state three results, which are important for the proof of Theorem 4.5 and Theorem 5.2. These results are consequences of the Propositions 8.8 to 8.10. Corollary 8.13. Assume that the conditions (H1), (H2) are satisfied. Assume also that for a ν ≥ 0, A+νI satisfies the condition (H3). If a function G : [0, ∞) × E → Lp (Z, ν; B) satisfies the Assumption 4.3, then (i) there exists a constant C < ∞ such that for all λ > 0 1

kAρ GukMp (R+ ;E) ≤ C RGp λ

δG +ρ− p2

λ

,

u ∈ Nλp (R+ ; E) .

(ii) Fix λ > 0, α ∈ (0, 1p − δG ) and 0 < λ0 < λ. Then, the map G : Nλp0 (R+ ; E) → Lp (Ω; Wα,p λ (R+ ; E)) is bounded. To be precise, there exists a constant C < ∞ such that 1

(δG +α)− p2

kG(u)kLp (Ω;Wα,p (R+ ;E)) ≤ C RGp λ0 λ,0

,

u ∈ Nλp0 (R+ ; E) .

(iii) there exists a constant C such that for all λ > 0 1

E sup e−λt |(Gu)(t)|B ≤ C (1 + T δ−δG ) RGp , 0≤t≤T

u ∈ Nλp (R+ ; E).

Proof. In order to show (i), (ii) and (iii) we put ξ(t, z) = G(t, u(t), z) for (t, z) ∈ R+ × Z in Proposition 8.8, Proposition 8.9 and Proposition 8.10. p R Then by the Assumption 4.3, E Z A−δG ξ(s; z) ν(dz) ≤ RG and therefore kAρ GukMp (R+ ;E) ≤ C2 RG λ(δG +ρ)p−2 . λ

This concludes the proof of (i), (ii) and (iii).

The next Corollary, although it is a simple consequence of the previous one, is important as a result in itself. Corollary 8.14. Assume that the conditions (H1), (H2) are satisfied. Assume also that for a ν ≥ 0, A + νI satisfies the condition (H3). Assume that q ∈ [p, ∞) and θ = 1 − pq and put X to be the complex interpolation space (E, B)[θ] . If a function G : [0, ∞) × E → Lp (Z, ν; B) satisfies the Assumption 4.3, then for all λ > ν the laws of the family {G(u) : u ∈ Nλp (R+ ; E)}, are tight on ILqλ (R+ ; X). Moreover, there exists

40


an r ∈ (0, p) such that (8.20)

sup

u∈Nλp (R+ ;E)

E kG(u)krILq (R+ ;X) < ∞. λ

Proof. Again, putting ξ(s, z) := G(s, u(s), z) on [0, ∞) × Z, and applying Lemma 8.11 give the assertion. By Proposition 8.10 and Proposition 8.10 we will show in the next Corollary, that the stochastic convolution term belongs to the Skorohod space D(R+ ; B). Corollary 8.15. Assume that the conditions (H1), (H2) are satisfied. Assume also that for a ν ≥ 0, A + νI satisfies the condition (H3). If a function G : [0, ∞) × E → Lp (Z, ν; B) satisfies the Assumption 4.3, then for all δ > δG + p1 the laws of the family Gu | u ∈ Nλp (R+ ; E)

are tight on D(R+ ; B).

Proof of Corollary 8.15: Corollary 8.15 can be shown by Corollary C.10 [?]. Putting ξ(s, z) := G(s, u(s), z), (s, z) ∈ [0, ∞) × Z, by Assumption 4.3 there exists a C > 0 and a r > 0 such that !r qG Z Z ∞ p C r p −ρG −λt ds ≤ RGp , u ∈ Nλp (R+ ; E). |A G(s, u(s); z)|E ν(dz) e E λ Z 0 Therefore, the process ξ satisfies the assumptions of Proposition 8.10, hence it satisfies Assumption (i) of Corollary Corollary C.10 [?]. Similarly, the process ξ satisfies the assumptions of Proposition 8.10, hence it satisfies Assumption (ii) of Corollary Corollary C.10 [?]. Therefore, Corollary 8.15 is shown. 9. Proof of Theorem 4.5 In this Section we will use the notation introduced in Theorem 4.5 and within Section 8. In particular, throughout this Section, p ∈ (1, 2], δI , δG and δF are fixed numbers, B is a Banach space of martingale type p, B (δ, p). Due to Remark 4.3 it δ > max(0, δF − 1 + p1 , δG + p1 , δI ), and E = DA is sufficient to assume that B is of martingale type p. Moreover, we will use the notation given in Paragraph 8.1 and Paragraph 8.2 on pages 30 and 31. Now, we will begin the proof of Theorem 4.5 by first defining a certain sequence of processes. Consider a sequence {xn } ⊂ E such that A−δI xn →


41

A−δI x in E as n → ∞. Define a function φn : [0, ∞) → [0, ∞) by φn (s) = 2kn , −n [2n s], s ≥ 0, where [t] is the if k ∈ N and 2kn ≤ s < k+1 2n , i.e. φn (s) = 2 integer part of t ≥ 0. Let us define a sequence {un } of adapted E–valued processes by

Z

un (t) = e (9.1)

t

e−(t−s)A F (s, uˆn (s)) ds xn + 0 Z tZ e−(t−s)A G(s, uˆn (s); z) η˜(dz; ds), + −tA

0

t ≥ 0,

Z

where u ˆn is defined by (9.2)

( xn , R φ (s) u ˆn (s) := 2n φnn(s)−2−n un (r) dr,

if s ∈ [0, 2−n ), if s ≥ 2−n .

Note, that u ˆ is a progressively measurable, piecewise constant, E-valued process. Between the grid points, Equation (9.1) is linear, therefore, un is well defined for all n ∈ N. Secondly, we define a sequence of Poisson random measures {ηn | n ∈ N} by putting ηn = η for all n ∈ N. Note, that, since MN (Z ×R+ ) is a separable metric space, by Theorem 3.2 of [55] the laws of the family {ηn , n ∈ N} are tight on MN (Z × R+ ). The proof of Theorem 4.5 will be divided into several steps. The first two steps are the following. Step (I) The laws of the family {un , n ∈ N} are tight on ILpλ (R+ ; E); Step (II) The laws of the family {un , n ∈ N} are tight on D(R+ ; B). We postponed the proofs of Step (I) and Step (II) at the end of this chapter and suppose for the time being that the proofs of those two steps have been accomplished. Hence, there exists a subsequence of {(un , ηn ), n ∈ N}, denoted by {(un , ηn ), n ∈ N}, and there exists a Borel probability measure µ∗ on D(R+ ; B) ∩ ILpλ (R+ ; E) × MN (Z × R+ ) such that L(un , ηn ) → µ∗ weakly. By the modified version of the Skorohod embedding Theorem, see Theo¯ F¯ , P) ¯ and ILp (R+ ; E)∩D(R+ ; B)× rem E.1, there exists a probability space (Ω, λ MN (Z × R+ )-valued random variables (¯ u1 , η¯1 ), (¯ u2 , η¯2 ), . . ., having the same law as the random variables (u1 , η1 ), (u2 , η2 ), . . ., and a ILpλ (R+ ; E) ∩ ¯ with ¯ F¯ , P) D(R+ ; B) × MN (Z × R+ )-valued random variable (u∗ , η∗ ) on (Ω,

42


L((u∗ , η∗ )) = µ∗ such that P a.s. (9.3)

(¯ un , η¯n ) −→ (u∗ , η∗ ) in ILpλ (R+ ; E) ∩ D(R+ ; B) × MN (Z × R+ ),

and η¯n = η∗ for all n ∈ N. Later on we will need the following fact. Step (III) The following holds (i) supn∈N k¯ un kMp (R+ ;E) < ∞ and (ii) k¯ un − u∗ kMp (R+ ;E) → 0 as n → ∞. Again, suppose for the time being that the proof of Step (III) has been ¯ = (F¯t )t≥0 be the filtration generated by the Poisson accomplished. Let F random measure η∗ , the processes {¯ un , n ∈ N} and the process u∗ . The next two Steps imply that the following two integrals over the filtered probability ¯ F, ¯ F, ¯ P) ¯ space (Ω, Z tZ

e−(t−s)A G(s, u¯n (s), z)) η˜¯n (dz, ds),

t ≥ 0,

Z tZ

e−(t−s)A G(s, u∗ (s), z)) η˜∗ (dz, ds),

t ≥ 0,

0

Z

and

0

Z

exist. Step (IV) The following holds (i) for every n ∈ N, η¯n is a time homogeneous Poisson random mea¯ F¯ , F, ¯ P) ¯ with intensity measure ν; sure on B(Z) × B(R+ ) over (Ω, (ii) η∗ is a time homogeneous Poisson random measure on B(Z) × ¯ P) ¯ with intensity measure ν; ¯ F, ¯ F, B(R+ ) over (Ω, Step (V) The following holds ¯ (i) for all n ∈ N, u ¯n is a F-progressively measurable process; ¯ (ii) the process u∗ is a F-progressively measurable process. Again, let us assume, for the time being, that the Steps (IV) and (V) have been established. For clarity, we will now introduce an operator K. ¯ F¯ , F, ¯ P) ¯ Let µ be a time homogeneous Poisson random measure over (Ω, ¯ P), ¯ such that ¯ F¯ , F, with intensity measure ν and let v be a process over (Ω,


43

A−δG v ∈ Nλp (R+ ; E) and let x ∈ B. Then we put Z t −tA e−(t−s)A F (s, v(s)) ds (K(x, v, µ))(t) := e x+ 0 Z tZ (9.4) e−(t−s)A G(s, v(s); z))˜ µ (dz, ds), t ≥ 0. + 0

Z

Here, as usual, µ ˜ denotes the compensated Poisson random measure of µ. Step (VI) k¯ un − K(xn , u ¯n , η¯n )kMp (R+ ;E) → 0 as n → ∞. λ

Step (VII) kK(xn , u ¯n , η¯n ) − K(x, u∗ , η∗ )kMp (R+ ;E) → 0, as n → ∞. λ

Again, let us assume, for the time being, that the Steps (VI) and (VII) have been established. Then by Steps (IV) to (VII) we infer that u ¯ n → u∗ , p u ¯n − K(xn , u ¯n , η¯n ) → 0 in Mλ (R+ ; E) and K(xn , u¯n , η¯n ) − K(x, u∗ , η∗ ) → 0 p in Mλ (R+ ; E). By uniqueness of the limit, we infer that u∗ = K(x, u∗ , η∗ ) ¯ in Mpλ (R+ ; E), which implies that P-a.s. u∗ = K(x, u∗ , η∗ ). It remains to ¯ show that P-a.s. u∗ (t) = K(x, u∗ , η∗ )(t),

∀t ∈ R+ ,

which will be shown in the last step. Step (VIII) P a.s. for all t ≥ 0 we have u∗ (t) = K(x, u∗ , η∗ )(t). Since the solution to (4.1) is a fix point of the operator K, it follows from Step (VIII) that u∗ is a martingale solution to Equation (4.1). The assertion (4.5) follows from Step (III). Therefore, in order to proof Theorem 4.5 we have to verify Step (I) to Step (VIII), which will be done in the following. Nevertheless, before, let us state the following two auxiliary results. Proposition 9.1. For n ∈ N, let gn be defined by gn (t) := e−tA xn ,

t ≥ 0.

Then (i) The family {gn , n ∈ N} is precompact in ILpλ (R+ ; E). (ii) The family {gn , n ∈ N} is precompact in C(R+ ; B). Proof of Proposition 9.1. Theorem 5.2-(d) of [56] implies that for any t0 > 0 the family {gn , n ∈ N} is precompact in ILpλ ([t0 , ∞); E) and C([t0 , ∞); B). Therefore, the critical point is at t = 0. Since A−δI xn → A−δI x in E and

44


δI < 1, the Young inequality for stochastic convolutions implies (i). Since δ > δI , xn → x ∈ B in B implies (ii). ˆn , n ∈ N} is Proposition 9.2. (i) For any γ ∈ (0, p1 − δG ) the set {Aγ u p bounded in Mλ (R+ ; E); (ii) There exists θ > 0 and C < ∞ such that kˆ un − un kMp (R+ ;E) ≤ C 2−θn , λ

n ∈ N.

Proof of Proposition 9.2. First, we will show part (i). First note that, since Law(ˆ u) = Law(un ) on ILpλ (R+ ; E), E kAγ u ˆn kpILp (R λ

kA

=

un kpILp (R ;E) + λ ∞ Z sn X k+1 k=1

≤

λ

k 2n ,

Let us put snk := γ

= E kAγ un kpILp (R

+ ;E)

=

Z

sn k+1

∞ Z X

sn k+1

∞

≤

k=1 ∞

≤

Z

0

sn k−1

λ

+ ;E)

,

n ∈ N.

γ

|A

0

e−λs e−λs

un (s)|pE

ds =

∞ Z X k=1

sn k+1

sn k

e−λs |Aγ un (s)|pE ds

p Z n Z sn sk k+1 n e−λs |Aγ xn |pE ds Aγ u(r) dr ds + 2 n n sk sk−1 E !p Z sn k |Aγ un (r)|E dr ds + C 2−n(1−(γ+δI )p) |A−δI xn |pE 2np sn k−1

sn k

Z

sn k−1

sn k

k=1 ∞ Z sn X k

−λs

e

n k=1 sk

≤

= kAγ un kpMp (R

k, n ∈ N. Then

sn k

∞ Z X

+ ;E)

"Z

e−λ(s−r) 2n e−λr |Aγ un (r)|pE dr ds + C 2−n(1−(γ+δI )p) |A−δI xn |pE

sn k+1

sn k

n −λ(s−r)

2 e

#

ds e−λr |Aγ un (r)|pE dr + C 2−n(1−(γ+δI )p) |A−δI xn |pE

e−λr |Aγ un (r)|pE dr + C 2−n(1−(γ+δI )p) |A−δI xn |pE

= kAγ un kpILp (R λ

+ ;E)

+ C 2−n(1−(γ+δI )p) |A−δI xn |pE .

Since, by Corollary 8.13-(i) and Remark 8.6, the set {Aγ un , n ∈ N} is bounded in Mpλ (R+ ; E)), the assertion (i) follows by Assumption 4.4.


45

To prove Part (ii), let p′ be the conjugate exponent to p, i.e. p1 + p1′ = 1. Let us put ρ = α + p1 . Then by the Hölder inequality p Z Z ∞ Z ∞ sn (un (s) − u ¯n (r)) dr ds e−λs |un (s) − u ˆn (s)|p ds ≤ 2np e−λs sn−1 0 0 ! ! p−1 Z sn Z sn Z ∞ |un (s) − un (r)|p np ρp′ −λs ≤ 2 |s − r| dr dr ds e |s − r|pρ sn−1 sn−1 0 ! Z sn Z ∞ p ρp+p−1 |u (s) − u (r)| n n ≤ 2np e−λs dr 2−n ds pρ |s − r| sn−1 0 −npα α = 2 ≤ 2n(p−ρp−p+1) |un |Wp,λ |un |Wp,αλ¯ .

By applying Corollary 8.13-(ii) we can conclude the part (ii).

9.1. Proof of Step (I). Set (9.5) (9.6)

fn (t) = F (t, u ˆn (t)), gn (t; z) = G(s, uˆn (t); z),

t ∈ R+ , t ∈ R+ ,

z∈Z

and (9.7)

vn (t) =

Z tZ 0

e−(t−s)A G(s, uˆn (s); z) η˜(dz; ds),

t ∈ R+ .

Z

By Proposition 9.2-(i), the family {ˆ un , n ∈ N} is bounded in Mpλ (R+ ; E). Therefore, in view of Assumption 4.2, we infer that the family {A−δF fn | n ∈ N} is bounded in ILpλ (R+ ; E). Hence, by Theorem 2.6 of [11], the laws of the family {Λ−1 fn | n ∈ N} are tight on ILpλ (R+ ; E). Again, by Proposition 9.2-(i), the family {ˆ un , n ∈ N} is bounded in p Mλ (R+ ; E). Hence, by Corollary 8.13, we infer that the laws of the family {vn | n ∈ N} are tight on ILp (R+ ; E). Finally, from Proposition 9.1 follows that the family of functions {e−·A xn , n ∈ N} is precompact in ILpλ (R+ ; E). Since un = vn + Λ−1 fn + e−·A xn ,

n ∈ N,

we conclude that the laws of the family {un | n ∈ N} are tight on ILpλ (R+ ; E). 9.2. Proof of Step (II). Again, we use the notation introduced in fromulae (9.5), (9.6) and (9.7). By Proposition 9.2-(i), the family {ˆ un , n ∈ N} is bounded in Mpλ (R+ ; E), and, by Assumption 4.2, we infer that the family {A−δF fn | n ∈ N} is bounded in Mpλ (R+ ; E). Now, for any δ > max(0, δF − 1 + p1 ), Corollary 8.7 implies that the laws of the family of {Λ−α A−δ Λ1−α fn | n ∈ N} are tight

46


on C(R+ ; E). Since the linear operators A and Λ commute, we infer that the laws of the family {Λ−1 fn | n ∈ N} are tight on C(R+ ; B). Since the embedding C(R+ ; B) ֒→ D(R+ ; B) is continuous (see Remark 8.2), it follows that the laws of the family {Λ−1 fn , n ∈ N} are tight on D(R+ ; B). By Proposition 9.2-(i), the family {ˆ un , n ∈ N} is bounded in Mpλ (R+ ; E). Therefore, it satisfies the assumptions of Corollary 8.15. Hence, we infer that the laws of the family {vn , n ∈ N} are tight on D(R+ ; B). Finally, by Proposition 9.1 we infer that the family of functions {e−·A xn , n ∈ N} is precompact in D(R+ ; B). Since un = vn + Λ−1 fn + e−·A xn , Proposition VI.1.23 of [41] gives that the laws of the family {un , n ∈ N} are tight on D(R+ ; B). 9.3. Proof of Step (III). Let us recall that by our construction which used the Skorohod embedding Theorem, the laws of un and u ¯n on ILpλ (R+ ; E) un kMp (R+ ;E). By are identical for any n ∈ N. Hence, kun kMp (R+ ;E) = k¯ λ λ Proposition 9.2-(i) the family {ˆ un , n ∈ N} is bounded in Mpλ (R+ ; E). Now, by Proposition 8.3 and Corollary 8.13-(i) we infer that supn kun kMp (R+ ;E) < λ ∞. This proves part (i). Part (ii) of Step (III) follows by part (i) of Step (III), by the Lebesgue ¯ dominated convergence Theorem and the fact that P-a.s. u ¯n → u∗ . 9.4. Proof of Step (IV). Before proofing (i) and (ii), let us recall first that the modified version of the Skorohod embedding Theorem, i.e. Theorem E.1, implies that η¯n (ω) = η ∗ (ω) for all ω ∈ Ω and n ∈ N. Let us recall secondly ¯ = (F¯t )t≥0 , where F¯t for t > 0 is defined by that for F F¯t := (9.8)

σ({1[0,t] Nη¯n , n ∈ N}, {1[0,t] u ¯n , n ∈ N}, 1[0,t] Nη∗ , 1[0,t] u∗ }),

Here for a process v = {v(t)}t≥0 and for any interval I ⊂ R+ we define 1I v : R+ × Ω ∋ (t, ω) 7→ 1I (t)v(t, ω). For any Poisson random measure ¯ ρ ∈ MN (S × R+ ) and for any A ∈ S let us define an N-valued process (Nρ (t, A))t≥0 by Nρ (t, A) := ρ(A × (0, t]), t ≥ 0. In addition, we denote by (Nρ (t))t≥0 the measure valued process defined by ¯ t ∈ R+ . Nρ (t) = {S ∋ A 7→ Nρ (t, A) ∈ N}, By Definition 2.2, an element ρ of MN (S × R+ ) is a time homogeneous Poisson random measure over (Ω, F, (Ft )t≥0 , P) with intensity ν, iff


47

(a) for any A ∈ S, the random variable Nρ (t, A) is Poisson distributed with parameter t ν(A); (b) for any disjoint sets A1 , A2 ∈ S, and any t ∈ R+ the random variables Nρ (t, A1 ) and Nρ (t, A2 ) are independent; (c) the measured valued process (Nρ (t))t≥0 is adapted to (Ft )t≥0 ; (d) for any A ∈ S the increments of (Nρ (t, A))t≥0 are independent from (Ft )t≥0 , i.e. for any t ∈ R+ , A ∈ S and any r, s ≥ t, the random variables Nρ (r, A) − Nρ (s, A) are independent of Ft . Proof of Step (IV)-(i): Fix n ∈ N. Since η¯n (ω) = η∗ (ω) = η¯m (ω) for all ¯ and m ∈ N, it is sufficient to show that η¯n satisfies properties (a), ω ∈Ω (b), (c) and (d) listed above. The properties (a) and (b) can be characterised by the law of η¯n on MN (Z × R+ ). In particular, (a) holds, if (ν(A) t)k , A ∈ S. k! Since Law(¯ ηn ) = Law(ηn ) on MN (S × R+ ) and Nη¯n (t, A) = iA×[0,t] ◦ η¯ and Nηn (t, A) = iA×[0,t] ◦η (see Paragraph 1.1), it follows that Law(Nη¯n (t, A)) = ¯ Since ηn is a time homogeneous Poisson random Law(Nηn (t, A)) on N. measure with intensity ν, Nηn (t, A) is a Poisson distributed random variable with parameter t ν(A). Hence Nη¯n (t, A) is also a Poisson distributed random variable with parameter t ν(A). This proves equality (9.9). Similarly, (b) holds iff for any two disjoint sets A1 , A2 ∈ S, (9.9)

¯ ( Nη¯n (t, A) = k) = P

(9.10) Eei(θ1 Nη¯n (t,A1 )+θ2 Nη¯n (t,A2 )) = Eei θ1 Nη¯n (t,A1 ) Eei θ2 Nη¯n (t,A2 ) . Again, since Law(¯ ηn ) = Law(ηn ) on MN (S ×R+ ) we infer that (9.10) holds. In order to show that η¯n is a time homogeneous Poisson random measure ¯ P), ¯ one needs to show that η¯ satisfies properties (c) and (d). ¯ F, ¯ F, over (Ω, ¯ it follows that one has for F ¯ = (F¯ = t)t≥0 Since η¯n (ω) = η∗ (ω) for all ω ∈ Ω, F¯t = σ( 1[0,t] Nη¯n , {1[0,t] u ¯m , m ∈ N}, 1[0,t] u∗ ),

t ∈ R+ .

¯ Moreover, η¯n is a Poisson random measure, and, Clearly, η¯n is adapted to F. so independently scattered. Hence, for any t0 ∈ R+ , Nη¯n (t0 )(:= {S ∋ A 7→ Nη¯n (t0 , A) ∈ N}) is independent from Nη¯m (r) − Nη¯m (s) for all r, s > t0 . Let us remind, that the filtration F is not only generated by η∗ but also by the family {¯ um , m ∈ N} and u∗ . Therefore, we have to show that for any t0 ≥ 0 the processes 1[0,t0 ] um , m ∈ N, and 1[0,t0 ] u∗ are independent from the increments of Nη¯m after time t0 , or, in other words, are independent from the process t 7→ 1(t0 ,∞) Nη¯m (t + t0 ) − Nη¯m (t0 ). Recall that Law((¯ um , η¯n )) =

48


Law((um , ηn )), and therefore Law((1[0,t0 ] u ¯m , η¯n )) = Law((1[0,t0 ] um , ηn )). Similarly, Law((1[0,t0 ] u ¯m , η¯n )) = Law((1[0,t0 ] um , ηn )) = Law((1[0,t0 ] um , ηm )), and, hence Law((1[0,t0 ] u ¯m , 1(t0 ,∞) Nη¯m (t + t0 ) − Nη¯m (t0 ))) = Law((1[0,t0 ] um , 1(t0 ,∞) Nηm (t + t0 ) − Nηm (t0 ))). Let us observe that due to the construction of the process um , the process is adapted to the σ algebra generated by ηm . Moreover, the random variable 1[0,t0 ] um and the process 1(t0 ,∞) Nηm (t + t0 ) − Nηm (t0 ) are independent. Since independence can be expressed in terms of the law, 1[0,t0 ] u ¯m and 1(t0 ,∞)Nη¯n (t + t0 ) − Nη¯n (t0 ) are independent. It remains to show that the random variable 1[0,t] u ¯∗ is independent from the process 1(t0 ,∞)Nη¯n (t + t0 ) − Nη¯n (t0 ). But, this is a subject of the next Lemma. Lemma 9.3. Let B be a Banach space, z and y∗ be two B-valued random variables over (Ω, F, P). Let {yn , n ∈ N} be a family of B valued random variables over a probability space (Ω, F, P) such that yn → y∗ weakly, i.e. for all φ ∈ B ∗ , eihφ,yn i → eihφ,yi . If for all n ≥ 1 the two random variables yn and z are independent, then z is also independent of y∗ . Proof. The random variables y∗ and z are independent iff Eei(θ1 z+θ2 y∗ ) = Eei θ1 z Eeiθ2 y∗ ,

θ1 , θ2 ∈ B ∗ .

The weak convergence and the independence of z and yn for all n ∈ N justify the following chain of equalities. Eei(θ1 z+θ2 y∗ ) = lim Eei(θ1 z+θ2 yn ) = lim Eeiθ1 z Eeθ2 yn = Eeiθ1 z Eeθ2 y . n→∞

n→∞

Since for every m ∈ N, 1[0,t] u ¯m is independent from 1(t0 ,∞) Nη¯n (t + t0 ) − Nη¯n (t0 ), Lemma 9.3 implies that 1[0,t] u∗ is independent from 1(t0 ,∞)Nη¯n (t + t0 ) − Nη¯n (t0 ). Proof of Step (IV)-(ii): We have to show that η∗ ∈ MI (S × R+ ) is a Poisson random measure with intensity ν, in particular, satisfies the properties (a), (b), (c) and (d) from above. But, since η∗ (ω) = η¯1 (ω) for all ω ∈ Ω, this follows from Step (III)-(i).


49

9.5. Proof of Step (V). Here, we have to show that for each n ∈ N, u ¯n ¯ and u∗ are F-progressively measurable. But, for fixed n ∈ N the process ¯ by the definition of F. ¯ By Step (III) the process u u ¯n is adapted to F ¯n is p bounded in Mλ (R+ ; E), hence, there exists a sequence of simple functions ¯n as m → ∞ in Mpλ (R+ ; E). In particularly, we ¯m u ¯m n →u n , m ∈ N such that u can consider the approximation by the shifted Haar projections used in [16]. ¯n It follows that u ¯n is progressively measurable, i.e. (i) holds. Since u ¯m n →u p p as m → ∞ in Mλ (R+ ; E) and u ¯n → u∗ as n → ∞ also in Mλ (R+ ; E), it p follows that u∗ is a limit in Mλ (R+ ; E) of some progressively measurable step functions. In particular, u∗ is also progressively measurable, i.e. (ii) holds. 9.6. Proof of Step (VI). Before proving (VI) we will do some preparations, that is we will verify that we can replace u ¯n and η¯n by un and ηn . First, recall that the family {(un , ηn ), n ∈ N} on (Ω, F, P) and the family ¯ have equal laws on ILp (R+ ; E)× MN (Z × R+ ). ¯ F¯ , P) {(¯ un , η¯n ), n ∈ N} on (Ω, λ Secondly, recall that we know how un is constructed. In order to do this, let us Let us define a mapping G : ILpλ (R+ ; E) → ILpλ (R+ ; E) by ( xn , if s ∈ [0, 2−n ), R G(u)(s) := φ (s) 2n φnn(s)−2−n u(r) dr, if s ≥ 2−n .

Note that G is a bounded and linear map from ILpλ (R+ ; E) into itself (see [15]). Therefore, for any n ∈ N, the two triplets of random variables ˆ¯n )}, where u {(un , ηn , u ˆn )} and {(¯ un , η¯n , u ˆn = G(un )) and have equal laws on p ˆ u ¯n = G(¯ un ), ILλ (R+ ; E)×MN (Z ×R+ )×ILpλ (R+ ; E). Let us define processes zñ and z˘n by Z t −tA e−(t−s)A F (un (s)) ds zñ (t) := e xn + 0 Z tZ (9.11) e−(t−s)A G(un (s, z))˜ η (dz, ds), t ≥ 0, + 0 Z Z t −tA e−(t−s)A F (ˆ un (s)) ds z˘n (t) := e xn + 0 Z tZ (9.12) e−(t−s)A G(ˆ un (s, z))˜ η (dz, ds), t ≥ 0. + 0

Z

Let us also define processes z˜¯n and z˜¯n by replacing un and u ˆn by u ¯n and ˆ u ¯n in formulae (9.11) and (9.12). According to Step (III) and (IV) the integrals in (9.11) and (9.12) with respect to the Poisson random measure ηn and η¯n can be interpreted as Ito integral. Due to Theorem 1, [16] and the continuity of G, i.e. Assumption 4.3, for any n ∈ N, the quintuples of

50


random variables {(un , ηn , u ˆn , zñ , z˘n )} and {(¯ un , η¯n , u ¯ˆn , z¯ñ , z˘¯n )} have equal p p p laws on ILλ (R+ ; E)×MN (Z ×R+ )×ILλ (R+ ; E)×ILλ (R+ ; E)×ILpλ (R+ ; E). In addition, since Law((un , z˘n )) is supported by the diagonal, Law((¯ un , z˘¯n )) is ˘¯n ) also supported by the diagonal. Since the diagonal is a Borel set, P(¯ un = u and, hence, we infer that kun − zñ kpMp (R λ

p

+ ,E)

= k¯ un − z˜¯n kMp (R+ ,E) . λ

After this preparations we can start with the actual proof of (VI) by showing that for any ̺ > 0, there exists a natural number n ¯ ∈ N such that kun − zn kpMp (R

(9.13)

λ

+ ;E)

≤ ̺,

n≥n ¯.

To show (9.13) fix ̺ > 0. Since, by the definition of ηn , we have ηn (ω) = η(ω) for all ω ∈ Ω, we write in the following η instead of η¯n . The definition of un and zn gives kun − zn kpMp (R ;E)) + λ Z t p Z ∞ −λt −(t−s)A e E e ≤ C [F (s, uˆn (s)) − F (s, un (s))] ds dt 0 0 E Z t Z p Z ∞ −λt −(t−s)A e E + C e [G(s, u ˆn (s); z) − G(s, un (s); z)] ds dt 0

0

Z

E

Z t p −(t−s)A e E e ≤ C [F (s, u ˆn (s)) − F (s, un (s))] ds dt 0 0 E Z t Z p Z ∞ −λt −(t−s)A + C e E e [G(s, uˆn (s); z) − G(s, un (s); z)] ds ν(dz) dt Z

∞

−λt

0

0

Z

E

=: S1n + S2n ,

where C is a generic positive constant. We will first deal with S1n . Young inequality and the Assumption 4.2 on F we get S1n ≤ C Z

0

∞

−λs −δF

e

s

ds

p Z

∞

−λs

e 0

E |F (s, uˆn (s)) − F (s, un (s))|−δF ds

The

p

.

For λ > 0 let Pλ = mλ ⊗ P be the product measure on (R+ × Ω, B(R+ ) ⊗ F), where mλ is a weighted Lebesgue measure defined by Z (9.14) e−λt dt , A ∈ B(R+ ). mλ (A) = λ A


51

By Eλ we will denote the expectation with respect to Pλ . Then, by the Fubini Theorem it follows that for any γ ≥ 0 Z Z Z p γ γ p |Aγ u(s, ω)|pE dP(ω) λe−λs ds |A u(s, ω)|E dPλ (s, ω) = Eλ (|A u|E ) = (9.15)

R+ ×Ω γ

= λ|A

R

Ω

u|pMp (R ;E) . + λ

Let R > 0 be an upper bound of {Aγ u ˆn , n ∈ N} and {Aγ un , n ∈ N} in Mpλ (R+ ; E), i.e. kAγ u ˆn kpMp (R

(9.16)

λ

+ ;E)

, kAγ un kpMp (R λ

+ ;E)

≤ R.

Then we choose a constant K > 0 such that 2RFp R ̺ ≤ , p K 6 and define, for each n ∈ N, a set n DK := {(t, ω) ∈ R+ × Ω, |Aγ u ˆn (t, ω)| , |Aγ un (t, ω)| ≤ K} .

By Hypothesis (H2-a), the operator A−1 is compact, and by Assumption 4.2 the map Bγ (0, K), where Bγ (0, K) is a ball of radius In particularly, there exists a constant ς x, y ∈ Bγ (0, K) then |F (s, x) − F (s, y)|p−δF n ∈ N, a second auxiliary set by

compact, therefore A−γ is also F is uniformly continuous on K in ths space D(AγE ). > 0 such that if |x − y| ≤ ς, ≤ ̺6 . Let us define, for each

Qnς := {(t, ω) ∈ R+ × Ω, such that |ˆ un (t, ω) − un (t, ω)| ≤ ς} . By the definition of u ˆn and un given in (9.1) and (9.2), we infer that Z ∞ p e−λs E A−δF [F (s, uˆn (s)) − F (s, un (s))] ds = Eλ |F (·, uˆn ) − F (·, un )|p−δF 0

=

Z

n )c (DK

+

Z

p −δF [F (·, uˆn ) − F (·, un )] dPλ A

n ∩(Qn )c DK δ

Z

n ∩Qn DK δ

p −δF [F (·, uˆn ) − F (·, un )] dPλ + A

p −δF [F (·, u ˆ ) − F (·, u )] A n n dPλ .

Since F is bounded by Assumption 4.2, we get Z p −δF n c [F (·, uˆn ) − F (·, un )] dPλ ≤ 2RFp Pλ ((DK ) ). A n )c (DK

n we infer By applying the Chebyschev inequality and the definition of DK that R n c Pλ ((DK ) ) ≤ p , n ∈ N, K

52


where R satisfies (9.16). Therefore, and by the choice of K, we obtain Z p ̺ −δF . A [F (·, u ˆ ) − F (·, u )] n n dPλ ≤ n 6 c (DK )

In order to handle the second integral, we note that Z p −δF [F (·, uˆn ) − F (·, un )] dPλ ≤ A n ∩(Qn )c DK δ

Z

c (Qn δ)

p −δF [F (·, uˆn ) − F (·, un )] ≤ Pλ . A

Now we can proceed as before. By Assumption 4.2 and by the Chebychev inequality, we get Z p −δF [F (·, u ˆ ) − F (·, u )] A n n dPλ ≤ n ∩(Qn )c DK ς

2RFp

Z

c (Qn ς)

dPλ =

2RFp Pλ

(Qnς )c

2RFp ≤ p Eλ |ˆ un − un |pE . ς

Since by Proposition 9.2, u ˆn −un → 0 in Mpλ (R+ ; E)), there exists a number n1 , such that 2RFp ̺ Eλ |ˆ un − un |p ≤ , ∀n ≥ n ¯1. p ς 6 It remains to analyse the term Z p −δF [F (·, uˆn ) − F (·, un )] dPλ . A n ∩Qn DK δ

n and Qn and by choice of K and ς, it follows By the definition of the sets DK ς that Z p ̺ −δF [F (·, uˆn ) − F (·, un )] dPλ ≤ . A n n 6 DK ∩Qδ

Summing up, we proved that there exists a natural number n ¯ 1 ≥ 0, such that Z ∞ p ̺ −λs −δF n , n≥n ¯1. [F (s, uˆn (s)) − F (s, un (s))] ds ≤ e E A S1 = 2 0 The term S2n can be dealt exactly in the same way only the conditions on K and ̺ have to be altered. To be precise, we first note that S2n ≤ ≤

Z

∞

−λs −δG p

e

Z0

0

∞

s Z

−λs

e

Z

ds ×

p −δG [G(s, uˆn (s); z) − G(s, un (s); z)] ν(dz)ds . E A


53

Let R be again a constant satisfying (9.16) and let K > 0 be chosen such that p 2RRG ̺ ≤ , p K 6

is satisfied. Then there exists a ς > 0 such that if |x−y| ≤ ς, x, y ∈ Bγ (0, K) such that

(9.17)

Z p ̺ −δG [G(x; z) − G(y; z)] ν(dz) ≤ . A 6 Z

Now proceeding in the same way as above, it can be shown that there exists an ¯≥n ¯ 1 such that Z ∞ e−λs S2n ≤ C 0 Z p ̺ −δG [G(s, uˆn (s)) − G(s, un (s))] ν(dz)ds ≤ , E A (9.18) 2 Z

n≥n ¯.

9.7. Proof of Step (VII). In Step (III) and Step (IV) we showed, that u∗ ¯ F¯ , F, ¯ P) ¯ and η∗ a time homogeis a progressively measurable process on (Ω, neous Poisson random measure. Hence, we can define a process u by Z

t

e−(t−s)A F (s, u∗ (s)) ds + u(t) := K(x, u∗ , η∗ )(t) = e x+ 0 Z tZ e−(t−s)A G(s, u∗ (s); z) η˜∗ (dz, ds), t ≥ 0. −tA

0

Z

Here, the stochastic integral in the last term is interpreted as Itô integral. We will proof that there exists a p0 ∈ (1, p] such that 0 E kK(xn , u ¯n , η¯n ) − K(x, u∗ , η∗ )kpM p0 (R λ

Observe, that for any n0 ∈ N

+ ;E)

→ 0 as n → ∞.

54


∞ p kK(xn , u ¯n , η¯n ) − e−λt e−tA [xn − x] E dt 0 Z t p Z ∞ −(t−s)A −λt ¯ e [F (s, u ¯n (s)) − F (s, u∗ (s))] ds dt e + E

K(x, u∗ , η∗ )kpMp (R ;E) + λ

n Z ¯ ≤C E

0

0

E

∞

Z t Z ¯ e−(t−s)A G(s, u ¯n (s); z)η˜¯n0 (dz, ds) e−λt +E 0 Z 0 Z tZ p e−(t−s)A G(s, u¯∗ (s); z)η˜¯n0 (dz, ds) dt − E 0 Z Z ∞ Z t Z ¯ e−(t−s)A G(s, u¯∗ (s); z)η˜¯n0 (dz, ds) e−λt E 0 Z 0 Z tZ p o e−(t−s)A G(s, u∗ (s); z)η˜¯∗ (dz, ds) dt − Z

=: C ( S0n + S1n +

0 n,n0 S2

Z

E

+ S3n,n0 ) .

Since A−δI xn → A−δI x in E, the Lebesgue DCT implies Z ∞ e−λt kAδI e−tA kL(E,E) |(AδI x − AδI xn )|pE dt −→ 0 as n → ∞. S0n ≤ C1 0

From the previous construction invoking the Skorohod embedding Theorem, we infer that (see (9.3)) lim (¯ un , η¯n ) = (u∗ , η∗ )

n→∞

in ILpλ (R+ ; E) ∩ D(R+ ; B) × MN (Z × R+ ),

¯ a.s. . P

Next, by the Young inequality we infer that Z t Z ∞ −λt e−(t−s)A F (s, u¯n (s)) − F (s, u∗ (s)) ds|p dt e | 0 0 Z ∞ e−λs |A−δF F (s, u ¯n (s)) − A−δF F (s, u∗ (s))|p ds =: ξn , ≤ C 0

R∞

¯ n , u∗ ∈ where C = 0 e−λs |AδF e−sA | ds . By Assumption (4.2) and since u p ILλ (R+ ; E) a.s. , there exists a C such that Z ∞ e−λt |A−δF F (s, u ¯n (s))|p ds ≤ C, 0

and

Z

∞

e−λt |A−δF F (s, u∗ (s))|p ds ≤ C.

0

The Lebesgue’s dominated convergence theorem and the continuity of F (see Assumption (4.2)) give ¯ |S n |p p + E 1 L (R ;E) → 0 as n → ∞.


55

To prove convergence of S2n,n0 , we proceed in a similar way as we have done to show convergence for S1n . In particular, applying the Burkholder inequality and the Fubini Theorem as well as (9.3), give Z ∞ Z tZ n,n0 −λt ¯ e S2 ≤E 0 0 Z −(t−s)A |e G(s, u ¯n (s); z) − G(s, u∗ (s); z) |p ν(dz) ds dt Z ∞ Z −λs e ≤ CE 0 −δG

|A

(9.19)

Z

G(s, u¯n (s); z) − A−δG G(s, u∗ (s); z)|p ν(dz) ds,

R∞ where C = 0 e−λs |AδG e−sA |p ds . By Assumption 4.3 there exists a C > 0 ¯ such that λ ⊗ P-a.e. Z (9.20) ¯n (s); z)|p ν(dz) ≤ C. supn e−λt |A−δG G(s, u Z

Hence and since the RHSs of (9.19) and (9.20) are independent of n0 , by the Lebesgue dominated convergence theorem we infer that ∀ε > 0, ∃n2 ∈ N such that S2n,n0 ≤ n,

∀n ≥ n2 , and ∀n0 ∈ N.

In order to treat the term S3n,n0 , we first note that by Theorem of E.1 η˜¯n0 = η˜∗ . Now, Proposition A.1 gives that a.s. S3n,n0 = 0. 9.8. Proof of Step (VIII). Proposition 8.10 gives p −λt ≤ E sup e |K(x, u∗ , η∗ )(t) − K(x, K(x, u∗ , η∗ ), η∗ )(t)|B 0≤t 0 and q¯ > q such that sup kvn kMq¯ (R+ ;E) < ∞. λ

n∈N

Step (II) The laws of the family {zn , n ∈ N} are tight on C(R+ ; X). In particular, for any T > 0 there exists a number r > 0 such that sup E sup |ˆ zn (t)|X < ∞.

n∈N

0≤t≤T

Finally, in order to use the Skorohod embedding theorem, we also need the following. Step (III) The laws of the family {vn , n ∈ N} are tight on D(R+ ; B). Step (IV) The laws of the family {ηn , n ∈ N} are tight on MI (Z ×R+ ). Suppose for the time being that the proofs of the previous four steps have been accomplished. From Steps (II), (III) and (IV) it follows that there exists a subsequence of {(zn , vn , ηn ) | n ∈ N}, also denoted by {(zn , vn , ηn ) | n ∈ N} and a C(R+ , X) × Lp (R+ , E) ∩ D(R+ , B) × MN (Z × R+ ) – valued random variable (z∗ , v∗ , η∗ ), such that the sequence of laws of (zn , vn , ηn ) converges to the law of (z∗ , v∗ , η∗ ). Moreover, by Theorem E.1, there exists a ˆ and C(R+ , X1 )×Lp(R+ , E)∩D(R+ , B)×MN (Z × ˆ Fˆ , P) probability space (Ω, ˆ R+ )- valued random variables (z∗ , v∗ , η∗ ), (ˆ zn , vˆn , ηˆn ), n ∈ N, such that P-a.s. (ˆ zn , vˆn , ηˆn ) → (z∗ , v∗ , η∗ )

(10.6)

on C(R+ , X)×Lp (R+ , E)∩D(R+ , B)×MN (Z×R+ ), ηˆn = η∗ , and L((ˆ zn , vˆn , ηˆn )) = ˆ = (Fˆt )t≥0 on (Ω, ˆ Fˆ ) L((zn , vn , ηn )) for all n ∈ N. We define a filtration F as the one generated by η∗ , z∗ , v∗ and the families {zn | n ∈ N} and {vn | n ∈ N}. The next two Steps imply that the following two Itô integrals over the filˆ P) ˆ ˆ Fˆ , F, tered probability space (Ω, Z tZ e−(t−s)A G(s, vˆn (s) + zˆn (s), z)) η˜ˆn (dz, ds), t ≥ 0, 0

Z

58


and

Z tZ 0

e−(t−s)A G(s, v∗ (s) + z∗ (s), z)) η˜∗ (dz, ds),

t ≥ 0,

Z

are well defined. Step (V) The following holds (i) for all n ∈ N, ηˆn is a time homogeneous Poisson random measure ˆ F, ˆ F, ˆ P) ˆ with intensity measure ν; on B(Z) × B(R+ ) over (Ω, (ii) η∗ is a time homogeneous Poisson random measure on B(Z) × ˆ t , P) ˆ with intensity measure ν; ˆ F, ˆ F B(R+ ) over (Ω, Step (VI) The following holds (i) for all n ∈ N, the processes vˆn and zˆn are with respect to (Fˆt )t≥0 progressively measurable; (ii) the processes z∗ and v∗ are with respect to (Fˆt )t≥0 progressively measurable. The proofs of Step (IV) and (V) are the same as the proofs of Step (IV) and (V) of Theorem 4.5. Also as earlier in the proof of Theorem 4.5 in order to complete the proof of Theorem 5.2 we have to show that (10.7)

u∗ (t) = K(x, u∗ , η∗ )(t),

∀t ∈ R+ ,

ˆ − a.s., P

where u∗ = z∗ + v∗ and the map K is again, with a slight abuse of notation, defined by Z t e−(t−s)A F (s, u(s)) ds K(x, u, η)(t) = e−tA x + 0 Z tZ e−(t−s)A G(s, u(s); z) η˜(dz; ds), t ≥ 0. + 0

Z

To prove the equality (10.7) it is sufficient to show that for any T > 0 (10.8)

E sup |u∗ (t) − K(x, u∗ , η∗ )(t)|B = 0. 0≤t≤T

To do this, we write first u∗ (t) − K(x, u∗ , η∗ )(t) = (u∗ (t) − zˆn (t) − vˆn (t)) + (ˆ zn (t) + vˆn (t) − Kn (x, zˆn + vˆn , ηˆn )(t)) + (Kn (x, u ˆn , ηˆn )(t) − Kn (x, u∗ , η∗ )(t)) + (Kn (x, u∗ , η∗ )(t) − K(x, u∗ , η∗ )(t)) , where Kn is an operator defined by the same formula as the operator K but with F replaced by Fn . Equality (10.7) will be shown by the four following steps.


59

Step (VII) There exists an r > 0 such that for any T > 0 E δ10T (ˆ zn (t) + vˆn (t), z∗ + v∗ )r → 0

as n → ∞.

Step (VIII) There exists an r > 0 such that for any T > 0 E sup |ˆ zn (t) + vˆn (t) − Kn (x, zˆn + vˆn , ηˆn )|r → 0 as n → ∞. 0≤t≤T

Step (IX) There exists an r > 0 such that for any T > 0 E sup |Kn (x, uˆn , ηˆn )(t) − K(x, u∗ , η∗ )|r → 0

as n → ∞.

0≤t≤T

With these four steps Therom 5.2 is shown.

gl:010 It remains to proof Step (I) to Step (IX), which is done in the following. Proof of Step (I). Step (I) is a simple consequence of Corollary 8.13, Corollary 8.14, inequality (8.20), and the continuity of the embedding X0 ֒→ X. Proof of Step (II). By Lemma 10.1 we infer that for any T ≥ 0 Z T e−k(t−s) a |vn (s)|X + |e−sA x|X ds sup |zn (t)|X ≤ 0≤t≤T

0

≤C

(10.9)

Z

0

T

1 + |vn (s)|qX + |e−sA x|qX ds.

From Step (I) it follows that there exists a number r > 0 such that sup E|vn |rILq (R+ ;X) < ∞.

(10.10)

n

λ

Therefore,

r q < ∞. sup E sup |zn (t)|X

n

0≤t≤T

Hence, we proved the second part of Step (II). The last inequality also imr

plies that supn E|zn |ILq q¯(R ;X) < ∞. Hence, by the identity zn = Λ−1 (vn +zn ) + and Corollary 8.7 we infer that the laws of the family {zn | n ∈ N} are tight on C(R+ , X). Consequently, since the embedding X ֒→ B is continuous, they are tight on C(R+ , B). Thus, we have finished the proof of Step (I). Proof of Step (III). By Corollary 8.14 and Corollary 8.15 the laws of the family {vn | n ∈ N} are tight on Lp (R+ , E) ∩ D(R+ , B). Thus, we have finished the proof of Step (III). 10δ denotes the Prohorov metric on D(R ; B), see e.g. Section B of [15]. +

60


Proof of Step (IV). Since En ηn (A, I) = Eη(A, I)11 for all A ∈ B(Z) and I ∈ B(R+ ), the laws of the random variables ηn , n ∈ N, are identical on MN (Z × R+ ). Therefore, and in view of Theorem 3.2 of [55], the laws of the family {ηn , n ∈ N} are tight on MN (Z × R+ ). Thus, we have finished the proof of Step (IV). Proof of Step (VII). Since zˆn , n ∈ N, and z∗ are continuous B-valued functions, δT (ˆ zn + vˆn , z∗ + v∗ ) ≤ δT (ˆ zn , z∗ ) + δT (ˆ vn , v∗ ) ≤

sup |zn (t) − z∗ (t)|B + δT (ˆ vn , v∗ ).

0≤t≤T

We first will show that there exists an r > 0 such that E sup |zn (t) − z∗ (t)|rB → 0 as

n → ∞.

0≤t≤T

Next, we will show that there exists an r > 0 such that E δT (ˆ vn , v∗ )r → 0 as

n → ∞.

By Step (II) and [64, Theorem 2.2.3, Part (2)] there exists an r > 0 such that the family n o |ˆ zn |rC([0,T ];B) , n ∈ N

is uniformly integrable. Furthermore, since zˆn → z∗ on C([0, T ]; B) P-a.s., |zn |rC([0,T ];B) → |z∗ |rC([0,T ];B)

P-a.s..

Hence, also |zn |rC([0,T ];B) → |z∗ |rC([0,T ];B) in probability. Thus, [64, Theorem 2.2.3, Part (1)] implies that E |z∗ |rC([0,T ];B) = lim E |zn |rC([0,T ];B) < ∞. n→∞

From the last inequality we infer, again by [64, Theorem 2.2.3, Part (2)], that there exists an r¯ > 0 such that the family n o ¯ r¯ |ˆ zn |rC([0,T + |z | , n ∈ N ∗ ];B) C([0,T ];B) is uniformly integrable. Again, since zˆn → z P-a.s. on C([0, T ]; B), |ˆ zn − z∗ |C([0,T ];B) → 0 in probability. Since ¯ ¯ r¯ |ˆ zn − z∗ |rC([0,T zn |rC([0,T ];B) ≤ C |ˆ ];B) + C |z∗ |C([0,T ];B) , 11E

n

n ∈ N,

denotes the expectation over Ωn with respect to the probability measure Pn .


61

¯ we infer that the family {|ˆ zn − z∗ |rC([0,T ];B) , n ∈ N} is uniformly integrable. Again, [64, Theorem 2.2.3, Part (1)] implies that ¯ lim |ˆ zn − z∗ |rC([0,T ];B) = 0.

n→∞

Next, we investigate the limit of the second term δ(ˆ vn , v∗ ). Let us note that, since δ(ˆ vn , v∗ ) → 0 P-a.s., it follows from the definition of the Prokorov 12 metric that there exists a sequence of functions λn ∈ Λ such that sup |ˆ vn ◦ λn (t) − v∗ (t)|B ≤ δ(ˆ vn , v∗ ),

n ≥ 0.

0≤t≤T

Since for any n ∈ N sup |ˆ v ◦ λ (t)| − sup |v (t)| n n ∗ B B ≤ 0≤t≤T 0≤t≤T vn ◦ λn (s)|B − |v∗ (s)|B ≤ sup |ˆ 0≤s≤T

vn ◦ λn (s)|B − sup |v∗ (t)|B sup |ˆ 0≤t≤T

0≤s≤T

sup vˆn ◦ λn (t) − v∗ (t) B < δ(ˆ vn , v∗ ),

0≤t≤T

[Appendix B] it follows from (10.6) that sup0≤t≤T |ˆ vn ◦λn (t)|B → sup0≤t≤T |ˆ vn (t)| P-a.s. Hence, sup |ˆ vn ◦ λn (t)|B → sup |ˆ vn (t)|

0≤t≤T

0≤t≤T

in probability. Since by Step (I) there exists an r > 0 such that sup0≤t≤T |ˆ vn (t)|r is uniformly integrable, the family {sup0≤t≤T |ˆ vn ◦ λn (t)|r , n ∈ N} is uniformly integrable. Hence, it follows by [64, Theorem 2.2.3, Part (1)] that E sup |v∗ (t)|rB = lim E sup |ˆ vn ◦ λn (t)|rB . 0≤t≤T

n→∞

0≤t≤T

Since δ(ˆ vn , v∗ ) ≤ δ(ˆ vn , 0) + δ(v∗ , 0) ≤ sup |ˆ vn (t)|B + sup |v∗ (t)|B , 0≤t≤T

0≤t≤T

by [64, Theorem 2.2.3, Part (2)] there exists an r¯ > 0 such that the family {δ(ˆ vn , v∗ )r¯, n ∈ N} is uniformly integrable. Now, recall that (10.6) implies δ(ˆ vn , v∗ ) → 0 P-a.s., therefore, δ(ˆ vn , v∗ ) → 0 in probability, and, therefore, by [64, Theorem 2.2.3, Part (1)] limn→∞ E δ(ˆ vn , v∗ )r¯ = 0. Thus, we have finished the proof of Step (VII). Proof of Step (VIII). In order to prove Step (VIII), let us recall that the laws of (ˆ zn , vˆn , ηˆn ) and (zn , vn , ηn ) are equal on C(R+ ; X) × Lp (R+ , E) ∩ D(R+ , B) × MN (Z × R+ ) and, moreover, the process un = zn + vn is a martingale solution to Problem (10.4). Hence, kun − Kn (x, un , ηn )kMp (R+ ,E) = 0. λ

12For the exact definition we refer to [15, Appendix B].

62


In particular, for any T > 0, E sup |un (t) − Kn (x, un , ηn )(t)|B = 0. 0≤t≤T

Therefore, by the main result of [16] we can conclude that for any T > 0 ˆ sup |ˆ E un (t) − Kn (x, u ˆn , ηˆn )(t)|B = 0. 0≤t≤T

Thus, we have finished the proof of Step (VIII). Proof of Step (IX). First we split the difference into the following differences Kn (x, u ˆn , ηˆn )(t) − Kn (x, u∗ , η∗ )(t) Z t e−(t−s)A [Fn (s, u ˆn (s)) − Fn (s, u∗ (s))] ds = 0 Z tZ e−(t−s)A G(s, uˆn (s); z) η˜ˆn − η˜∗ (dz; ds) + 0 Z Z tZ e−(t−s)A [G(s, u ˆn (s); z) − G(s, u∗ (s); z)] η˜∗ (dz; ds). + 0

Z

The aim is to tackle the first term similarly to Step (VII) and the last term by the Lebesgue dominated convergence Theorem. Note, that, according to Proposition A.1, the second term is zero. We begin with the first term. Let us note that Z t ˆ e−(t−s)A [Fn (s, uˆn (s)) − F (s, u∗ (s))] ds E sup 0≤t≤T

(10.11)

B

0

ˆ sup ≤ E

0≤t≤T

ˆ sup ≤ E 0≤t≤T

(10.12)

ˆ ≤ E

Z

T 0

Z

0

Z

0

t

e−(t−s)A [Fn (s, u ˆn (s)) − F (s, u∗ (s))] B ds

t

Fn (s, uˆn (s)) − F (s, u∗ (s)) ds B

Fn (s, u ˆn (s)) − F (s, u∗ (s)) B ds.

By (10.2) it follows that there exists a constant C > 0 such that for any n≥0 Z T Z T (1 + |ˆ vn (s)|qX ) ds. Fn (s, u ˆn (s)) ds ≤ C B

0

0

By Step (I) and [64, Theorem 2.2.3, Part (2)] for any q < q¯ the family

(1 + |ˆ vn (s)|qX , n ≥ 0 ,


63

is uniformly integrable over (Ω × [0, T ], F × B([0, T ]), P × λ). Therefore, the family Z T Fn (s, u ˆn (s)) ds B , n ∈ N . 0

is also uniformly integrable over (Ω × [0, T ], F × B([0, T ]), P × λ). Since u ˆn → u∗ in D([0, T ]; B) P-a.s., there exists a sequence {λn , n ∈ N} ⊂ Λ13, such that u ˆn ◦ λn → u∗ uniformly on [0, T ]. Assumption 5.2 implies that F (ˆ un ◦ λn (·), ·) − F (u∗ (·), ·) B → 0, P × λ-a.s. .

Therefore, we infer that F (ˆ un ◦ λn (·), ·) − F (u∗ (·), ·) B → 0

in probability over (Ω×[0, T ], F ×B([0, T ]), T1 P×λ). By [64, Theorem 2.2.3, Part (1)] Z T ˆ Fn (s, u ˆn (s)) − F (s, u∗ (s)) B ds → 0. E 0

Hence,

ˆ sup E 0≤t≤T

Z

0

t

Fn (s, uˆn (s)) − F (s, u∗ (s)) ds → 0. B

The second summand is zero by Proposition A.1. In order to tackle the last summand, we apply Corollary 8.13-(iii). Again, by the Lebesgue dominated convergence Theorem almost sure convergence implies the convergence of the expectation. Note here, that, in contrary to F by Assumption 4.3, the function G is bounded. Appendix A. Stochastic Integration - a useful result As seen in [14], the stochastic integral is an extension of the set of simple functions to progressively measurable functions. In the same way we will show the following Proposition. Proposition A.1. Assume that (S, S) is a measurable space, ν ∈ MS+ is a non-negative measure on (S, S) and P = (Ω, F, (Ft )t≥0 , P) is a filtered probability space. Assume also that η1 and η2 are two time homogeneous Poisson random measures over P, with the intensity measure ν. Assume that p ∈ (1, 2] and E is a martingale type p Banach space and 13For the defintion of Λ we refer to [15, Appendix B].

64


ξ ∈ Mp (0, ∞, Lp (S, ν; E)). If P-a.s. η1 = η2 on MN (S, R+ ), then for any t ≥ 0 P-a.s. Z t Z t (A.1) ξ(s, z) η˜2 (dz, ds). ξ(s, z) η˜1 (dz, ds) = 0

0

Proof of Proposition A.1. In order to show the equality, we will go back to the definition of the stochastic integral by step functions. First, put PI,J I = (a, b]. We may suppose that f = j=1,i=1 fji 1Aji ×Bi with fji ∈ E, Aji ∈ Fa and Bi ∈ S. For fixed i we may suppose that the finite family of sets {Aji × Bi , j = 1, . . . , J} are pairwise disjoint and that the family of sets {Bi , i = 1, . . . I} are also pair-wise disjoint and ν(Bi ) < ∞. Let us notice that for i = 1, . . . , I Z J X 1Aji η˜(Bi × I) fji , f (x) η˜1 (dx, I) = Bi

j=1

and Z

f (x) η˜2 (dx, I) = Bi

J X

1Aji η˜(Bi × I) fji .

i=1

R Since P-a.s. η1 = η2 on MN (S, R+ ), it follows that a.s. S f (x)η˜1 (dx, I) = R S f (x)η˜2 (dx, I). It remains to investigate the limit as done in [14, Appendix C]. But if xn = yn for all n ∈ N and xn → x, yn → y in a Banach space X, then x = y. Appendix B. A Compactness result Let B be a separable Banach space, I ⊂ R+ be an interval. Recall that Z p p −λt e |u(t)|B dt < ∞ . ILλ (I; B) = u : I → B : u measurable and I

Wα,p λ (I; B)

Given p > 1, α ∈ (0, 1), let be the Sobolev space of all u ∈ p ILλ (R+ ; B) such that Z Z |u(t) − u(s)|pB e−λ(t+s) ds dt < ∞. |t − s|1+αp I I equipped with the norm kukWα,p (I;B) := λ

Z Z I

I

−λ(t+s)

e

u(t) − u(s)|pB ds dt |t − s|1+αp

p1

.

We begin with recalling Theorem 2.1 from [37, Gatarek and Flandoli] which is valid for finite time horizon.


65

Theorem B.1. Let B0 ⊂ B ⊂ B1 be Banach spaces, B0 and B1 reflexive, with compact embedding of B0 in B. Let p ∈ (1, ∞) and α ∈ (0, 1) be given. Let X be the space X = ILp0 ([0, T ]; B0 ) ∩ Wα,p 0 ([0, T ]; B1 ). Then the embedding of X in Lp0 ([0, T ]; B) is compact. Since for λ > 0 the space Lpλ ([0, T ]; B) is isomorphic to a space Lp ([0, T ]; B) via a map λ

− · Φλ : Lpλ ([0, T ]; B) ∋ f 7→ f˜ = e p f (·) ∈ Lp ([0, T ]; B),

we get the following result. Theorem B.2. Let B0 ⊂ B ⊂ B1 be Banach spaces, B0 and B1 reflexive, with compact embedding of B0 in B. Let λ > 0. Let p ∈ (1, ∞) and α ∈ (0, 1) be given. Let X be the space X = ILpλ (R+ ; B0 ) ∩ Wα,p λ (R+ ; B1 ). Then the embedding of X in Lpλ (R+ ; B) is compact. Proof. The proof is similar to the proof of Flandoli and Gatarek [37] resp. Dunford and Schwartz [32, Chapter IV.8.20]. Similarly, let us first define a family of operators Ja in Lp (R+ ; Bi ), i = 0, 1, a > 0, by setting Z s+a 1 g(t) dt, s ∈ R+ , Ja g(s) = 2a s−a

where, we put g(t) := 0 if t > 0. First, note that for any a > 0, Ja maps Lp (R+ ; Bi ) to C(R+ ; Bi ), i = 0, 1. Taking into account the finiteness R of the measure B(R+ ) ∋ A 7→ A e−tλ dt, by [32, Theorem 18, Chapter IV.8, p. 297] a bounded set K of Lpλ (R+ ; R) is precompact in Lpλ (R+ ; R), iff lima→0 Ja f = f uniformly on K. Now, let G ⊂ X be a bounded set. We have to prove that G is relatively compact in Lpλ (R+ ; B). Taking into account that e−λ(t+s) ≤ e−λs e−λa , for all t ∈ (s − a, s + a), we get Z ∞ Z Z eλa ∞ a −λ(t+s) p −λs e |Ja g(s) − g(s)|B1 ds ≤ e |g(t + s) − g(s)|pB1 dt ds. 2a 0 0 −a Now, arguing as Flandoli and Gatarek we obtain Z ∞ 1 e−λs |Ja g(s) − g(s)|pB1 ds ≤ eλa aαp |g|pWα,p (R ;B ) . + 1 λ 2 0

Therefore, lima→0 Ja f = f uniformly on G in Lpλ (R+ ; B1 ). Next, since G is bounded in Lpλ (R+ : B1 ) and B1 ֒→ B compactly, G is compact in Lpλ (R+ : B).

66


Appendix C. Some Results in Interpolation Theory It follows from the Riesz Thorin Theorem, that if p0 , p1 ≥ 1, 0 < θ < 1 θ and 1p = 1−θ p0 + p1 , then [Lp0 , Lp1 ]θ 14 = Lp

(with equal norms).

It is possible to generalize this result to vector–valued Lp –spaces. Let A be a Banach space and ILp (U, dν; A) the space of all strongly measurable functions f : U → A such that Z |f (x)|pA dν(x) < ∞, U

where 1 ≤ p < ∞. We shall denote by L∞ (U, dν; A) the completion in the sup–norm of all functions X s(x) = (C.1) ak 1Ek (x), ak ∈ A, k

where the sum is finite and 1Ek is the characteristic function of the measurable disjoint sets Ek . If ν(Ek ) < ∞, function of the form (C.1) are called simple functions. The completion in L∞ (R+ ; A) of the norm of the simple functions is denoted by L∞ 0 (U, dν; A). Now, we can formulate the following Theorem: Theorem C.1. (See Bergh and L¨ ofstr¨ om [7], Theorem 5.1.2, p. 107) Assume A0 and A1 are Banach spaces and that 1 ≤ p0 , p1 < ∞, 0 < θ < 1 and (1−θ) θ 1 p = p0 + p1 . Then [ILp0 (U, dν; A0 ), ILp1 (U, dν; A1 )]θ = ILp (U, dν; [A0 , A1 ]θ ) If in addition 1 ≤ p0 < ∞ with

1 p

=

(1−θ) p0 ,

(with equal norms).

then

[ILp0 (U, dν; A0 ), ILp1 (U, dν; A1 )]θ = ILp (U, dν; [A0 , A1 ]θ ) . Remark C.2. It follows from the Riesz–Thorin interpolation Theorem, see e.g. Theorem 1.1.1 Bergh and L¨ ofstr¨ om, that under the assumptions of Theorem C.1 θ kxkILp (U,ν;[A0,A1 ]θ ) ≤ kxk1−θ x ∈ ILp U, ν; (A0 , A1 )[θ] . ILp0 (U,ν;A0 ) kxkILp1 (U,ν;A1 ) , 14For θ ∈ (0, 1) and two separable Banach spaces A and A , [A , A ] denotes the 1 2 1 2 θ

complex interpolation functor.


67

In order to characterise regularity of the functions we used real interpolation spaces between B and D(A). If the underlying space B is a Hilbert B (θ, 2) for any θ ∈ (0, 1). If space and A is selfadjoint, then [B, D(A)]θ = DA B is a Banach space, this is not necessarily true. Nevertheless, the following Theorems are valid. Theorem C.3. (see Bergh and L¨ ofstr¨ om [7, Theorem 4.7.1, p.102]) Let 15 (A1 , A2 ) be a Banach couple . Then the following inclusion holds (A1 , A2 )θ,1 16 ⊂ [A1 , A2 ]θ ⊂ (A1 , A2 )θ,∞ , if 0 < θ < 1. Moreover, if A1 ⊂ A2 , then for 1 ≤ p < q ≤ ∞ and 0< ε < 1−θ (A1 , A2 )θ,q ֒→ (A1 , A2 )θ,p ֒→ (A1 , A2 )θ−ε,∞ continuously. In general, the real method and the complex method leads to different results. Neither of the indices 1 and ∞ can be replaced with a q, 1 < q < ∞. However, in case of reiteration one can mix up both methods to a certain extent. Theorem C.4. (see Bergh and L¨ ofstr¨ om [7, Theorem 4.7.2, p.103 and Theorem 3.4.1, p.46]) Let (A1 , A2 ) be a compatible Banach couple. If 0 < θ1 < θ2 < 1, θ = (1 − η)θ1 + ηθ1 and 0 < p ≤ ∞ then ([A1 , A2 ]θ1 , [A1 , A2 ]θ2 )η,p = (A1 , A2 )θ,p If 1 ≤ pi ≤ ∞ (i = 1, 2) and

1 p

(equivalent norms).

= (1 − η) p11 + η p12 then

[(A1 , A2 )θ1 ,p1 , (A1 , A2 )θ2 ,p2 ]η = (A1 , A2 )θ,p

(equivalent norms).

Given two compatible couples (A0 , A1 ) and (B0 , B1 ), an linear operator T : A0 → B0 compactly and T : A1 → B1 continuous, it is still an open problem in interpolation theory to determine whether or not an interpolated operator by the complex method is compact. Nevertheless, in certain specific cases, one can verify if the interpolated operator is compact or not. Theorem C.5. (see [18, Theorem 9]) Let (A0 , A1 ) and (B0 , B1 ) be two compatible Banach couples and T : (A0 , A1 ) → (B0 , B1 ). If A0 is a UMDBanach space, then for any 0 < θ < 1, the interpolated operator T : [A0 , A1 ]θ → [B0 , B1 ]θ is compact. 15Two topological vector spaces are called compatible, iff there exists a Hausdorff topological vector space A such that A1 and A2 are subspaces of A. If we want to underline the space A, then we say with respect to A compatible. 16For θ ∈ (0, 1), 1 ≤ p ≤ ∞ and two Banach spaces A and A , (A , A ) 1 2 1 2 θ,p denotes the real interpolation functor.

68


Appendix D. Besov spaces and their properties An introduction to Besov spaces are given in Runst and Sickl [61]. We are interested is the continuity of the mapping G described in 4.3. To be precise, we will prove the following result. Proposition D.1. If f ∈ S(Rd ) then for each a ∈ Rd the Schwartz distrid

−d

p (Rd ) and moreover there exists bution f δa belongs to the Besov space Bp,∞ a constant C = C(d, p) independent of f such that Z |f δa |p d −d (D.1) da = |f |pLp (Rd ) .

Rd

p Bp,∞ (Rd )

Thus, there exists a unique bounded linear map d

−d

p (Rd )) Λ : Lp (Rd ) → Lp (Rd , Bp,∞

such that [Λ(f )](a) = f δa , f ∈ S(Rd ), a ∈ Rd . Moreover, for any s ≥ 0 there exists a constant C such that Z |f δa |p d −d−s da ≤ C |f |pW −s (Rd ) . (D.2) Rd

p Bp,∞

(Rd )

p

In the following we denote the value of Λ(f ) at a, where f ∈ Lp (Rd ) by f δa . Let us recall the definition of the Besov spaces as given in [61, Definition 2, pp. 7-8]. First we choose a function ψ ∈ S(Rd ) such that 0 ≤ ψ(x) ≤ 1, x ∈ Rd and 1, if |x| ≤ 1, ψ(x) = 0 if |x| ≥ 32 . Then put   φ0 (x) = ψ(x), x ∈ Rd , φ (x) = ψ( x2 ) − ψ(x), x ∈ Rd ,  1 φj (x) = φ1 (2−j+1 x), x ∈ Rd , j = 2, 3, . . . .

−1 being the Fourier With the choice of φ = {φj }∞ j=0 as above and F and F and the inverse Fourier transformations (acting in the space S ′ (Rd ) of Schwartz distributions) we have the following definition.

Definition D.2. Let s ∈ R, 0 < p ≤ ∞ and 0 < q ≤ ∞. If 0 < q < ∞ and f ∈ S ′ (Rd ) we put  1 q ∞ X q sjq −1   2 F [φj Ff ] Lp = |f |Bp,q , s j=0

|f |Bp,∞ s

We denote by

s (Rd ) Bp,q

= sup 2sj F −1 [φj Ff ] Lp . j∈N

is finite. the space of all f ∈ S ′ (Rd ) for which |f |Bp,q s


69

Lemma D.3. If h ∈ S(Rd ), g ∈ S(Rd ) a ∈ Rd , then |(hδa ) ∗ g|Lp (Rd ) = |h(a)||g|Lp (Rd ) . Proof. Let us recall that hδb ∈ S ′ (Rd ) assigns to a function g ∈ S(Rd ) a value δa (hg) = h(a)g(a). Since by the definition of a convolution of a distribution with a test function, where gˇ(·) = g(− ·) [(hδa )∗g](x) = (hδa )(τx gˇ) = h(a)(τx gˇ)(a) = h(a)ˇ g (a−x) = h(a)g(x−a), x ∈ Rd , we have |(hδa ) ∗ g|pLp (Rd )

=

Z

Rd

|g(x − a)h(a)|p dx = |g|pLp (Rd ) .

Lemma D.4. If ϕ ∈ S(Rd ), λ > 0 and g(x) := ϕ(λx). x ∈ Rd , then |F −1 g|Lp (Rd ) = λ

d( p1 −1)

|F −1 ϕ|Lp (Rd ) .

Proof. Simple calculations.

Proof of Proposition D.1. Obviously it is enough to prove the first part of the Proposition. We will use the definition of the Fourier transform as in [61, p. 6]. In particular, with h·, ·i being the scalar product in Rd , we put −d/2

(Ff )(ξ) := (2π)

Z

e−ihx,ξi f (x) dx, ξ ∈ Rd , f ∈ S(Rd ). Rd

Let us fix f ∈ S(Rd ). Since F −1 (ϕu) = (2π)−d/2 (F −1 ϕ) ∗ (F −1 u), ϕ ∈ S, u ∈ S ′ we infer that for j ∈ N∗ , |F −1 [ϕj F(f δa )|Lp (Rd ) = (2π)−d/2 |(F −1 ϕj ) ∗ (f δa )|Lp (Rd ) = (2π)−d/2 |f (a)|F −1 ϕj |Lp (Rd ) d( p1 −1) −jd( p1 −1)

= (2π)−d/2 2

2

d( 1 −1)

|f (a)||F −1 ϕ1 |Lp (Rd ) .

p Hence, f δa belongs to the Besov space Bp,∞ (Rd ) as requested and the equality (D.1) follows immediately. In order to show (D.2) note, that by

γ kf kBp,∞ s+γ (Rd ) = kI(s)f kBp,∞ (Rd ) ,

70


s where I(s) = F −1 1 + |ξ|2 2 F (see Proposition 1-(ii) [61, p. 18]). hence, we obtain

Z

−1 s 2 2

F δ f da) F( 1 + |ξ| ϕ a j

p d

Rd L (R )

Z s

−1 −1 2 2

δa f da) F( ≤ F ϕj ∗ F 1 + |ξ|

Rd

Z

−1

−1 s 2 2

≤ F ϕj L1 (Rd ) F 1 + |ξ| F(

Moreover,

Z

−1 s 2 2

F F( 1 + |ξ|

Lp (Rd )

Rd

δa f da)

.

Lp (Rd )

δa f da)

p d Rd L (R ) p Z Z Z s ihy,ξi 2 2 −iha,ξi dy e 1 + |ξ| = e f (a) dξ da d Rd R Rd p Z Z Z 2s −iha,ξi ihy,ξi 2 e 1 + |ξ| = e f (a) da dξ dy Rd Rd Rd p Z Z Z p h i i h s s −1 ihy,ξi 2 f (ξ) dξ 2 f F F (I + ∆) (y) dy = e F (I + ∆) = dy d d d R R R

h i s s

≤ F −1 F (I + ∆) 2 f p d = (I + ∆) 2 f p d = kf kHps,1 (Rd ) . L (R )

L (R )

Summing up, one can show that for any s ∈ R, there exists a constant C > 0 such that

Z

0 d

d f δa da s+ dp −d d ≤ C kf kWps (Rd ) , f ∈ Cb (R ). R

Bp

(R )

For s ≤ 0 the operator I(s) is positive. Therefore, it follows Z kf δa k s+ dp −d da ≤ C kf kWps (Rd ) , f ∈ Cb0 (Rd ), f ≥ 0.. Rd

Bp

(Rd )

Theorem 5.1.2 in [7] gives inequality (D.2).

Appendix E. A modified version of the Skorohod embedding Theorem Within the proof we are considering the limit of pairs of random variables. For us it was important that certain propetries of the pairs are conserved by the Skorohod embedding Theorem. Therefore, we had to use a modified version wich is stated below. Theorem E.1. Let (Ω, F, P) be a probability space and E1 , E2 be two separable Banach spaces. Let χn : Ω → E1 × E2 , n ∈ N, be a family of random


71

variables, such that the sequence {Law(χn ), n ∈ N} is weakly convergent on E1 × E2 . For i = 1, 2 let πi : E1 × E2 be the projection onto Ei , i.e. E1 × E2 ∋ χ = (χ1 , χ2 ) 7→ πi (χ) = χi ∈ Ei . Finally let us assume that there exists a random variable ρ : Ω → E1 such that Law(π1 ◦ χn ) = Law(ρ), ∀n ∈ N. ¯ a family of E1 × E2 – ¯ F¯ , P), Then, there exists a probability space (Ω, ¯ F¯ , P) ¯ and a random variable valued random variables {χ ¯n , n ∈ N}, on (Ω, ¯ → E1 × E2 such that χ∗ : Ω (i): Law(χ ¯n ) = Law(χn ), ∀n ∈ N; (ii): χ ¯n → χ∗ in E1 × E2 a.s. ¯ (iii): π1 ◦ χ ¯n (¯ ω ) = π1 ◦ χ∗ (¯ ω ) for all ω ¯ ∈ Ω. Proof of Theorem E.1: The proof is a modification of the proof of [28, Chapter 2, Theorem 2.4]. For simplicity, let us put µn := Law(χn ), µ1n := Law(π1 ◦ χn ), n ∈ N, and µ∞ := limn→∞ L(χn ). We will generate families of partitions of E1 and E2 . To start with let {xi , i ∈ N} and {yi , i ∈ N} be dense subsets in E1 and E2 , respectively, and let {rn , n ∈ N} be a sequence of natural numbers convergent to zero. Some additional condition on the sequence will be given below. k−1 1 Ai for k ≥ 2. SimNow, let A11 := B(x1 , r1 )17, A1k := B(xk , r1 ) \ ∪i=1 k−1 1 Ci for k ≥ 2. Inductively, ilarly, C11 := B(y1 , r1 ), Ck1 := B(yk , r1 ) \ ∪i=1 we put ∩ B(xik , rk ), Aki1 ,...,1 := Aik−1 1 ,...,ik−1 Aki1 ,...,ik

ik −1 k , ∩ B(x , r ) \ ∪ A := Aik−1 i k i ,...,i ,j k j=1 ,...,i 1 k−1 1 k−1

k ≥ 2,

and similarly, where we will replace ”A” by ”C” ∩ B(yik , rk ), Cik1 ,...,1 := Cik−1 1 ,...,ik−1 Cik1 ,...,ik

ik −1 k ∩ B(y , r ) \ ∪ C := Cik−1 ik k j=1 i1 ,...,ik−1 ,j , 1 ,...,ik−1

k ≥ 2.

For simplicity, we renumerate for any k ∈ N these families and call them now (Aki )i∈N , and (C k )j∈N . ¯ be the Lebesgue measure on [0, 1) × [0, 1). In ¯ := [0, 1) × [0, 1) and P Let Ω the first step, we will construct a family of partition consisting of rectangles ¯ in Ω. 17For r > 0 and x let B(x, r) := {y, |y| ≤ r}.

72


Definition E.2. Suppose that µ is a Borel probability measure on E = E1 × E2 and µ1 is the marginal of µ on E1 , i.e. µ1 (A) := µ(A × E2 ), A ∈ B(E1 ). Assume that (Ai )i∈N and (Ci )i∈N are partitions of E1 and E2 , respectively. Define the following partition of the square [0, 1) × [0, 1). For i, j ∈ N we put " ! i [ i−1 [ Aα , µ1 Iij := µ1 Aα α=1

α=1 j−1 [

! j [ 1 1 Cα , Cα . µ Ai × µ Ai × × µ1 (Ai ) µ1 (Ai ) α=1 α=1 "

(E.1)

Remark Obviously, if µ1 (Ai ) = 0 for some i ∈ N, then we have Iij = ∅ for all j ∈ N. l,n Next for fixed l ∈ N and n ∈ N∪{∞}, we will define a partition Iij

i,j∈N

¯ = [0, 1) × [0, 1) corresponding to partitions (Al )i∈N and (C l )i∈N of the of Ω i i spaces E1 and E2 , respectively. We denote by µ(A|C) the conditional probability of A under C. Then, we have for n ∈ N ∪ {∞} 1,n I1,1 := 0, µ1n (A11 ) × 0, µn (E1 × C11 | A11 × E2 ) , 1,n := µ1n (A11 ), µ1n (A11 ) + µ1n (A12 ) × 0, µn (E1 × C11 | A12 × E2 ) I2,1

... ... 1,n Ik,1

:=

" k−1 X

µ1n (A1k ),

k X

m=1

m=1

!

µ1n (A1k )

× 0, µn (E1 × C11 | A1k × E2 ) ,

k ≥ 2,

and 1,n I1,2 := 0, µ1n (A11 ) × µn (E1 × C11 | A11 × E2 ), µn (E1 × C11 | A11 × E2 ) + µn (E1 × C21 | A11 × E2 , 1,n I2,2 := µ1n (A11 ), µ1n (A11 ) + µ1n (A12 ) × µn (E1 × C11 | A11 × E2 ), µn (E1 × C11 | A11 × E2 ) + µn (E1 × C21 | A11 × E2 .

More generally, for k ∈ N ! " k−1 k X X 1,n µ1n (A1k ) × µ1n (A1k ), Ik,2 :=

m=1 µn (E1 × C11

m=1

|

A11

× E2 ), µn (E1 × C11 | A11 × E2 ) + µn (E1 × C21 | A11 × E2 ) ,


and, for k, r ∈ N ! " k−1 k X X 1,n 1 1 1 1 µn (Am ) × µn (Am ), Ik,r := m=1

m=1

" r−1 X

73

r X

1 µn (E1 × Cm | A1k × E2 ),

!

µn (E1 × Cr1 | A1k × E2 ) .

m=1

m=1

Finally, for k, l, r ≥ 2 (E.2) l,n Ik,r :=

" k−1 X

µ1n (Alm ),

k X

!

µ1n (Alm )

m=1

m=1 " r−1 X

l | Alk × E2 ), µn (E1 × Cm

×

r X

!

l | Alk × E2 ) . µn (E1 × Cm

m=1

m=1

l , k, r ∈ N} are Let us observe that for fixed l ∈ N, the rectangles {Ik,r l , k, r ∈ N} is a covering of Ω. ¯ Therefore, pairwise disjoint and the family {Ik,r we conclude that for any n ∈ N ∪ {∞} we have µn (E1 × E2 ) = 1 and P l l m∈N µn (E1 × Cm | Ak ) = 1. Hence, as a consequence, it follows that for fixed l, n ∈ N the family of sets {Ik,r , k, r ∈ N} is a covering of [0, 1) × [0, 1) and consists of disjoint sets. are probability measures. ¯ → E1 × E2 , The next step is to construct the random variables χ ¯n : Ω such that Law(χ ¯n ) = Law(χn ). We assume that rm is chosen in such a way, that the measure of the boundaries of the covering (Aj )j∈N and (Cj )j∈N are m zero. Now, in each non-empty sets int(Am j ) and int(Cj ) we choose points m xm j and yj , respectively, from the dense subsets {xi , i ∈ N} and {yi , i ∈ N} and define the following random variables. First, we put for m ∈ N m,n 1 ¯ ∈ Ik,r , Zn,m (¯ ω ) = xm k if ω m,n 2 Zn,m (¯ ω ) = yrm if ω ¯ ∈ Ik,r ,

n ∈ N ∪ {∞},

and then, for n ∈ N ∪ {∞} χ ¯1n (¯ ω) =

m→∞

1 lim Zn,m (¯ ω ),

χ ¯2n (¯ ω) =

m→∞

2 lim Zn,m (¯ ω ).

Due to the construction of the partition, the limits above exist. To be ¯ we have precise, for any n ∈ N ∪ {∞} and ω ¯ ∈ Ω, (E.3)

i i (¯ ω )| ≤ rm , |Zn,m (¯ ω ) − Zn,k

k ≥ m, i = 1, 2,

¯ = [0, 1) × and therefore (Zn,m (¯ ω ))m≥1 is a Cauchy-sequence for all ω ¯ ∈Ω [0, 1). Hence, for i, m Zni (¯ ω ) is well defined. Furthermore, χn is measurable,

74


i ω) are simple functions, hence measurable. Therefore, for i, m Zni (¯ since Zn,m 1 2 is a random variable. Let χ ¯n := (χ ¯n , χ ¯n ). Finally we have to proof that the random variables χ ¯ and χ ¯n have the following three properties:

(i): Law(χ ¯n ) = Law(χn ), ∀n ∈ N, (ii): χn → χ a.s. in E1 × E2 , (iii): π1 ◦ χn (ω) = π1 ◦ χ∗ (ω). Proof of (i): The following identity for the rectangle Alk × Ckl holds l,n ) = µ1n (Alk ) × µn (E1 × Crl | Alk × E2 ) λ χ ¯n ∈ Alk × Ckl = λ(Ik,r

= µn (Alk × E2 ) × µn (E1 × Crl | Alk × E2 ) = µn (u1 ∈ Alk and u2 ∈ Crl ) = µ1 ((u1 , u2 ) ∈ Alk × Crl ).

Now, one has to use that the set of rectangles of [0, 1) × [0, 1) form a π system in B([0, 1) × [0, 1)). Moreover, λ and µ1 are identical on the set of rectangles. Therefore, by Lemma 1.17 [42, Chapter 1], λ and µ are equal on B([0, 1) × [0, 1)). Proof of (ii): We will first prove that there exists a random variable χ = (χ1 , χ2 ) such that χ ¯1n → χ1 and χ ¯2n → χ2 λ-a.s. for n → ∞. For this it ¯2n , n ∈ N} are is enough to show that the sequences {χ ¯1n , n ∈ N} and {χ λ–a.s. Cauchy sequences. From the triangle inequality we infer that for all n, m, j ∈ N, i = 1, 2 i i i i i i χ + Zm,j − χ + Zn,j − Zm,j ¯im . ¯n − Zn,j ¯n − χ ¯im ≤ χ

1 ,j ∈ Let us first observe that by (E.3), for any n ∈ N, the sequences {Zn,j 2 , j ∈ N} converge uniformly on Ω ¯ to χ N} and {Zn,j ¯1n and χ ¯2n , respectively Therefore, it suffices to show that for all ε > 0 there exists a number n0 such that (see [31, Lemma 9.2.4])

λ ({ω ∈ [0, 1) × [0, 1) | Zn,l (ω) 6= Zm,l (Ω), n, m ≥ n0 }) ≤ ε Since {µn , n ∈ N} converges weakly, for any δ > 0 there exists a number n0 ∈ N such that18. ρ(µn , µm ) ≤ ε for all n, m ≥ n0 . Hence, for any δ > 0 we can find a number n0 ∈ N such that (E.4) i,l,n i,l,m i,l,m i,l,n ak,r − ak,r , bk,r − bk,r ≤ δ,

k, r, l ∈ N, i = 1, 2, n, m ≥ n0 ,

18ρ denotes the Prohorov metric on measurable space (E, E ), i.e. ρ(F, G) := inf{ε >

0 : F (Aε ) ≤ G(A) + ε, A ∈ E }


75

l,n 1,l,n 2,l,n 2,l,n l,m 1,l,m 1,l,m 2,l,m 2,l,m where Ik,r = [a1,l,n k,r , bk,r )×[ak,r , bk,r ) and Ik,r = [ak,r , bk,r )×[ak,r , bk,r ). l,n l we have In fact, by the construction of Ik,r and Ik,r k−1 l 1 ∪ A a1,l,n = µ n j=1 j , n, m ∈ N, r, k ∈ N, k,r k−1 l 1 ∪ A a1,l,m = µ j=1 j , n, m ∈ N, r, k ∈ N. k,r

Let us fix δ > 0 and let us choose n0 such that ρ(µn , µm ) ≤ δ, n, m ≥ n0 . Then δ k−1 l k−1 l k−1 l 1 1 Aj + δ, r, k, l ∈ N. ∪j=1 Aj ≤ µ1m ∪j=1 µn ∪j=1 Aj ≤ µn On the other hand, by symmetricity of the Prohorov metric, δ k−1 l k−1 l k−1 l 1 1 Aj + δ, r, k ∈ N. ≤ µ1n ∪j=1 ∪j=1 Aj µm ∪j=1 Aj ≤ µm

Hence, we infer that 1,l,n 1,l,m (E.5) ak,r − ak,r ≤ δ,

r, k, l ∈ N,

n, m ≥ n0 .

The second inequality in (E.4) can be shown similarly. Since the sequence {µn , n ∈ N} is tight on E1 × E2 we can find a compact set K1 × K2 such that ε sup (µn ((E1 × E2 ) \ (K1 × K2 ))) ≤ . 2 n Let us fix l ∈ N. Since the set K1 × K2 is compact, from the covering (Alk × Crl )k∈N,r∈N of E1 × E2 there exists a finite covering (Alk × Crl )k=1,··· ,K,j=1,··· ,R of K1 × K2 . Next, observe that the estimate (E.4) is uniformly for all n, m ≥ n0 . Therefore we can use estimate (E.4) with δ = ε/2(KR) and infer that K,R ε X l,n l,m λ Ik,r △ Ik,r , n, m ≥ n0 ≤ . 2 k,r=1

Moreover, since

λ ({ω ∈ [0, 1) × [0, 1) : Zn,l (ω) 6= Zm,l (ω), n, m ≥ n0 }) ≤ X l,n l,m λ Ik,r △ Ik,r , n, m ≥ n0 + λ ((E1 × E2 ) \ (K1 × K2 )) , k,r=1

it follows that

ε ε + . 2 2 Summarizing, we proved that for any ε > 0 there exists a number n0 ∈ N such that {ω ∈ [0, 1) × [0, 1) | Zn,l (ω) 6= Zm,l (Ω), n, m ≥ n0 } ≤ ε. Applying then [31, Lemma 9.2.4] we infer (ii). λ ({ω ∈ [0, 1) × [0, 1) : Zn,l (ω) 6= Zm,l (ω), n, m ≥ n0 }) ≤

76


Proof of (iii): Let us denote ! " k−1 k X X l,n 1 l 1 l (E.6) µn (Ak ) , µn (Ak ), Jk := m=1

k = 1, . . . , Nl1 .

m=1

Since the laws of π1 ◦ χn and π1 ◦ χn , n, m ∈ N ∪ {∞} are equal, Jkl,n = Jkl,m , n, m ∈ N ∪ {∞}. Let us denote these (equal) sets Jkl . Since for each set Jkl we can find a set Alk satisfying the relation (E.6) for all n ∈ N ∪ {∞}, we infer that for any m ∈ N, 1 1 Zn,m (¯ ω ) = Z1,m (¯ ω ),

ω ¯ ∈ Ω, n ∈ N ∪ {∞}.

1 (¯ ω ), m ∈ Let n ∈ N be fixed. Considering now the limit for the sequence {Zn,m 1 ω ), m ∈ N} is a Cauchy N} for m → ∞ and keeping in mind that {Z1,m (¯ sequence implies the assertion (iii).

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