Modelling Levy space-time white noises

Based on the theory of independently scattered random measures, we introduce a natural generalisation of Gaussian space-time white noise to a Levy-type setting, which we call Levy-valued random measures. We determine the subclass of cylindrical Levy processes which correspond to Levy-valued random measures, and describe the elements of this subclass uniquely by their characteristic function. We embed the Levy-valued random measure, or the corresponding cylindrical Levy process, in the space of general and tempered distributions. For the latter case, we show that this embedding is possible if and only if a certain integrability condition is satisfied. Similar to existing definitions, we introduce Levy-valued additive sheets, and show that integrating a Levy-valued random measure in space defines a Levy-valued additive sheet. This relation is manifested by the result, that a Levy-valued random measure can be viewed as the weak derivative of a Levy-valued additive sheet in the space of distributions.


Introduction
Gaussian random perturbations of partial differential equations are most often modelled either as a cylindrical Brownian motion or a Gaussian space-time white noise. The choice usually depends on the exploited method, in which one follows either a semi-group approach, based on the work by Da Prato and Zabczyk in [13], or a random field approach, originating from the work by Walsh in [42]. It is well known that both models essentially result in the same dynamics as established by Dalang and Quer-Sardanyons in [16].
Cylindrical Brownian motions can be naturally generalised to cylindrical Lévy processes by exploiting the theory of cylindrical measures and random variables. This was accomplished by one of us together with Applebaum in [2]. In the random field approach, Gaussian spacetime white noise is generalised to Lévy space-time white noise as an infinitely divisible random measure, often represented by integrals with respect to Gaussian and Poisson random measures. Both generalisations, cylindrical Lévy processes and Lévy space-time white noises, serve as a model for random perturbations of complex dynamical systems. These applications can be found for cylindrical Lévy processes, for example, in the monograph in Peszat and Zabczyk [34] or in Kumar and Riedle [30], and for Lévy space-time white noise in Applebaum and Wu [3], Chong [11], Chong and Kevei [12] and Dalang and Humeau [15], among many others. Another approach to model such perturbed dynamical systems, for example, parabolic stochastic partical differential equations, is provided by the recently introduced ambit fields, presented in the monograph [6] by Barndorff-Nielsen, Benth and Veraart, and their relations to SPDE investigated in [7] by the same authors.
The main objectives of our work are the comparison of cylindrical Lévy processes and Lévy space-time white noises, as well as their embeddings in the space of general and tempered (Schwartz) distributions. It turns out that these results significantly differ from the Gaussian situation. Only the standard cylindrical Brownian motion corresponds to the Gaussian section, Section 5, is devoted to the comparison of cylindrical Lévy processes and Lévy-valued random measures. Our main results here characterise exactly the subclass of cylindrical Lévy processes which correspond to Lévy-valued random measures. In the last section, Section 6, we complete the picture by establishing Lévy-valued random measures as the weak derivative of Lévy-valued additive sheets. The Lebesgue measure on B(R d ) is denoted by leb. The closed unit ball in R d is denoted by Throughout the paper, we fix a probability space (Ω, A, P ). The space of P -equivalence classes of measurable functions f : Ω → R is denoted by L 0 (Ω, P ), and of pth integrable functions by L p (Ω, P ) for p > 0. These spaces are equipped with their standard metrics and (quasi-)norms.

Lévy-valued random measures
Our definition of Lévy-valued random measures is based on the work [36] by Rajput and Rosinski. Instead of general δ-rings, it is sufficient for us to restrict ourselves to the δ-ring   For the notion of measures on a ring, see, for example, [23]. We call the triple (γ, Σ, ν) the characteristics of M . Furthermore, we may extend the total variation γ TV of γ and Σ to σ-finite measures on B(O). In this case, the mapping defines a σ-finite measure, which is called the control measure of M . We note that λ( We extend Definition 2.1 to include a dynamical aspect, that is, a time variable. This extension can be thought of as a similar construction to that of Walsh in [42].
and n ∈ N, the stochastic process is a Lévy process in R n . We shall write M (t, A) := M (t)(A).
Let (M (t) : t 0) be a Lévy-valued random measure on B b (O), and suppose (γ, Σ, ν) and λ are the characteristics and control measure, respectively, of the infinitely divisible random measure M (1). Then, it follows from the stationary increments of the process (M (t, A) : t 0) that for each t 0 the characteristics of the infinitely divisible random measure M (t, A) are given by (tγ, tΣ, tν), and the control measure of M (t) is given by tλ. We shall refer to (γ, Σ, ν) as the characteristics of M and λ as the control measure of M .
Our definition above of Lévy-valued random measures assigns a special role to the time domain although this is not necessary for infinitely divisible random measures in general. However, as we will later compare Lévy-valued random measures with cylindrical Lévy processes, which are naturally carrying a time domain as generalised stochastic processes, we found it more illustrative to have the time domain distinguished. Indeed, the following theorem shows that a Lévy-valued random measure corresponds to an infinitely divisible random measure on the product space of time and space domain if the stationarity in the time domain is described by the control measure accordingly.
Proof. This can be proved similarly as [39,Theorem 3.2]  The relation between random measures and models of Lévy-type noise utilising a Lévy-Itô decomposition is well known. We rigorously formulate this result in our setting in the following proposition; for a converse conclusion, see Remark 3.6.
Proposition 2.5. Let ζ be a σ-finite Borel measure on B(O) and (U, U , ν) a σ-finite measure space. Assume that: is Poisson random measure with intensity leb ⊗ ζ ⊗ ν, independent of W , and with compensated Poisson random measure N .
Then for any functions: we define a mapping M : Then we obtain a Lévy-valued random measure on B b (O) by the prescription Proof. The existence of the Gaussian integral is guaranteed by [42,Theorem 2.5] and that of the Poisson integrals by [27,Lemma 12.13]. The characteristic function of M ([0, t] × A), see, for example, in [40,Proposition 19.5], shows that M is an infinitely divisible random measure, and thus applying Proposition 2.3 completes the proof.
Example 2.6. The class of α-stable random measures is introduced, for example, in [38,Section 3.3]. These can be obtained from Proposition 2.5 by defining for bounded sets B in for some p + q = 1 (for the case α = 1, it is required that p = q = 1 2 ); see Balan [5] for this construction. Proposition 2.5 guarantees that, by defining M (t, A) := M ((0, t] × A) for t 0 and A ∈ B b (R d ), we obtain a Lévy-valued random measure M on B b (R d ). Direct calculation shows that for α = 1, the characteristic function of M (t, A) is given by, for t 0, where β := p − q, and thus we see the characteristics of M are (β α 1−α leb, 0, leb ⊗ ν α ). The control measure is given by For the case α = 1, the characteristic function of M (t, A) is given by Example 2.7. Mytnik, in [31], considers a martingale-valued measure (M (t, A) : t 0, A ∈ B b (R d )) in the sense of Walsh [42], such that for any A ∈ B b (R d ), the process (M (t, A) : t 0) is a real-valued α-stable process (α ∈ (1, 2)), with Laplace transform The author terms M an α-stable measure without negative jumps.

Lévy-valued additive sheets
Just as the Brownian sheet is the generalisation of a Brownian motion to a multidimensional index set, additive sheets are defined as the corresponding generalisation of an additive process. Adler et al. [1] first defined additive random fields on R d , and termed them 'Lévy processes' should they be stochastically continuous. In [17], Dalang and Walsh discuss Lévy sheets in R 2 . Additive fields with stationary increments are considered by Barndorff-Nielsen and Pedersen in [8] and are called 'homogeneous Lévy sheets'. Herein we present our definition based on the deposition of Dalang and Humeau in [14], which extends [1], and results from Pedersen [33]. For a function f : where c j (0) = b j and c j (1) = a j . For example, in the case d = 2, we have Δ b The càdlàg property is generalised to random fields in the following way: a function f : R d → R has limits along monotone paths (lamp) if for every x ∈ R d and any sequence (x n ) n∈N ⊆ R d converging to x with either x n,j < x j or x n,j x j for all n ∈ N and j ∈ {1, . . . , d} where x = (x 1 , . . . , x d ) and x n = (x n,1 , . . . , x n,d ), the limit f (x n ) exists as n → ∞ and furthermore f is right-continuous if f (x n ) → f (x) as n → ∞ for all sequences with x x n for all n ∈ N. We note that this property is a path-based property, and thus in contrast to random measures we define our sheets as mappings from R d × Ω → R.  For relaxing the requirements in Definition 3.1, we refer to [1], for example, to capture arbitrary initial conditions or sheets which are not continuous in probability. In particular, it is shown that Conditions (a)-(c) guarantee the existence of a lamp and rightcontinuous modification.
If (X(x) : x ∈ I) is an additive sheet, then for fixed x ∈ I the random variable X(x) is infinitely divisible; see Adler [1, Theorem 3.1]; let its characteristics be denoted by (p x , A x , μ x ). The additive sheet is said to be natural if the mapping x → p x , which is necessarily continuous, is of bounded variation, or equivalently, if there exists an atomless signed measure γ with p x = γ((0, x]) for all x ∈ I; here, we use the convention (0, Similarly as for infinitely divisible random measures, we introduce a dynamical aspect in the following definition: is called a Lévy-valued additive sheet if for every x 1 , . . . , x n ∈ R d and n ∈ N, the stochastic process is a Lévy process in R n .
The wording 'Lévy-valued additive sheet' is motivated by the following result: Proof. The domain of definition and Conditions (a), (b) and (d) of Definition 3.1 are clearly met. Regarding stochastic continuity, let (t n , x n ) n∈N be a sequence in R + × R d converging to (0, x). For each n ∈ N, the random variable X(1, x n ) is infinitely divisible, say with characteristics (p xn , V xn , μ xn ). As X(1, ·) is a natural, additive sheet, there exists a signed measure γ such that p xn = γ((0, x n ]). Since the Lévy process (X(t, x n ) : t 0) has stationary increments, it follows that each X(t, x n ) has characteristics (tp xn , tV xn , tμ xn ) for every t 0. Theorem 3.1 in [1] implies that there exist a measure Σ on B(R d ) such that V xn = Σ((0, x n ]), and a measure ν on B(R d × R) such that, for each B ∈ B(R), the mapping ν(· × B) is a measure on B(R d ), and μ xn = ν((0, x n ] × ·). Therefore, the Lévy symbol of X(t n , x n ) is given by, for u ∈ R, As the set {x n : n ∈ N} is bounded, there exists a bounded box I ⊆ R d containing every box (0, x n ], n ∈ N. Thus, we obtain for each u ∈ R that Finiteness of the right side follows from the fact that the measures are finite on I. Therefore, it follows that X(t n , x n ) → 0 in probability as (t n , x n ) converges to (0, x). If (t n , x n ) is an arbitrary sequence converging to (t, x), stationary increments imply for each c > 0 that . Consequently, the above established continuity in probability shows the general case.
The fact that X(z) is natural can be seen from the form of the characteristic function, where we have p z = tγ((0, x]) for z = (t, x).
We are now able to state the link between Lévy-valued random measures and Lévy-valued additive sheets by formulating a result from Pedersen in [33] in our setting.

Then any lamp and right-continuous modification of the stochastic process
Proof. See [33].
Remark 3.6. Theorem 3.5 and its proof enables us to conclude a converse implication of Proposition 2.5. If M is a Lévy-valued random measure with atomless control measure λ, then it satisfies a Lévy-Itô decomposition of the form Furthermore, we see that one does not achieve larger generality by allowing an arbitrary measure space (U, U , ν) in Proposition 2.5, as the Poissonian components can be represented as integrals over R.

Lévy-valued measures in the space of distributions
In this section, we embed the Lévy-valued random measure into the spaces of distributions and of tempered distributions. These embeddings are based on the integration theory for independently scattered infinitely divisible measures developed by Rajput and Rosinski in [36]. The multiplicative relation between the characteristics of the infinitely divisible random measures M (1) and M (t), remarked after Definition 2.2, enables us to apply directly the integration theory for infinitely divisible random measures to Lévy-valued random measures (M (t) : t 0) on B b (O): for a simple function for α k ∈ R and pairwise disjoint sets An arbitrary measurable function f : O → R is said to be M -integrable if the following hold.
(1) There exists a sequence of simple functions (f n ) n∈N of the form (4.1) such that f n converges pointwise to f λ-a.e., where λ is the control measure of M .
(2) For each A ∈ B(O) and t 0, the sequence ( A f n (x) M (t, dx)) n∈N converges in probability.
In this case, the integral of f is defined as It is clear, by the stationarity of the increments of Lévy processes, that Condition (2) Here, (γ, Σ, ν) denotes the characteristics of M . The measure Furthermore for all t 0, the mapping For an open set O ⊆ R d , let D(O) denote the space of infinitely differentiable functions with compact support. We equip D(O) with the inductive topology, that is, the strict inductive limit of the Fréchet spaces D( is called the space of distributions, which we equip with the strong topology, that is the topology generated by the family of seminorms Analogously as locally integrable functions and measures are identified with distributions, we proceed to relate a Lévy-valued random measure M on B b (O) to a distribution-valued process. For this purpose, we define for each t 0 the integral mapping In the proof of Theorem 4.1 below, we show that D(O) is continuously embedded in L M (O, λ), and thus the mapping J D (t) is well defined.
In the following theorem as in the reminder of the article, we use the phrase genuine Lévy process in a space F to emphasise that this is a Lévy process in the space  Our proof of this theorem relies on the following two Lemmas.
is a Lévy process in R n .
Proof. Let f k for k = 1, . . . , n be simple functions of the form for α k,j ∈ R and A k,j ∈ B b (O) with A k,1 , . . . , A k,m k disjoint for each k ∈ {1, . . . , n}. By taking the intersections of all possible permutations of the sets A k,j , we can assume that for all k = 1, . . . , n, whereα k,j ∈ R and disjoint setsÃ 1 , . . . ,Ã m ∈ B b (O) for some m ∈ N. For each 0 t 1 < · · · < t n , we obtain by the definition in (4.2) that .
and thus the stochastic continuity of J(·)f implies that of ((J(t)f 1 , . . . , J(t)f n ) : t 0). Consequently, the latter is verified as an n-dimensional Lévy process. Proof. Denote the characteristics of M by (γ, Σ, ν). Note, that for arbitrary g ∈ L p (O, λ) and p ∈ [1, 2], we have Furthermore we obtain from the definition of U and (4.9), recalling that Let C := sup{|y| : (x, y) ∈ K}. Define for n ∈ N, x ∈ O and y ∈ R functions g n (x, y) := f n (x). Since (f n ) converges in L 1 (O, λ) it follows from (4.9) that (g n ) converges to 0 in L 1 (O × B c R , ν), and thus in ν 1 -measure. Consequently, there exists N ∈ N such that, for n N , which shows the claim.
Consequently, it follows from (4.10) that (f n ) converges in L M (Φ, λ), which completes the proof.
Proof of Theorem 4.1. We first show that the space D(K) is continuously embedded in L M (O, λ) for each compact K ⊆ O. Trivially, the space D(K) is continuously embedded in L ∞ (K, λ). As K ∈ B b (O), the control measure λ is finite on K, and it follows that L ∞ (K, λ) is continuously embedded in L 2 (K, λ). The latter is continuously embedded in L M (K, λ) by Lemma 4.3. Because whenever supp(f ) ⊆ K, we have In the second part of this section, we embed the Lévy-valued random measure into the space of tempered distribution S * (R d ). We introduce the Schwartz space Define for each t 0 the integral mapping   In this case, the mapping J S (t) as defined in (4.11) is well defined and continuous for each t 0. Furthermore, there exists a genuine Lévy process (Y (t) : λ). Let (f n ) n∈N ⊆ S(R d ) be a sequence converging to 0 in S(R d ). As the convergence is uniform in x, we have the existence of another K > 0 such that (

Proof. We begin by showing the implication (b) ⇒ (a), for which we suppose there exists
which completes the proof of the implication (b) ⇒ (a). Conversely, suppose S(R d ) is continuously embedded in L M (R d , λ). Thus, the identity mapping ι : λ) is continuous. Then, there exists a neighbourhood for some k ∈ N and δ > 0 such that ι maps U (0; k, δ) into the open unit ball of L M (R d , λ). Let (f n ) n∈N ⊆ S(R d ) be any sequence such that f n S k → 0. Then, (f n ) is eventually in U (0; k, δ) and thus (ιf n ) is eventually in the unit ball and so is bounded in L M (R d , λ). By [25,Proposition 4,p. 41], we have the continuity of ι in the semi-norm · S k , and thus we may extend ι by continuity to the completion of S(R d ) in this semi-norm. We thus obtain the integrability condition by observing that the C ∞ (R d ) mapping x → (1 + |x| 2 ) r has finite semi-norm · S k for r −k.
As in the proof of Theorem 4.1, an application of [21, Lemma 4.2 and Theorem 3.8] establishes the existence of the Lévy process Y in S(R d ).
Remark 4.5. In Kabanava [26], it is shown that a Radon measure ζ can be identified with a tempered distribution in S * (R d ) if and only if there is a real number r such that r is integrable over R d with respect to ζ. Our condition for the mapping J S in Theorem 4.4 is analogous. Remark 4.7. In a series of papers, for example, [4, 14 18, 19], Dalang, Humeau, Unser and co-authors have studied the Lévy white noise Z defined as a distribution. Here, Z is defined as a cylindrical random variable in D * (R d ), that is, a linear and continuous mapping Z : D(R d ) → L 0 (Ω, P ), with characteristic function for some constants p ∈ R and σ 2 ∈ R + and a Lévy measure ν 0 on R.
Let M be a Lévy-valued random measure on B b (R d ) with characteristics (γ, Σ, ν) and J D (t) the corresponding operator defined in (4.7) for t 0. By comparing the Lévy symbol in (4.6) with (4.12), it follows that, for fixed t 0, the mapping J D (t) is a Lévy white noise in the above sense, if and only if γ = p · leb, Σ = σ 2 · leb, ν = leb ⊗ ν 0 , for some p ∈ R, σ 2 ∈ R + and a Lévy measure ν 0 on R. It follows that M (t, A) D

= M (t, B) for any sets A, B ∈ B b (R d ) with leb(A) = leb(B). In this case, we call M stationary in space.
Dalang and Humeau have shown in [14] that a Lévy white noise in D * (R d ) with Lévy symbol (4.12) takes values in S * (R d ) P -a.s. if and only if This result is analogous to our Theorem 4.4. However, as Lévy-valued random measures are not necessarily stationary in space, our condition is more complex. For example, even in the pure Gaussian case with characteristics (0, Σ, 0), the measure Σ must be tempered; cf. Remark 4.5.
Regularity of the Lévy white noise Z in terms of Besov spaces is studied in [4]. Their results can be applied to a Lévy-valued random measure if it is additionally assumed to be stationary in space, that is, which can be considered as a Lévy white noise in the above sense. We illustrate such an application in the following example.
Example 4.8. Let M be the α-stable random measure, α ∈ (0, 2), described in Example 2.6. For simplicity we consider the symmetric case, that is, p = q = 1 2 . As the characteristics of M is given by (0, 0, leb ⊗ ν α ), it follows that M is stationary in space. Thus, for a fixed time t 0, the mapping J D (t) or, equivalently the random variable Y (t), where Y denotes the Lévy process derived in Theorem 4.1, can be considered as a Lévy white noise in D * (R d ); see Remark 4.7. Furthermore, since R (|y| ε ∧ |y| 2 ) ν α (dy) < ∞ for ε < α, we have that Y (t) is in S * (R d ) P -a.s. By applying the results from [4], we obtain the following: for p ∈ (0, 2) ∪ 2N ∪ {∞} and for all t 0, we have, almost surely: where B τ p (R d , ρ) is the weighted Besov space of integrability p, smoothness τ and asymptotic growth rate ρ. Furthermore, a modification of Y is a Lévy process in any Besov space satisfying (4.13), since its characteristic function is continuous in 0, guaranteeing stochastic continuity.

Cylindrical Lévy processes
The concept of cylindrical Lévy processes in Banach spaces is introduced in [2]. It naturally generalises the notation of cylindrical Brownian motion, based on the theory of cylindrical measures and cylindrical random variables. Here, a cylindrical random variable Z on a Banach space F is a linear and continuous mapping Z : In many cases, we will choose F = L p (O, ζ) for some p 1 and an arbitrary locally finite Borel measure ζ. In this case, F * = L p (O, ζ) for p := p p−1 . The characteristic function of a cylindrical Lévy process (L(t) : t 0) is given by for all t 0. Here, Ψ L : F * → C is called the (cylindrical) symbol of L, and is of the form where a : F * → R is a continuous mapping with a(0) = 0, the mapping Q : F * → F * * is a positive, symmetric operator and μ is a finitely additive measure on Z(F ) satisfying Here, Z(F ) is the algebra of all sets of the form {g ∈ F : ( g, f 1 , . . . , g, f n ) ∈ B} for some f 1 , . . . , f n ∈ F * , B ∈ B(R n \ {0}) and n ∈ N. We call (a, Q, μ) the (cylindrical) characteristics of L.
defines a cylindrical Lévy processes L in F . In this case, the characteristics (a, Q, μ) of L is given by , dy), Proof. Lemma 4.2 shows that L is a cylindrical Lévy process in F . The claimed characteristics follows from (4.6) after rearranging the terms accordingly.
The integration theory developed in [36] and briefly recalled in Section 4 guarantees that (5.1) is well defined for every f ∈ L M . However, in order to be in the framework of cylindrical Lévy processes, we need that the domain of L(t) is the dual of a Banach space (or alternatively of a nuclear space). Since the Musielak-Orlicz space L M is not in general the dual of a Banach space, for the hypothesis of Theorem 5.2 we require the existence of the Banach space F with F * continuously embedded in L M . If the control measure λ of M is finite on O, then Lemma 4.3 gives us that L 2 (O, λ) is continuously embedded in L M (O, λ). It is possible as illustrated in the following example to relax the condition on finiteness of λ, but also the same example shows that there are cases where the finiteness of λ is necessary for any L p space to be continuously embedded. Assume α ∈ (1, 2). Then Theorem 5.2 implies that (5.1) defines a cylindrical Lévy process L in F = L α (O, leb), and its symbol is given by We now turn to the question of which cylindrical Lévy processes induce Lévy-valued random measures. For this purpose, we introduce the following:  Proof. If L is independently scattered, then Theorem 5.5 implies that L defines a Lévyvalued random measure M by (5.2). Denote the characteristics of M by (γ, Σ, ν) and its control measure by λ. For a simple function f of the form (4.1), we obtain For an arbitrary function f ∈ L p (O, ζ), let (f n ) n∈N be a sequence of simple functions converging to f both pointwise ζ-almost everywhere and in L p (O, ζ). We note that, as L(t)1 A = 0 whenever ζ(A) = 0, ζ-null sets have null λ-measure, and thus we have (t, dx). We obtain the stated form of the characteristic function of L by (4.6). Conversely, if the Lévy symbol is given by (5.3), then this form implies for any disjoint sets for all u 1 , . . . , u n ∈ R.
Applying Theorem 5.5 to a given cylindrical Lévy process L on L p (O, ζ) gives the corresponding Lévy-valued random measure M , say with control measure λ. The first part of the proof of Theorem 5.6 shows that L p (O, ζ) is a subspace of L M (O, λ). The following result guarantees that the embedding is continuous in non-degenerated cases. Let (f n ) be a sequence in L p (O, ζ) converging to f ∈ L p (O, ζ) and assume that ιf n converges to some g ∈ L M (O, λ). As lim n→∞ J(t)(ιf n ) = J(t)g and lim n→∞ L(t)f n = L(t)f = J(t)(ιf ), we derive J(t)(g − ιf ) = 0. Since J(t) is injective, we conclude g = ιf λ-a.e., and the closed graph theorem implies the continuity of ι.
where N is a Poisson random measure on R + × O × R with intensity leb ⊗ ζ ⊗ μ for a Lévy measure μ on B(R); see also [2,Example 3.6]. Since its symbol is given by Theorem 5.6 guarantees that L is independently scattered.
The Lévy measure of the Lévy process ((L(t)1 A , L(t)1 B ) : t 0) in R 2 is given by μ • π −1 1A,1B . As L(1)1 A and L(1)1 B are independent, it follows from the uniqueness of the characteristic functions that where μ • π −1 1A is the Lévy measure of (L(t)1 A : t 0) and μ • π −1 1B is the Lévy measure of (L(t)1 B : t 0). It follows in particular that On the other hand, [37,Lemma 4.2] implies where r k : R → R 2 is defined by r k (x) = ( 1 A , e k x, 1 B , e k x). It follows from (5.5) that which results in a contradiction.

Weak derivative of a Lévy-valued random measure
In this last section, we establish the relation between a Lévy-valued random measure and a Lévy-valued additive sheet. For this purpose, we introduce a stochastic integral of deterministic functions f : R d → R with respect to a Lévy-valued additive sheet. Instead of following the standard approach starting with simple functions and extending the integral operator by continuity, we utilise the correspondence between Lévy-valued additive sheets and Lévy valued random measures, established in Theorem 3.5, and refer to the integration for the latter developed in Rajput and Rosinksi [36] as presented in Section 4. For a Lévy-valued additive sheet (X(t, x) : t 0, x ∈ R d ), let M denote the corresponding Lévy-valued random measure on B b (R d ) with control measure λ. Then we define for all f ∈ L M (R d , λ), A ∈ B(R d ) and t 0: In other words, if we neglect the embedding by the operators I D and J D , we could interpret this result that M is the weak derivative of X. This is not very surprising, since, if we adapt notions from classical measure theory, the relation M (t, (0, x]) = X(t, x) derived in Theorem 3.5, can be seen that X is the cumulative distribution function of the random measure M . Proof. We show that, for each f ∈ D(O), the process (I D (t)f : t 0) has a càdlàg modification. First we consider a sequence (t n ) decreasing monotonically to some t 0. Let K be the support of f . Then, as (t n ) is bounded, there exists a C > 0 such that t n ∈ [t, t + C] for each n. The lamp property of X implies that X is bounded on the compact set [t, t + C] × K.
Thus, since X(t n , x) converges to X(t, x) in probability for each x ∈ O, Lebesgue's dominated convergence theorem (for a stochastically convergent sequence) implies To show (6.4), we use ideas from [14]. By the fundamental theorem of calculus, as f has compact support,