Wavelet analysis of the Besov regularity of Lévy white noise

We characterize the local smoothness and the asymptotic growth rate of the Lévy white noise. We do so by characterizing the weighted Besov spaces in which it is located. We extend known results in two ways. First, we obtain new bounds for the local smoothness via the Blumenthal-Getoor indices of the Lévy white noise. We also deduce the critical local smoothness when the two indices coincide, which is true for symmetric-α-stable, compound Poisson, and symmetric-gamma white noises to name a few. Second, we express the critical asymptotic growth rate in terms of the moment properties of the Lévy white noise. Previous analyses only provided lower bounds for both the local smoothness and the asymptotic growth rate. Showing the sharpness of these bounds requires us to determine in which Besov spaces a given Lévy white noise is (almost surely) not. Our methods are based on the wavelet-domain characterization of Besov spaces and precise moment estimates for the wavelet coefficients of the Lévy white noise.


Introduction and main results
We study the Besov regularity of Lévy white noises. We are especially interested in identifying the critical local smoothness and the critical asymptotic growth rate of those random processes for any integrability parameter p ∈ (0, ∞]. In a nutshell, our contributions are as follows.
1. Wavelet Methods for Lévy White Noises. First appearing in the eighties, especially in the works of Y. Meyer [59], I. Daubechies [24], and S. Mallat [58], wavelet techniques have become primary tools in functional analysis [80]. As such, they are a natural choice to study random processes, as is done, for instance, with fractional Brownian motion [60], SαS processes [64], and with solutions of singular stochastic partial differential equations [43,44]. In this paper, we demonstrate that wavelet methods are also adapted to the analysis of the Lévy white noise. In particular, all our results are derived using the wavelet characterization of weighted Besov spaces.
2. New Moment Estimates for Lévy White Noise. The wavelet method allows us to obtain lower and upper bounds for the moments of a Lévy white noise as a function of the wavelet scale. Our moment estimates, which are new contributions to the rich literature on the moments of Lévy and Lévy-type processes [25,53,54,56,57], are fundamental for our study of the Besov regularity of the Lévy white noise.
3. Besov Regularity of Lévy White Noises. Regularity properties are usually stated in terms of the inclusion of the process in some weighted Besov spaces (positive result). In order to show that such a characterization is sharp, it is of interest to identify the smoothness spaces in which the process is not included (negative result). To the best of our knowledge, very little is known in this direction. A precise answer to this question requires a more evolved analysis as compared to positive results. We achieve this goal thanks to the use of wavelets.
It is worth noting that our analysis requires the identification of a new index associated to a Lévy white noise, characterized by moment properties. By relying on this index, our negative results suggest moreover that some of the previous state-of-the-art inclusions are not sharp. We are then able to improve some of these results, in particular for the growth properties of the Lévy white noise.

Critical Local Smoothness and Asymptotic
Rate. The combination of positive and negative results allows us to determine the critical Besov parameters of a Lévy white noise, both for the local smoothness and the asymptotic behavior. The results are summarized in Theorem 1.1. Two consequences are the characterization of the critical Sobolev and Hölder-Zigmund regularities of the Lévy white noise in Corollary 1.2.

Local smoothness and asymptotic rate of tempered generalized functions
We construct random processes as random elements in the space S (R d ) of tempered generalized functions from R d to R (see Section 2.1). We therefore describe their local and asymptotic properties as we would for a (deterministic) tempered generalized function. To do so, we rely on the family of weighted Besov spaces. They are embedded in S (R d ) and allow for the joint study of the local smoothness and the asymptotic behavior of a generalized function.
Besov spaces are denoted by B τ p,q (R d ), with τ ∈ R the smoothness, p ∈ (0, ∞] the integrability parameter, and q ∈ (0, ∞] a secondary parameter. In this paper, we focus on the case p = q and we use the simplified notation B τ p,p (R d ) = B τ p (R d ) for those spaces which are also referred to as Slobodeckij spaces after [78]. See [63] or [81, Section 2.2.1] for more details. We say that f is in the weighted Besov space B τ p (R d ; ρ) with weight exponent ρ ∈ R if · ρ × f is in the classic Besov space B τ p (R d ), with the notation x = (1 + x 2 ) 1/2 . We precisely define weighted Besov spaces in Section 2.4 in terms of wavelet expansions. For the time being, it is sufficient to remember that the space of tempered generalized functions satisfies [51,Proposition 1] Wavelet analysis of the Besov regularity of Lévy white noise for which this is true. For a fixed p, Besov spaces are continuously embedded in the sense that, for τ, τ 0 , τ 1 ∈ R and ρ, ρ 0 , ρ 1 ∈ R such that τ 0 ≥ τ 1 and ρ 0 ≥ ρ 1 , we have B τ0 p (R d ; ρ) ⊆ B τ1 p (R d ; ρ) and B τ p (R d ; ρ 0 ) ⊆ B τ p (R d ; ρ 1 ). (1.2) To characterize the properties of f ∈ S (R d ), the key is to determine the two critical exponents τ p (f ) ∈ (−∞, ∞] and ρ p (f ) ∈ (−∞, ∞] such that • if τ < τ p (f ) and ρ < ρ p (f ), then f ∈ B τ p (R d ; ρ); while • if τ > τ p (f ) or ρ > ρ p (f ), then f / ∈ B τ p (R d ; ρ).
The case τ p (f ) = ∞ corresponds to infinitely smooth functions, and ρ p (f ) = ∞ means that f is rapidly decaying. The quantity τ p (f ) measures the local smoothness and ρ p (f ) the asymptotic rate of f for the integrability parameter p. When ρ p (f ) ≤ 0 (which will be the case for the Lévy white noise), we talk about the asymptotic growth rate of f .

Local smoothness and asymptotic growth rate of Lévy white noises
The complete family of Lévy white noises, defined as random elements in the space of generalized functions, was introduced by I.M. Gel'fand and N.Y. Vilenkin [41]. Recently, R. Dalang and T. Humeau completely characterized the Lévy white noises located in the space S (R d ) of tempered generalized functions [20]. We briefly introduce in this section the concepts required to state our main results. A more complete exposition is given in Section 2.
Our main contributions concern the inclusion of a Lévy white noise w in weighted Besov spaces. It includes positive (w is almost surely in a given Besov space) and negative (w is almost surely not in a given Besov space) results. In order to characterize the local smoothness τ p (w) and the asymptotic growth rate ρ p (w), let X = w, 1 [0,1] d be the random variable that corresponds to the integration of the Lévy white noise w over the domain [0, 1] d . The characteristic exponent Ψ of w is the logarithm of the characteristic function of X. More precisely, for every ξ ∈ R, Ψ(ξ) = log E e iξ w,1 [0,1] d = log E e iξX . (1.3) We associate to a Lévy white noise its Blumenthal-Getoor indices, defined as β ∞ = inf p > 0 lim |ξ|→∞ |Ψ(ξ)| |ξ| p = 0 , (1.4) β ∞ = inf p > 0 lim inf |ξ|→∞ |Ψ(ξ)| |ξ| p = 0 . (1.5) The distinction is that β ∞ considers the limit, whileβ ∞ deals with the inferior limit. In general, one has that 0 ≤β ∞ ≤ β ∞ ≤ 2. The Blumenthal-Getoor indices are linked to the local behavior of Lévy processes and Lévy white noises (see Section 2.3 for more details). In addition, we introduce the moment index of the Lévy white noise w as (1.6) which is closely related-but in general not identical-to the Pruitt index (see Section 2.3).
As we shall see, p max ∈ (0, ∞] fully characterizes the asymptotic growth rate of w. The class of Lévy white noises is rich, and includes Gaussian and compound Poisson white noises. We summarize the results of this paper in Theorem 1.1. We use the convention that 1/p = 0 when p = ∞. Theorem 1.1. Consider a Lévy white noise w with Blumenthal-Getoor indices 0 ≤β ∞ ≤ β ∞ ≤ 2 and moment index 0 < p max ≤ ∞. We fix 0 < p ≤ ∞.
• If w is a Gaussian white noise, then, almost surely, (1.7) • If w is a compound Poisson white noise, then, almost surely, . (1.8) • If w is a Lévy white noise and not a Gaussian white noise, then, almost surely, In particular, if p ≥ β ∞ , then τ p (w) = d/p − d. • If w is a Lévy white noise and not a Gaussian white noise, then, almost surely, if p ∈ (0, 2), p is an even integer, or p = ∞, , (1.10) and for any 0 < p ≤ ∞, In a nutshell, Theorem 1.1 provides: 1. A full characterization of the local smoothness and the asymptotic growth rate for Gaussian and compound Poisson white noises; 2. A characterization of the asymptotic growth rate for any Lévy white noise for integrability parameter p ≤ 2, p an even integer, or p = ∞; and 3. A full characterization of the local smoothness of a Lévy white noise for which β ∞ = β ∞ ; that is, for any p ∈ (0, ∞], (1.12) We discuss the remaining cases-the local smoothness forβ ∞ < β ∞ and the asymptotic growth rate for p > 2, p / ∈ 2N-in Section 8.3. Two direct consequences are the identification of the Sobolev (p = 2) and Hölder-Zigmund (p = ∞) regularity of Lévy white noises. Corollary 1.2. Let w be a Lévy white noise in S (R d ) with Blumenthal-Getoor indices 0 ≤β ∞ ≤ β ∞ ≤ 2 and moment index 0 < p max ≤ ∞. Then, the Sobolev local smoothness and asymptotic growth rate (p = 2) are .

Local smoothness of Lévy processes
It is worth noting that the Sobolev regularity is the same-τ 2 (w) = −d/2-for any Lévy white noise. We also observe that the Hölder-Zigmund regularity of any non-Gaussian Lévy white noise is (−d) (which is also the regularity of a Dirac impulse), the Gaussian case being different and reaching a smoothness of (−d/2). In the one-dimensional setting (d = 1), this is reminiscent to the fact that the Brownian motion is the only continuous random process with independent and stationary increments, the other Lévy processes being only càdlàg (French acronym for functions that are right continuous with left limits at every points) [7]. Using Theorem 1.1, we deduce the local smoothness of Lévy processes in Corollary 1. (1.17) In the general case, we have almost surely that, for any 0 < p ≤ ∞, .

Related works on Lévy processes and Lévy white noises
In this section, for comparison purposes, we reinterpret all the results in terms of the critical smoothness and asymptotic growth rate of the considered random processes.
Lévy Processes. Most of the attention has been so far devoted to classic Lévy processes. The Wiener process was studied in [15,16,47,68,77,85], while [16] also contains results on the Besov regularity of fractional Brownian motions and SαS processes. By exploiting the self-similarity of the stable processes, Ciesielski , (1.20) for any p ≥ 1, where s Gauss is the Brownian motion and s α is the SαS process with parameter 1 < α < 2.
The complete family of Lévy processes-and, more generally, of Lévy-type processeshas been considered by R. Schilling in a series of papers [71,72,73] synthesized in [11, Chapter V] and by V. Herren [45]. To summarize, Schilling has shown that, for a Lévy process s with indices 0 ≤ β ∞ ≤ 2 and 0 < p max ≤ ∞, ≤ ρ p (s). (1.22) We observe that (1.19), (1.20), and (1.21) are consistent with Corollary 1.3. Moreover, our results provide an improvement by showing that the lower bounds of (1. 19) and (1.20) are actually sharp. Finally, we significantly improve the upper bound of (1.21) for general Lévy processes.
In contrast to the smoothness, the growth rate (1.22) of the Lévy process s does not seem to be related to the one of its derivative the Lévy white noise w = s by a simple relation. In particular, the rate of s is expressed in terms of the Pruitt index β 0 = min(p max , 2), conversely to p max for w (see Section 2.3). This needs to be confirmed by a precise estimation of ρ p (s) for which only a lower bound is known.
Lévy White Noises. M. Veraar extensively studied the local Besov regularity of the d-dimensional Gaussian white noise. As a corollary of [86,Theorem 3.4], one then deduces that τ p (w Gauss ) = −d/2. This work is based on the specific properties of the Fourier series expansion of the random process under the Gaussianity assumption, and cannot be directly adapted to Lévy white noises.
In our own works, we have investigated the question for general Lévy white noises in dimension d in the periodic [35] and global settings [31]. We obtained the lower bounds ≤ρ p (w). (1.24) These estimates are improved by Theorem 1.1, which provides an upper bound for τ p (w) and shows that (1.23) is sharp when β ∞ =β ∞ . It is also worth noticing that the lower bound of (1.24) is sharp if and only if p max ≤ 2.

Sketch of proof and the role of wavelet methods
Our techniques are based on the wavelet characterization of Besov spaces, as presented by H. Triebel in [80]. We shall see that wavelets are especially relevant to the analysis of Lévy white noises.
We briefly present the strategy of the proof of Theorem 1.1 when the ambiant dimension is d = 1. The general case d ≥ 1 is analogous and will be comprehensively addressed in the rest of the paper. Let (ψ M , ψ F ) be the (mother, father) Daubechies wavelets of a fixed order (the choice of the order has no influence on the results as soon as it is large enough). For j ∈ N and k ∈ Z, we define the rescaled and shifted functions ψ F,k = ψ F (·−k) and ψ j,M,k = 2 j/2 ψ M (2 j ·−k). Then, the family (ψ F,k ) k∈Z ∪(ψ j,M,k ) j∈N,k∈Z forms an orthonormal basis of L 2 (R) [24]. For a given one-dimensional Lévy white noise w, one considers the family of random variables ( w, ψ F,k ) k∈Z ∪ ( w, ψ j,M,k ) j∈N,k∈Z .
(1. 25) We then have that w = k∈Z w, ψ F,k ψ F,k + j∈N k∈Z w, ψ j,M,k ψ j,M,k , where the convergence is almost sure in S (R).
Then, for 0 < p < ∞ (the case p = ∞ will be deduced by embedding and is not discussed in this section) and τ, ρ ∈ R, the random variable s. We then fix 0 < p < ∞. We assume that we have guessed the values τ p (w) and ρ p (w) introduced in Section 1. Here are the main steps leading to the proof that these values are effectively the critical ones.
• For τ < τ p (w) and ρ < ρ p (w), we show that w B τ p (R;ρ) < ∞ a.s. For p < p max (see (1.6)), we establish the stronger result E w p B τ p (R;ρ) < ∞. This requires moment estimates for the wavelet coefficients of a Lévy white noise, which gives a precise estimation of the behavior of E[| w, ψ j,M,k | p ] as j goes to infinity. When p > p max , the random variables w, ψ j,M,k have an infinite pth moment and the present method is not applicable. In that case, we actually deduce the result using embedding relations between Besov spaces. It turns out that this approach is sufficient to obtain sharp results.
• For τ > τ p (w), we show that w B τ p (R;ρ) = ∞ a.s. To do so, we only consider the mother wavelet and truncate the sum over k to yield the lower bound for some constant C such that 2 −j k ρp ≥ C for every j ∈ N and 0 ≤ k < 2 j . We then need to show that the wavelet coefficients w, ψ j,M,k cannot be too small altogether using Borel-Cantelli-type arguments. Typically, this requires us to control the evolution of quantities such as P(| w, ψ j,M,k | > x) with respect to j and is again based on moment estimates.
• For ρ > ρ p (w), we show again that w B τ p (R;ρ) = ∞ a.s. This time, we only consider the father wavelet in (1.26) and use the lower bound (1.28) A Borel-Cantelli-type argument is again used to show that the | w, ψ F,k | cannot be too small altogether, and that the Besov norm is a.s. infinite.
The rest of the paper is dedicated to the proof of Theorem 1.1. The required mathematical concepts-Lévy white noises as generalized random processes and weighted Besov spaces-are laid out in Section 2. In Sections 3, 4, and 6, we consider the case of Gaussian white noises, compound Poisson white noises, and finite-moments Lévy white noises, respectively. Section 5 provides some new moment estimates for Lévy white noises that are preparatory to the upcoming sections. The general case is deduced in Section 7, where we provide the proof of Theorem 1.1. Finally, we discuss our results and give important examples in Section 8.

Lévy white noises as generalized random processes
The theory of generalized random processes was initiated independently by K. Itô [48] and I.M. Gel'fand [40] in the 50's and corresponds to the probabilistic counterpart of the theory of generalized functions of L. Schwartz. It was later brought to light by Gel'fand himself together with N.Y. Vilenkin in [41,Chapter III]. In this framework, a generalized random process is characterized by its effects against test functions. This allows to consider random processes that are not necessarily defined pointwise, as is the case for the Lévy white noise. The theory of generalized random processes, besides being very general, appears to be very flexible for the construction and analysis of random processes. It is a powerful alternative to more classic approaches, as argumented in [13,38]. The theory of generalized random processes is used as the ground for generalized CARMA processes [12] and fields [5,6], for conformal field theory in statistical physics [1], for studying the solutions of stochastic differential PDEs [21,49,87], and as random models in signal processing [10,19,30,83].
We shall define random processes as random elements of the space S (R d ), that we introduce now. Let S(R d ) be the space of rapidly decaying smooth functions from R d to R. It is endowed with its natural Fréchet nuclear topology [79]. Its topological dual is the space of tempered generalized functions S (R d ). It is endowed with the strong topology and B(S (R d )) denotes the Borelian σ-field for this topology. Note that S (R d ) can be endowed with other natural σ-fields: the one associated to the weak-* topology or the cylindrical σ-field generated by the cylinders {u ∈ S (R d ), ( u, ϕ 1 , . . . , u, ϕ N ) ∈ B} for N ≥ 1, ϕ n ∈ S(R d ), and B a Borelian subset of R N . However, these different σ-fields are known to coincide in this case [8, Proposition 3.8 and Corollary 3.9]. 1 See also Itô's [49] and Fernique's monographs [38] for general discussions on the measurable structures of function spaces. Throughout the paper, we fix a complete probability space (Ω, F, P). Definition 2.1. A measurable function s from (Ω, F) to (S (R d ), B(S (R d ))) is called a generalized random process. Its probability law is the probability measure on S (R d ) defined for B ∈ B(S (R d )) by P s (B) = P({ω ∈ Ω, s(ω) ∈ B}). The characteristic functional of s is the functional P s : It turns out that the characteristic functional is continuous, positive-definite over S(R d ), and normalized such that P s (0) = 1. The converse of this result is also true: if P is a continuous and positive-definite functional over S(R d ) such that P(0) = 1, then it is the characteristic functional of a generalized random process in S (R d ). This is known as the Bochner-Minlos theorem. It was initially proved in [61] and uses the nuclearity of S (R d ). See [75,Theorem 2.3] for an elegant proof based on the Hermite expansion of tempered generalized functions [76]. It means in particular that one can define generalized random processes via the specification of their characteristic functional. Following Gel'fand and Vilenkin, we use this principle to introduce Lévy white noises.
We consider functionals of the form P(ϕ) = exp R d Ψ(ϕ(x))dx . It is known that P is a characteristic functional over the space D(R d ) of compactly supported smooth functions if and only if the function Ψ : R → C is continuous, conditionally positivedefinite, with Ψ(0) = 0 [ where µ ∈ R, σ 2 ≥ 0, and ν is a Lévy measure, which means a positive measure on R such that ν({0}) = 0 and R inf(1, t 2 )dν(t) < ∞. The triplet (µ, σ 2 , ν) is unique and called the Lévy triplet of Ψ.
In our case, we are only interested in the definition of Lévy white noises over S (R d ).
This requires an adaptation of the construction of Gel'fand and Vilenkin. We say that for every ϕ ∈ S(R d ), where Ψ is a characteristic exponent that satisfies the -condition. The Lévy triplet of w is denoted by (µ, σ 2 , ν). Then, we say that w is a Gaussian white noise if ν = 0, a compound Poisson white noise if µ = σ 2 = 0 and ν = λP , with λ > 0 and P a probability measure on R such that P ({0}) = 0, and a Lévy white noise with finite moments if E[| w, ϕ | p ] < ∞ for any ϕ ∈ S(R d ) and p > 0.
The -condition is extremely mild. Lévy white noises in S (R d ) include stable white noises, symmetric-gamma white noises, and compound Poisson white noises whose jumps probability measure P admits a finite moment ( R |t| P (dt) < ∞ for some > 0) [ to. This is done by approximating a test function ϕ with functions in S(R d ) and showing that the underlying sequence of random variables converges in probability to a random variable that we denote by w, ϕ . This principle is developed with more generality in [32] by connecting the theory of generalized random processes to independently scattered random measures in the sense of B.S. Rajput and J. Rosinski [66]; see also [42].
In particular, as soon as ϕ ∈ L 2 (R d ) is compactly supported, the random variable w, ϕ Remark. The random variable w, ϕ can be interpreted as a stochastic integral with respect to a Lévy sheets s : We recall that Lévy sheets are multivariate generalizations of the Lévy processes [20,22,42]. In that case, we have the formal relation w, ϕ = R d ϕ(x)ds(x), whose precise meaning has been investigated in [20,32].

The Lévy-Itô decomposition of Lévy white noises
The Lévy-Itô decomposition is a fundamental result of the theory of Lévy processes.
It reveals that a Lévy process s = (s(t)) t∈R can be decomposed as s = s 1 + s 2 + s 3 , where s 1 is a Wiener process, s 2 is a compound Poisson process, and s 3 is a square integrable pure jump martingale, which corresponds to the small jumps of s [ [22,26,62]. This includes Lévy sheets, that we already mentioned and for which the Lévy-Itô decomposition has been extended for Lévy sheets in [2,Theorem 4.6].
Using the Lévy-Itô decomposition of Lévy sheets, we are able to provide an identical result for the Lévy white noise. This is based on the connection between Lévy sheets and Lévy white noises, which is one of the main contribution of [20]. Indeed, the Lévy white noise w satisfies the relation for some Lévy sheet s in S (R d ) [ with w 1 a Gaussian white noise, w 2 a compound Poisson white noise, and w 3 a Lévy white noise with finite moments, the three being independent.
Proof. According to (2.5 where w 1 is a Gaussian white noise, w 2 is a compound Poisson white noise, and w 3 is a Lévy white noise with finite moments. This last point is indeed ensured by the fact that the Lévy measure ν 3 associated to s 3 and therefore w 3 has a compact support. Hence, we have that R |t| p dν 3 (t) < ∞ for any p > 0. This implies that E[| w 3 , ϕ | p ] < ∞ for any ϕ ∈ S(R d ) and p > 0 according to [70,Theorem 25.3] (see also Proposition 2.5 thereafter). Note moreover that w 1 , w 2 , and w 3 are independent, because the corresponding Lévy sheets are.

Indices of Lévy white noises
We introduce various indices associated to Lévy white noises. First of all, we exclude Lévy white noises with dominant drift via the following classic notion that appears for instance in [25,72].  This condition ensures that no drift is dominating the Lévy white noise. For instance, the deterministic Lévy white noise w = µ = 0 a.s., which corresponds to the Lévy triplet (µ, 0, 0), is such that Ψ(ξ) = iµξ and does not satisfy the sector condition. This is also the case for w = µ + w α where w α is a SαS process with α ∈ (0, 2]. It is worth noting that the characteristic exponent of a symmetric Lévy white noise is real, and therefore satisfies the sector condition. In the rest of the paper, we will always assume that the sector condition is satisfied without further mention. In Theorem 1.1, the smoothness and growth rate of Lévy white noises is characterized in terms of the indices (1.4), (1.5), and (1.6). We give here some additional insight about these quantities. The index β ∞ was introduced by R. Blumenthal and R. Getoor [9] to characterize the behavior of Lévy processes at the origin. This quantity appears to be related to many local properties of random processes driven by Lévy white noises, including the Hausdorff dimension of the image set [11], the spectrum of singularities [26,50], the Besov regularity [11,31,71,73] and more generally sample path properties [14,55,67], the local self-similarity [34], or the local compressiblity [36]. Finally, the index β ∞ plays a crucial role in the specification of negative results, such as the identification of the Besov spaces in which the Lévy white noises are not. It satisfies moreover the In [65], W. Pruitt proposed the index as the asymptotic counterpart of β ∞ . This quantity appears in the asymptotic growth rate of the supremum of Lévy(-type) processes [72] and the asymptotic self-similarity of random processes driven by Lévy white noises [34]. The Pruitt index differs from the index p max that appears in Theorem 1.1. Actually, the two quantities are linked by the relation β 0 = min(p max , 2). This is shown by linking β 0 to the Lévy measure [11] and knowing that β 0 ≤ 2 (see the appendix of [25] for a short and elegant proof). This means that β 0 < p max when the Lévy white noise has some finite pth moments fo p > 2, and one cannot recover p max from β 0 in this case. It is therefore necessary to introduce the index p max in addition to the Pruitt index in our analysis. Note moreover that the moment index fully characterizes the moment properties of the Lévy white noise in the following sense.
Proposition 2.5. Let 0 < p < ∞ and w be a Lévy white noise with moment index p max ∈ (0, ∞]. We also fix a compactly supported and bounded test function ϕ = 0. (2.11) Proposition 2.5 can be deduced from more general results presented in [66] and [32], us, it is enough to know that the result is true for compactly supported bounded test functions, which includes Daubechies wavelets. We provide a proof thereafter for the sake of completeness, since this result is not exactly stated as such in the literature and known results require to introduce tools that are unnecessary for this paper. The first part (2.10) allows one to consider the moments of w, ϕ for any p < p max . The second part (2.11) shows that some moments are infinite and will appear to be useful later on.
Proof of Proposition 2.5. The proof relies on the link between the moments of w and the moments of its Lévy measure ν. According to [70,Theorem 25.3], a random variable X with Lévy measure ν is such that Applying this to X = w, 1 [0,1) d (whose Lévy measure is indeed ν)) and using the definition of p max in (1.6), we deduce that (2.14) Let K be the compact support of ϕ. Moreover, the test function being non identically zero, there exists m > 0 such that Leb({|ϕ| ≥ m}) > 0 where Leb is the Lebesgue measure. Set p < p max , then for every t ∈ R and every x ∈ K, we have that Therefore, according to the left side of (2.13), proving (2.10). Moreover, if p > p max , then, using the right side of (2.13) and the inequality 1 |tϕ(x)|>1 ≥ 1 |t|m>1 for every x ∈ {|ϕ| ≥ m}, we have and (2.11) is proved.
We now summarize how the indices of the Lévy-Itô decomposition of a Lévy white noise behave in Proposition 2.6. Proposition 2.6. Let w be a Lévy white noise w and let w = w 1 + w 2 + w 3 be its Lévy-Itô decomposition according to (2.6) in Proposition 2.3, where w 1 is Gaussian, w 2 is compound Poisson, and w 3 have finite moments.
Proof. Let Ψ be the characteristic exponent of w, and (µ, σ 2 , ν) be its Lévy triplet. We assume that µ = 0, what has no impact on the indices. The Lévy-Itô decomposition corresponds to the following sum for the characteristic exponent: , (2.20) where, Ψ 1 , Ψ 2 , and Ψ 3 are the characteristic exponents of w 1 , w 2 , and w 3 respectively, with respective triplets (0, we have seen in the proof of Proposition 2.3 that the moments of w 3 are finite, hence p max (w 3 ) = ∞. It is moreover clear that the moments of the Gaussian white noise are finite, hence p max (w 1 ) = ∞ and the relations (2.17) are proved.
(ii) Assume that w 1 = 0. The characteristic exponent Ψ 2 is bounded by some constant

. The right inequality gives the other inequalities for the Blumenthal-Getoor indices and thereforeβ
For the moment index, we recall that |a + b| p ≤ 2 p−1 (|a| p + |b| p ) (by convexity of x → x p on R + ) for every a, b ∈ R and p ≥ 1 and that |a + b| Hence, w and w 2 have the same moment index.

Weighted Besov spaces
As we have seen in Section 1.4, Besov spaces are natural candidates for characterizing the regularity of Lévy processes and Lévy white noises. We define the family of weighted Besov spaces based on wavelet methods, as exposed in [80]. Besov spaces have a long history in functional analysis [81]. They were successfully revisited by the introduction of wavelet methods following the works of Y. Meyer [59] and applied to the analysis of stochastic processes, including the Brownian motion [15,16,68], the fractional Brownian motion [39,60], sparse random processes [30,36,64,83], and general solutions of SPDEs [17,18].
Essentially, weighted Besov spaces are subspaces of S (R d ) that are characterized by weighted sequence norms of the wavelet coefficients. Following H. Triebel, we use the compactly supported wavelets discovered by I. Daubechies [23], which we introduce first. The scale and shift parameters of the wavelets are respectively denoted by j ∈ N and k ∈ Z d . The symbols M and F refer to the gender of the wavelet (M for the mother wavelets and F for the father wavelet). Consider two functions ψ M and ψ F ∈ L 2 (R). We G ∈ G j , and k ∈ Z d , we define ψ j,G,k := 2 jd/2 ψ G (2 j · −k). (2.24) We shall also use the notations F = (F, . . . , F ) and M = (M, . . . , M ) for the purely father and purely mother genders. It is known that, for any r 0 ≥ 1, there exists two functions ψ M , ψ F ∈ L 2 (R), called Daubechies wavelets, that are compactly supported, with at least r 0 continuous derivatives and vanishing moments up to order at least (r 0 − 1), and such that the family 2 {ψ j,G,k } (j,G,k)∈N×G j ×Z d is an orthonormal basis of L 2 (R d ) [ We now introduce the family of weighted Besov spaces B τ p (R d ; ρ). Traditionally, Besov spaces also depend on the additional parameter q ∈ (0, ∞] (see for instance [80,Definition 1.22]). We shall only consider the case q = p in this paper, so that we do not refer to this parameter.
We introduce weighted Besov spaces in Definition 2.7 relying on the wavelet decomposition of (generalized) functions. This construction is equivalent to the more usual Fourier-based definitions, as proved in [80,Theorem 1.26]. We use the notation (x) + = max(x, 0). Definition 2.7. Let τ, ρ ∈ R and 0 < p ≤ ∞. Fix an integer r 0 > max(τ, d(1/p − 1) + − τ ) and consider a family of Daubechies wavelets {ψ j,G,k } (j,G,k)∈N×G j ×Z d , where ψ M and ψ F have at least r 0 continuous derivatives and ψ M has vanishing moments up to order at least (r 0 − 1). The weighted Besov space B τ p (R d ; ρ) is the collection of tempered generalized functions f ∈ S (R d ) that can be written as f = j∈N G∈G j k∈Z d c j,G,k ψ j,G,k , (2.25) where the c j,G,k satisfy j∈N 2 j(τ p−d+ dp and where the convergence (2.25) holds on S (R d ). The usual adaptation is made for p = ∞; that is, sup The integer r 0 in Definition 2.7 is chosen such that the mother wavelet has enough vanishing moments and the mother and father wavelets are regular enough to be applied to a function of B τ p (R d ; ρ). We refer the reader to [    f, ψ j,G,k is well defined and we have c j,G,k = f, ψ j,G,k . Moreover, for p < ∞, the is finite for any f ∈ B τ p (R d ; ρ) and specifies a norm (a quasi-norm, respectively) on the and B τ ∞ (R d ; ρ) is a Banach space for this norm. Proposition 2.8 (Embeddings between weighted Besov spaces). Let 0 < p 0 ≤ p 1 ≤ ∞ and τ 0 , τ 1 , ρ 0 , ρ 1 ∈ R.
As a simple example, we obtain the Besov localization of the Dirac distribution. This result is of course well-known (an alternative proof can be found for instance in [74]) but we provide a new proof for two reasons: (1) it illustrates how to use the wavelet-based characterization of Besov spaces and (2) the result will be used to obtain sharp results for compound Poisson white noises. Proposition 2.9. Let 0 < p < ∞, τ ∈ R, and ρ ∈ R. Then, the Dirac impulse δ is in if and only if τ ≤ −d. We remark that the weight ρ ∈ R plays no role in Proposition 2.9. This is a simple consequence of the fact that δ is compactly supported, and therefore insensitive to the weight, as will appear in the proof.
Proof of Proposition 2.9. We first treat the case p < ∞. The wavelet coefficients of δ are c j,G,k = 2 jd/2 ψ G (−k), hence the Besov (quasi-)norm of the Dirac impulse is given by We first introduce some notations. The 2 d wavelets ψ G with gender G describing G 0 are bounded, hence the constant b = max G∈G 0 ψ G ∞ is finite. We denote by K the set of multi-integers k ∈ Z d such that ψ G (−k) = 0 for some gender G ∈ G 0 . The set K is finite because the 2 d wavelets are compactly supported and we set n 0 = Card(K). Moreover, the set K is non empty; otherwise, (2.32) would imply that δ p B τ p (R d ;ρ) = 0, hence δ = 0, which is absurd. We fix some element k 0 ∈ K and G 0 a gender such that ψ G0 (−k 0 ) = 0. We set a = |ψ G0 (−k 0 )| > 0. Then, there exists a constant M > 0 such that 2 −j k ≤ M for any j ∈ N and (−k) ∈ K. In particular, for such k and j, we have that 1 ≤ 2 −j k = (1 + 2 −j k 2 ) 1/2 ≤ M , and therefore, (2.34) We recall that Card(G j ) ≤ 2 d . Then, we also have the upper bound (2.35) Combining (2.34) and (2.35), we therefore deduce that The sum converges for (τ p − d + dp) < 0 and diverges otherwise, implying the result.
We now adapt the argument to the case p = ∞ for which the Besov norm is given by We have that, for any j ∈ N, (2.38) Therefore, we deduce that

Gaussian white noise
Our goal in this section is to prove the Gaussian part of Theorem 1.1. Without loss of generality, we focus on the Gaussian white noise with zero mean and unit variance.
The Gaussian case is much simpler than the general one since the wavelet coefficients of the Gaussian white noise are independent and identically distributed. We present it separately for three reasons: (i) it can be considered as an instructive toy problem that already contains some of the technicalities that will appear for the general case; (ii) it cannot be deduced from the other sections, where the results are based on a careful study of the Lévy measure; and (iii) the localization of the Gaussian white noise in weighted Besov spaces has not been addressed in the literature, to the best of our knowledge. We first state three simple lemmata that will be useful throughout the paper.
, it is then sufficient to demonstrate Lemma 3.1 for µ < ∞, which we do now. Let (W k ) k≥1 be a family of i.i.d. random variables whose common law is the one of the Y i k and define W k = 1 The weak law of large numbers implies that P(|W k − µ| ≥ x) vanishes when k → ∞ for any x > 0. Taking x = µ/2, we readily deduce that P(W k ≥ µ/2) goes to 1 when k → ∞. Moreover, we have the equality Z k (L) = W k , therefore, we also have that P(Z k ≥ µ/2) −→ k→∞ 1.

(3.2)
This implies in particular that k≥1 P(Z k ≥ µ/2) = ∞. The events {Z k ≥ µ/2} are moreover independent due to the independence of the Y i k . Using the Borel-Cantelli lemma, we deduce that Z k ≥ µ/2 for infinitely many k a.s. An obvious consequence is then that k≥1 Z k = ∞ a.s..
As a consequence of Lemma 3.1, we deduce Lemma 3.2.
Proof. First of all, the result for any dimension d is easily deduced from the onedimensional case. Moreover, |k| and k are equivalent asymptotically, hence it is 3) by applying Lemma 3.1 with N k = 2 k−1 and Y i k = X 2 k−1 +i−1 and observing that k≥1 Finally, we state the last lemma that deals with supremum of i.i.d. sequences of random variables.  Proof. Let M > 0. The assumption P(|X k | ≥ M ) > 0 and the fact that the events {|X k | ≥ M } are independent implies, thanks to the Borel-Cantelli lemma, that there exists almost surely (infinitely many) k ≥ 1 such that |X k | ≥ M . Hence, sup k≥1 |X k | ≥ M almost surely. This being true for every M > 0, we deduce (3.4).
We characterize the Besov regularity of the Gaussian white noise in Proposition 3.4.   [41, Section 2.5]. The family of functions {ψ j,G,k } (j,G,k)∈N×G j ×Z d being orthonormal, the random variables w, ψ j,G,k are therefore i.i.d. with law N (0, 1). Case p < ∞, τ < −d/2, and ρ < −d/p. For p > 0, we denote by C p the pth moment of a Gaussian random variable with zero mean and unit variance. In particular, we have that E [| w, ψ j,G,k | p ] = C p for any j ∈ N, G ∈ G j , k ∈ Z d , and therefore The last inequality is due to Card(G j ) ≤ 2 d . Since ρp < −d and 2 −j k ∼ k →∞ 2 −j k , we have that In particular, the series j 2 j(τ p+ dp 2 ) 2 −jd k 2 −j k ρp converges if and only if the series j 2 j(τ p+ dp 2 ) does; in other words, if and only if τ < −d/2. Finally, if τ < −d/2 and ρ < −d/p, we have shown that E[ w p B τ p (R d ;ρ) ] < ∞ and therefore w ∈ B τ p (R d ; ρ) almost surely.
Case p < ∞ and τ ≥ −d/2. Then, we have 2 j(τ p−d+dp/2) ≥ 2 −jd . We aim at establishing a lower bound for the Besov norm of w and we restrict to the purely mother wavelet with gender G = M = (M, . . . , M ) ∈ G j for any j ∈ N. For k = (k 1 , . . . , k d ) ∈ Z d such that 0 ≤ k i < 2 j for every i = 1, . . . , d, we have that The random variables w, ψ j,M ,k are i.i.d. We can therefore apply Lemma 3.1 with blocks of size 2 jd , which goes to infinity when j → ∞ to conclude that w p  (3.10) with the notation φ k = φ(· − k). Finally, the random variables w, φ k being i.i.d., Lemma 3.2 applies and w p B τ p (R d ;ρ) = ∞ almost surely. Case p = ∞, τ < −d/2, and ρ < 0. This case is deduced using the embeddings between Besov spaces. First of all, for ≤ min(−d/2 − τ, −ρ), we have that using (1.2) first for the weight and then for the smoothness parameters. It therefore suffices to show the existence of 0 . Fix such an . Then, for every p < ∞, we already proved that w ∈ B s. Applying this to p = 2d/ , we then remark that  (3.13) concluding this case.
Case p = ∞ and τ ≥ −d/2. Note that the case τ > −d/2 can be deduced from the results for p < ∞ by embedding, but one cannot deduce the case τ = −d/2. By keeping only the purely mother wavelet ψ M with M = (M, . . . , M ) and the shift parameter k = 0, the Besov norm (2.29) applied to the Gaussian white noise is (3.14) The Gaussian random variables w, ψ j,M ,0 are independent and verifies the conditions of Lemma 3.3 (since P(| w, ψ j,M ,0 | ≥ M ) > 0 for every M ≥ 0). This implies that sup j∈N | w, ψ j,M ,0 | = ∞ a.s., and therefore w / ∈ B τ ∞ (R d ; ρ) a.s. due to (3.14). Case p = ∞ and ρ ≥ 0. Again, the case ρ > 0 can be deduced from the results for p < ∞ by embedding, but the case ρ = 0 cannot. Using that k ρ ≥ 1 for ρ ≥ 0 and keeping only the father wavelet φ = ψ F with F = (F, . . . , F ) ∈ G 0 and the scale j = 0, the Besov norm (2.29) of w is lower bounded by Again, Lemma 3.3 applies to the Gaussian random variables w, φ k and max k∈Z | w, φ k | = ∞ a.s., implying that w B τ ∞ (R d ;ρ) = ∞ a.s. due to (3.15). The proof of Proposition 3.4 for the case ρ ≥ −d/p uses an argument that can be easily adapted to any Lévy white noise. We hence state this result in full generality. Proposition 3.6. Fix 0 < p ≤ ∞ and τ, ρ ∈ R. If w is a non-constant Lévy white noise, then w / ∈ B τ p (R d ; ρ) as soon as ρ ≥ −d/p.
Proof. The proof is very similar to the one of Proposition 3.4 for ρ ≥ −d/p and p < ∞, and for ρ ≥ 0 and p = ∞, except that we only consider father wavelet φ = ψ F and its shifts φ k = ψ 0,F ,k with k = k 0 Z d , where k 0 ∈ N\{0} is chosen such that the φ k have disjoint supports. In particular, this implies the random variables w, φ k , are independent for k ∈ k 0 Z d (the support of the test functions being disjoint) and independent (the Lévy white noise being stationary). As a consequence, (3.10) becomes

Compound Poisson white noise
Compound Poisson white noises are almost surely made of countably many Dirac impulses, what will be crucial in their analysis. Our positive results are based on a careful estimation of the moments of the compound Poisson white noise presented in Proposition 4.1.  for every j ∈ N, G ∈ G j , and k ∈ Z d .
Proof. We recall that the Lebesgue measure is denoted by Leb. Let λ > 0 and P be respectively the sparsity parameter and the law of the jumps of w. Then, we have that where the a k are i.i.d. with law P , and the x k , independent from the a k , are randomly located such that Card{k ∈ N, x k ∈ B} is a Poisson random variable with parameter λLeb(B) for any Borel set B ⊂ R d with finite Lebesgue measure. For a demonstration that the right term in (4.2) specifies a compound Poisson white noise in the sense of a generalized random process with the adequate characteristic functional, we refer the reader to [82,Theorem 1].
Let ψ ∈ L 2 (R d )\{0} be a compactly supported function and I ψ be the closed convex hull of its support. In particular, 0 < Leb(I ψ ) < ∞. We set N (ψ) = Card{k ∈ N, x k ∈ I ψ }, which is a Poisson random variable with parameter λLeb(I ψ ). We denote by a n and x n , n = 1, . . . , N (ψ), the weights and Dirac locations of the compound Poisson white noise w on I ψ . That is, w, ψ = N (ψ) n=1 a n ψ(x n ). By conditioning on N (ψ), we then have that where (i) uses the relation N n=1 y n p ≤ N max(0,p−1) N n=1 |y n | p , valid for any p > 0 and y n ∈ R [33,Eq.(50)], (ii) is due to E |a n | p |N (ψ) = N = E |a n | p , a n and N (ψ) being independent, and (iii) comes from N n=1 E |a n | p = N E |a 1 | p , the a n sharing the same law.
Our goal is now to apply (4.3) to ψ = ψ j,G,k . For fixed j ≥ 1 and k ∈ Z d , the Lebesgue measure of the convex hull I ψ j,G,k of the support of the ψ j,G,k is Leb(I ψ j,G,k ) = 2 −jd Leb(I ψ G ). Therefore, N (ψ j,G,k ) = Card{k ∈ N, x k ∈ ψ j,G,k } is a Poisson random variable with parameter 2 −jd λLeb(I ψ G ). As a consequence, we have where we used that 2 −jdN ≤ 2 −jd and e −2 jd λLeb(I ψ G ) ≤ 1. We have moreover the relation ψ j,G,k p ∞ = 2 jpd/2 ψ G p ∞ . (4.5) Applying inequalities (4.4) and (4.5) in (4.3) with ψ = ψ j,G,k , we finally deduce (4.1) (for Proposition 4.2. Fix 0 < p ≤ ∞ and τ, ρ ∈ R. Let w be a compound Poisson white noise with index p max ∈ (0, ∞]. If 0 < p < ∞, then, w is where C is the constant appearing in (4.1), and using that Card(G j ) ≤ 2 d . Then, is therefore finite if and only if j 2 j(τ p−d+dp) < ∞, which happens here due to our assumption (τ p−d+dp) < 0. This shows that w ∈ B τ p (R d ; ρ) almost surely.
Case p max ≤ p, τ < d/p − d, and ρ < −d/p max . We prove that w ∈ B τ p (R d ; ρ) a.s. using the embeddings between Besov spaces and the study of the case p < p max before. From the conditions on τ and ρ, one can find p 0 ∈ (0, p max ), τ 0 ∈ R, and ρ 0 ∈ R such that Then, in particular, p 0 < p, τ 0 −τ > d/p 0 −d/p, and ρ 0 > ρ, so that B τ0 p0 (R d ; ρ 0 ) ⊂ B τ p (R d ; ρ) (according to (2.30)). Moreover, p 0 < p max , τ 0 < d/p 0 − d, and ρ 0 < −d/p 0 . We are therefore back to the first case, for which we have already shown that w ∈ B τ0 p0 (R d ; ρ 0 ) a.s. In conclusion, w ∈ B τ p (R d ; ρ) a.s. Combining these first two cases, we obtain that w ∈ B τ p (R d ; ρ) if τ < d/p − d and ρ < −d/ min(p, p max ) for every p ∈ (0, ∞]. Case p < ∞ and τ ≥ d/p − d. We use the representation (4.2) of the compound Poisson white noise. Assume that w is in B τ p (R d ; ρ) for some ρ ∈ R. Then, the product of w by any compactly supported smooth test function ϕ is well-defined and also in B τ p (R d ; ρ). Choosing a (random) test function ϕ ∈ S(R d ) such that ϕ(x 0 ) = 1 and ϕ(x n ) = 0 for n = 0, we get where a 0 = 0 a.s. This is absurd due to Proposition 2.9, proving that w / ∈ B τ p (R d ; ρ) for all ρ ∈ R.
Case p = ∞ and τ > −d. The same argument than for the case p < ∞ and τ ≥ d/p−d applies, using this time that w / ∈ B τ ∞ (R d ; ρ) for any ρ > 0, again due to Proposition 2.9. Case ρ ≥ −d/p. This case has been treated in full generality in Proposition 3.6. Case p > p max and ρ > −d/p max . This means in particular that p max < ∞. We treat the case ρ < 0, the extension for ρ ≥ 0 clearly follows from the embedding relations between Besov spaces. We set q := −d/ρ > p max . In particular, according to Proposition 2.5, we have that E[| w, ϕ | q ] = ∞ for any compactly supported and bounded function ϕ = 0. Proceeding as for (3.16), we have that where k 0 ≥ 1 is chosen such that the functions φ k = φ(· − k) have disjoint supports. Then, the random variables X k = w, φ k are i.i.d. The independence implies that the events A k = {X k ≥ k d/q } are independent themselves. Then, the X k having the same law, we have The choice of q implies moreover that E[|X 0 | q ] = E[| w, φ | q ] = ∞ due to Proposition 2.5.

Moment estimates for the Lévy white noise
The proof of Theorem 1.1 will be based on new estimates for the moments E[| w, ϕ | p ] of Lévy white noises. In Section 5.1, we consider the case p = 2m where m ≥ 1 is an integer. This will be critical when dealing with Lévy white noises with finite moments. In Section 5.2, we determine lower bounds for the moments, which is the main technicality for the negative Besov regularity results of Lévy white noises.

Moment estimates for p = 2m ≥ 2
We estimate the evolution of the even moments of the wavelet coefficients of a Lévy white noise w with the scale j. Most of the moment estimates in the literature deal with pth moments with the restriction p ≤ 2 [25,31,53,57], and we shall see that the extension to higher-order moments calls for some technicalities.
Proposition 5.1. Let w be a Lévy white noise with finite moments and m ≥ 1 be an integer. We assume that the moment index of w satisfies p max > 2m. Then, there exists a constant C > 0 such that, for every j ∈ N, G ∈ G j , and k ∈ Z d , Proof. Consider a test function ϕ ∈ S(R d ) and set X = w, ϕ . The characteristic function of X is [28,Proposition 2.12] P X (ξ) = exp R d Ψ(ξϕ(x))dx := exp(Ψ ϕ (ξ)).  [83,Proposition 9.11], we obtain the bound, with C > 0 a constant. We now apply (5.4) to ϕ = ψ j,G,k . Since we have 2 jd 1≤v≤2m (nv(v/2−1)) , (5.6) where C is a new constant independent from j, G, k. Finally, since v n v v = 2m and v n v ≥ 1, we have v (n v (v/2 − 1)) ≤ (m − 1). Therefore, we obtain (5.1) for an adequate C > 0.

Lower bound for moment estimates
In our previous moment estimates, we gave upper bounds for the quantity E[| w, ϕ | p ] (see not only Propositions 4.1 and 5.1, but also Theorem 2 in [31]). This allows one to identify in which Besov space is w. We now address the following problem: Can we bound E[| w, ϕ | p ] from below with the moments of ϕ? Theorem 5.2 answers positively to this question and is crucial for the proof of Theorem 1.1.
Theorem 5.2. Let w be a Lévy white noise whose indices satisfy 0 <β ∞ ≤ β ∞ < p max , ψ = 0 a bounded, and compactly supported test function, and p an integrability parameter such that 0 < p < β ∞ . Then, for > 0 small enough, there exists constants A, B > 0 independent from j ∈ N and k ∈ Z d such that for any j ∈ N and k ∈ Z d , where we recall that ψ j,k = 2 jd/2 ψ(2 j · −k).
Proof. First of all, the shift parameter k in (5.7) can be omitted since w is stationary.
We also remark that the upper bound of (5.7) has already been proven [31,Corollary 1], where the conditions p < β ∞ < p max are required. Actually, [31] does not consider the index p max and distinguishes between the conditions β ∞ < β 0 < 2 and β ∞ ≤ β 0 = 2 with finite variance. These two scenarios cover the condition β ∞ < p max of Theorem 5.2.
Hence, we focus on the lower bound.

Lévy white noise with finite moments
We consider Lévy white noises whose all the moments are finite, which means that p max = ∞. Their specificity is that one can use the finiteness of the pth moments of the wavelet coefficients of the Lévy white noise w for any p > 0. Thanks to the moment estimates in Section 5, we have all the tools to deduce the Besov regularity of white noises with finite moments.
As was the case for the Gaussian and compound Poisson cases, Proposition 6.1 allows to deduce the asymptotic growth rate of Lévy white noises with finite moments. Moreover, we obtain lower and upper bounds for the local smoothness in terms of the Blumenthal-Getoor indices of the Lévy white noise. Corollary 6.2. Let 0 < p ≤ ∞ and w be a Lévy white noise with finite moments and Blumenthal-Getoor indices 0 ≤β ∞ ≤ β ∞ ≤ 2. Then, we have that For p ∈ (0, 2), p an even integer, or p = ∞, we moreover have that Proof of Proposition 6.1. We only treat the case p < ∞. For p = ∞, the result is obtained using embeddings (with p = 2m, m → ∞ for positive results and p → ∞ for negative results) following the same arguments than for the Gaussian case in Proposition 3.4.
If τ < d/ max(p, β ∞ ) − d and ρ < −d/p. We first remark that for p ≤ 2, we have that ρ < − d min(p,2,pmax) = − d min(p,pmax) , and the result is a consequence of our previous work [31,Theorem 3]. We can therefore assume that p = 2m with m ≥ 1 an integer. Then, p = 2m ≥ 2 ≥ β ∞ , and the conditions on τ and ρ become τ < d 2m − d and ρ < − d 2m . Due to Proposition 5.1, we have This part of the proof is actually valid for any Lévy white noise. It uses the decomposition w = w 1 + w 2 with w 1 a nontrivial compound Poisson white noise and w 2 a Lévy white noise with finite moments (see Proposition 2.3, here w 2 combines the Gaussian part and the finite-moment part of the Lévy-Itô decomposition). The main idea is that the jumps of the compound Poisson part are by themselves enough to make the Besov norm infinite.
One writes that w 1 (L) = k∈Z a k δ(· − x k ), as in (4.2). Then, almost surely, all the a k are nonzero. Let m > 0 be such that P(|a 0 | ≥ m) > 0. One can assume without loss of generality that m = 1 (otherwise, consider the white noise w/m). Then, there exists almost surely k ∈ Z such that |a k | ≥ 1 (it suffices to apply the Borel-Cantelli argument using the independence of the a k ). For simplicity, one reorders the jumps such that this is achieved for k = 0, so that |a 0 | ≥ 1.
We first introduce preliminary notations. Let [a, b] be a finite interval on which the mother Daubechies wavelet is strictly positive. In particular, min x∈[a,b) |ψ M (x)| := c > 0. Then, setting K = [a, b) d and C = c d , we have that |ψ M (x)| ≥ C for every x ∈ K, with M = (M, . . . , M ). For each scale j ≥ 0, we define k j ∈ Z d as the unique multi-integer such that 2 j x 0 − k j ∈ [a, a + 1) d . If b ≥ a + 1, then 2 j x 0 − k j ∈ K. Otherwise, due to the law of the location x 0 , there is almost surely an infinity of scale j ≥ 0 such that 2 j x 0 − k j ∈ [a, b) d = K. We denote by J the (random) ensemble of such j. Then, for each j ∈ J, using that |a 0 | ≥ 1 and 2 j x 0 − k j ∈ [a, b) d = K, we deduce that Moreover, on each finite interval, there are almost surely finitely many jumps x k . In particular, the random variable inf k∈Z\{0} x k − x 0 is a.s. strictly positive. This implies that there exists a (random) integer j 0 ∈ N such that 2 j0 x k − x 0 > diam(Supp(ψ M )) for any k = 0, where diam(B) is the diameter of a Borelian set B ⊂ R d , understood as the Lebesgue measure of its closed convex hull. We therefore have that ψ M (2 j x k − k j ) = 0 for any j ≥ j 0 and any k = 0. From these preparatory considerations, one has a.s. that, for j ≥ j 0 , j ∈ J, Let now focus on the Lévy white noise w 2 . Since w 2 has a finite variance, we have that, using the Markov inequality, where σ 2 0 is the variance of w 2 such that E[ w 2 , ϕ 2 ] = σ 2 0 ϕ 2 2 for any test function ϕ. In particular, From a new Borel-Cantelli argument, we know that, almost surely, only finitely many j satisfy w 2 , ψ M (2 j · −k j ) ≥ C/2. In particular, there exists a (random) integer j 1 such that, for any j ≥ j 1 , w 2 , ψ M (2 j · −k j ) ≤ C/2. Combining this to (6.5), we then have that, for j ∈ J such that j ≥ max(j 0 , j 1 ), Note moreover that, by definition of k j , 2 j x 0 − k j ∈ [a, a + 1) d , hence we have that x 0 − 2 −j k j ∞ ≤ M/2 j where M = max(|a| , |a + 1|). Then, there exists a (random) integer j 2 ≥ 1 such that, for every j ≥ j 2 , M/2 j ≤ x 0 ∞ . We have moreover that 2 −j k j 2 ≤ d 1/2 2 −j k j ∞ ≤ d 1/2 (M/2 j + x 0 ∞ ). Therefore, recalling that ρ < 0, for every j ≥ j 2 , we have Putting the pieces together, we can now lower bound the Besov norm of w by keeping only the mother wavelet ψ M , a scale j ∈ J such that j ≥ max(j 0 , j 1 , j 2 ), and the corresponding shift parameter k j . Then, combining (6.7) and (6.8), we obtain the almost-sure lower This is valid for any j ∈ J such that j ≥ max(j 0 , j 1 , j 2 ) and because J is infinite and (τ p − d + dp) > 0, one concludes that w p B τ p (R d ;ρ) = ∞ almost surely. Case 0 < p <β ∞ and (d/β ∞ − d) < τ < (d/p − d). Assume that, under those assumptions, we prove that w / ∈ B τ p (R d ; ρ) a.s. Then, together the case τ > (d/p − d) considered below and using embeddings, we deduce the expected result for τ > p(d/ max(β ∞ , p) − d).
As soon as f / ∈ B τ p (R d ; ρ) for some p > 0, we also have that f / ∈ B τ + q (R d ; ρ) for any q > p and > 0 (see Figure 1). A crucial consequence for us is that it suffices to work with arbitrarily small p in order to obtain the negative result we expect. We assume here . (6.10) Note that the right inequality in (6.10) simply means that p < ∞ (i,e., no restriction) whenβ ∞ = β ∞ . We fix k 0 ∈ N\{0} such that, for any gender G, the functions Ψ 0,G,k0k have disjoint support for every k ∈ Z d . Then, at fixed G and j, the random variables ( w, Ψ j,G,k ) k∈k0Z d are independent. By restricting the range of k and the gender to G = M , we have that with C = inf x ∞ ≤k0 x ρ > 0 is such that 2 −j k ≥ C for any k ∈ k 0 {0, . . . 2 j − 1} d and any j ≥ 0. We set X j,k = 2 which is an average among 2 jd independent random variables.
Recall that p <β ∞ /2. Moreover, since all the moments are finite, p max = ∞ > β ∞ . Hence, one can apply Theorem 5.2 with integrability parameters q = p and q = 2p, respectively. There exists > 0 that can be choosen arbitrarily small and constants m q , M q such that m q 2 −j 2 its Lévy measure is not identically zero. In particular, w can have a Gaussian part in the Lévy-Itô decomposition (see Proposition 2.3). Proposition 7.1 characterizes the Besov regularity of non-Gaussian Lévy white noises, the Gaussian white noise having already been treated in Section 3. We conclude this section with the proof of Theorem 1.1.

Application to subfamilies of Lévy white noises
We apply Theorem 1.1 to deduce the local smoothness and asymptotic growth rate of specific Lévy white noises. We consider Gaussian, symmetric-α-stable [69], symmetric Gamma (including Laplace) [52], compound Poisson, inverse Gaussian [4], and layered stable white noises [46]. All the underlying laws are known to be infinitely divisible [46,70]. In Table 1 The layered stable white noises have the particularity of describing the complete spectrum of possible couples (α 1 , α 2 ) = (β ∞ , p max ) ∈ (0, 2) 2 . The characteristic exponent of a layered stable white noise is Ψ α1,α2 (ξ) = R (cos(tξ) − 1) 1 |t|≤1 |t| −(α1+1) + 1 |t|>1 |t| −(α2+1) dt. We also provide a visualization of our results in terms of Triebel diagrams. In Figures 2 to 5, we plot the local smoothness 1 p → τ p (w) and asymptotic growth rate 1 p → ρ p (w) for different Lévy white noises (with the exception of τ p (w) which is not fully determined for the general case in Figure 5; here, we represent the lower and upper bounds of (1.9)). A given noise is almost surely in a Besov space B τ p (R d ; ρ) if the points (1/p, τ ) and (1/p, ρ) are in the lower shaded green regions. A contrario, the Lévy white noise is almost surely not in B τ p (R d ; ρ) if (1/p, τ ) or (1/p, ρ) are in the upper shaded red region. In Figure 5, the white region corresponds to the case where we do not know if the Lévy white noise is or is not in the corresponding Besov spaces, a situation that is examplified in Section 8.2 and discussed in Section 8.3. In this diagrams, we moreover assume that our lower bound (1.10) is valid for any p > 0, including no even integers when p ≥ 2. This conjectural point is discussed in Section 8.3.

Conclusive remarks and open questions
We have obtained new results on the localization of Lévy white noises in weighted Besov spaces, summarized in Theorem 1.1. This includes the identification of the local  Figure 6: Lévy white noise with 2/3 = β ∞ < β ∞ = 2 and p max = 2/3 smoothness in many cases (including the examples presented in Section 8.1), and lower and upper bounds for the general case. We also identify the asymptotic growth rate of the Lévy white noise in many situations, significantly improving known results. However, some questions remain open for a definitive answer regarding the Besov regularity of Lévy white noise.
• Moment estimates: the case p > 2. Our growth results in Theorem 1.1 present some restriction on the integrability parameter p, since relation (1.10) presently excludes the case p > 2, p / ∈ 2N. Our derivation makes extensive use of (5.8), which is only valid for p < 2. As is classic in moment estimation, the case p > 2 adds tehnical difficulties and deserve a specific treatment. Nevertheless, we conjecture that the derived formulas remains true for any 0 < p ≤ ∞, i.e., that our lower bound (1.11) is sharp for any integrability parameter.
• Blumenthal-Getoor indices and the local smoothness. Whenβ ∞ < β ∞ and p ≤ β ∞ , we have distinct lower and upper bounds for the local smoothness in Theorem 1.1.
The identification of τ p (w) for these cases is unknown at this stage. It is actually not even clear if τ p (w) can be expressed in terms of the indices considered so far.
We believe that a precise answer to this question requires the development of new tools to capture the precise behavior of the moments in relation with the scale j. A first step in this direction will be to consider the examples presented in Section 8.2.
• Critical values. We did not investigate the localization of a general Lévy white noise for the critical values τ = τ p (w) or ρ = ρ p (w). However, partial answers have been given for compound Poisson white noises (Proposition 4.2) and finite-moment white noises (Proposition 6.1). A complete characterization was given in the Gaussian case (Proposition 3.4). For the general case, we conjecture that w / ∈ B τ p (R d ; ρ) as soon as τ = τ p (w) or ρ = ρ p (w), in accordance with known results.