text:
\section{Introduction}

	\subsection{Three Constructions for the L\'evy White Noise}\label{subsec:3constructions}

This paper is dedicated to the study of $d$-dimensional L\'evy white noises.
These entities have been defined in several ways and are available in the literature as follows.

\begin{itemize}

	\item \textit{L\'evy white noise as a generalized random process:}
		A generalized random process can be \emph{a priori} observed though  test functions $\varphi$ living in the space ${\mathcal{D}}( \mathbb{R}^d)$ of compactly supported and smooth functions. The construction of L\'evy white noises as generalized random processes is established by I.M. Gelfand and N.Y. Vilenkin in \cite[Chapter III]{GelVil4}.
		It means that the L\'evy white noise $\dot{X}$ is defined by the collection of random variables $\langle \dot{X}, \varphi \rangle$ with $\varphi \in {\mathcal{D}}( \mathbb{R}^d)$,
		as presented in Section \ref{sec:GRP}.

		The distributional point of view of Gelfand and Vilenkin offers the advantage of allowing a proper definition of  L\'evy white noise as a valid (random) generalized function. It can then be used as the driving term of a linear (stochastic) partial differential equation of the form $\Lop s = \dot{X}$.

	\item \textit{L\'evy white noise as an independently scattered random measure:}
		In this framework, the L\'evy white noise is specified as a random measure; that is, a collection of random variables $\langle \dot{X}, \One_A\rangle$ indexed by Borelian subsets $A \subset  \mathbb{R}^d$ with finite Lebesgue measure. Independently scattered random measures are investigated by B.S. Rajput and J. Rosinski in \cite{Rajput1989spectral}. We provide a recap in Section \ref{sec:domain}.

	\item \textit{L\'evy white noise and It\^o-type stochastic integrals:}
	This approach relies on the L\'evy-It\^o decomposition of L\'evy processes.
	The starting point is the $d$-parameter L\'evy sheet $X = (X_t)_{t\in  \mathbb{R}^d}$ that generalizes L\'evy processes in higher dimensions.
	The It\^o theory of stochastic integration relies on the L\'evy-It\^o decomposition of a L\'evy sheet (see Theorem \ref{lidd}).
	It gives a meaning to quantities of the form $\int_{ \mathbb{R}^d}  f(s) \mathrm{d}X_s$ for well-suited deterministic functions $f$.
	In particular, with $f = \One_{[0,t]}$, one recovers the original L\'evy sheet $\int_{ \mathbb{R}^d} \One_{[0,t]}(s) \mathrm{d}X_s = X_t$.

	Strictly speaking, the L\'evy white noise is not constructed but rather only suggested in the notation $\mathrm{d}X$.
	It can nevertheless be defined as the partial derivative (in the sense of generalized functions) $\dot{X} = \frac{\partial^d}{\partial _1 \cdots \partial_d} X$ of $X$.

	This construction was for instance presented in \cite{Dalang2015Levy} and is reviewed in Section \ref{sec:integral}.

\end{itemize}

It can be argued that the specification of the L\'evy white noise as an independently scattered random measure is more informative than its description as a generalized random process.
Indeed, random measures are random generalized functions, while the converse in generally not true.
However, the framework of generalized random processes is especially adapted for the study of linear stochastic partial differential equation.
In particular, the (weak) derivative together with the integration (when it is well-defined) of a generalized random process is still a generalized random process, what is false in general for independently scattered random measures.

Despite these differences in formalisms, these three constructions are deeply interconnected and implicitly specify the same mathematical object, although observing it from different perspectives.
They range from more to less general, in the sense that measures generalize functions, and generalized functions generalize measures \cite[Chapter 1]{Schwartz1966distributions}. This remains valid for the \textit{random} functions, measures, and generalized functions.
Each approach brings its own advantages.
It is therefore interesting to precisely connect them in order to benefit from their strengths when applied to different contexts.

	\subsection{The Domain of Definition of the L\'evy White Noise}\label{subsec:whydomain}

Since we are interested in the study of linear stochastic partial differential equations driven by L\'evy white noise, we start from a L\'evy white noise defined as a generalized random process.
Any L\'evy white noise $\dot{X}$ can \emph{a priori} be observed  through a test function $\varphi$ living in the space of compactly supported and smooth functions.
However, it is often desirable in practice to extend the definition of the family of random variables $\langle \dot{X} , f \rangle$ to functions $f$ that are possibly neither smooth nor compactly supported. Hereafter, we provide some motivations in that direction.

\begin{itemize}

	\item \emph{Expansion of   L\'evy white noise  into orthonormal bases:}
	Consider an orthonormal basis $(f_n)$ of $L^2( \mathbb{R}^d)$ and a L\'evy white noise $\dot{X}$. We would like to characterize when it is reasonable to consider the family of coefficients $\langle \dot{X} , f_n \rangle$. As a motivational example, we mention our recent works~\cite{Fageot2017multidimensional} where we use the Daubechies wavelets coefficients of a L\'evy white noise to accurately estimate its regularity. Daubechies wavelets are compactly supported, but have a limited smoothness~\cite{Daubechies1988orthonormal}. We shall see that the expansion on any Daubechies wavelet basis is possible for any L\'evy white noise.
	More generally, we are interested in considering bases with elements that are neither compactly supported nor smooth.

	\item \emph{Localizing the probability law of the L\'evy white noise:}
	The domain of definition of L\'evy white noise coincides with the domain of continuity of its characteristic functional. There are strong connections between the continuity properties of the characteristic functional and the localization of the process in appropriate Sobolev spaces, as exploited in \cite[Proposition 4]{Fageot2017besov}. The more we can extend the domain of definition, the more we  learn about the regularity of the L\'evy white noise.

	\item \emph{Construction of solutions of linear SDEs driven by L\'evy white noise:}
	By extending the domain of definition of the L\'evy white noise, one weakens the conditions on a differential operator $\Lop$ such that the stochastic differential equation $\Lop s = \dot{X}$ admits a solution $s$ as a generalized random process. Indeed, we have formally that, for $\varphi \in {\mathcal{D}}( \mathbb{R}^d)$,
	\begin{equation} \label{eq:formalrequirement}
		\langle s,  \varphi \rangle = \langle \Lop^{-1} \dot{X} , \varphi \rangle = \langle \dot{X} , (\Lop^{-1})^{*} \{\varphi\} \rangle,
	\end{equation}
	where $(\Lop^{-1})^{*}$ is the adjoint of $\Lop^{-1}$.
	We therefore see that $(\Lop^{-1})^{*} \{\varphi\} $ must  belong to the domain of definition of $\dot{X}$ in order to give a meaning to \eqref{eq:formalrequirement}.

\end{itemize}

The previous examples show the interest of extending the domain of definition of the L\'evy white noise.
We also want to go further and identify the broadest set of test functions such that the random variable $\langle \dot{X}, f \rangle$ is well-defined.
It turns out that the theory of Rajput and Rosinski is especially relevant to achieving this goal.

	\subsection{Contributions and Outline}\label{subsec:contribution}

The primary contributions of this paper are as follows.

\begin{itemize}
	\item The connection of the L\'evy white noise in ${\mathcal{D}}'( \mathbb{R}^d)$ with the theory of independently scattered random measures investigated by  B.S. Rajput and J. Rosinski \cite{Rajput1989spectral}. We rely on the work of those authors to identify in full generality the domain of definition of  the L\'evy white noise, along with the domain of definition with finite $p$-th moment (Section \ref{sec:domain}).
	\item The interpretation of this identification in terms of stochastic integration (Section \ref{sec:integral}). We show that there exists an integral representation for functions that are part of the domain of definition, and that, in general, this representation strictly includes the usual space of It\^o-integrable functions associated to a L\'evy white noise.
	\item The specification of practical criteria for the identification of the domain of definition of large classes of L\'evy white noise (Section \ref{sec:criteria}). This includes Gaussian, S$\alpha$S, Laplace, and compound Poisson white noise.
	\item The application of our results for solving non-Gaussian linear stochastic partial differential equations (Section \ref{sec:application}).
\end{itemize}

\section{Generalized Random Processes and L\'evy White Noises} \label{sec:GRP}

The theory of generalized random processes was initiated in the 50's independently by I.M. Gelfand~\cite{Gelfand1955generalized} and K. It\^o~\cite{Ito1954distributions}. It has the advantage of allowing the construction of a broad class of random processes, including many instances that do not admit a pointwise representation. Generalized random processes are typically used as a general framework for the scaling limits of statistical models in conformal field theory~\cite{Abdesselam2013qft,Abdesselam2015towards}, where the continuous-domain limit fields are typically too irregular to admit a pointwise representation~\cite{Camia2015planar,Chelkak2012conformal}.
The framework also lends itself to the construction of the $d$-dimensional Gaussian white noise, as is exploited in white noise analysis~\cite{Hida1980brownian,Hida2004} or for more general classes of Gaussian processes~\cite{Lodhia2016fractional,Simon1979functional,Walsh1986introduction}.
More generally, one can describe $d$-dimensional L\'evy white noises---including   Gaussian ones---as   random elements in the space of generalized functions  \cite{GelVil4}.  We can then study linear stochastic partial differential equations driven by L\'evy white noise, whose solutions are defined as generalized random processes.
This framework has been applied in signal processing in order to specify stochastic models for sparse signals~\cite{Fageot2014,Unser2014sparse}.

	\subsection{Generalized Random Processes}

We denote by ${\mathcal{D}}( \mathbb{R}^d)$ the   space of infinitely differentiable and compactly supported functions, associated with its usual nuclear topology  \cite{Treves1967}. In particular, $\varphi_n \rightarrow \varphi$ in ${\mathcal{D}}( \mathbb{R}^d)$ as $n \rightarrow \infty$ if there exists a compact $K \subset  \mathbb{R}^d$ that contains the support of all the $\varphi_n$  and if $\lVert \mathrm{D}^{\bm{\alpha}} \{ \varphi_n - \varphi \} \rVert_{\infty} \rightarrow \infty$ for every multi-index $\bm{\alpha} \in  \mathbb{N}^d$. The topological dual of ${\mathcal{D}}( \mathbb{R}^d)$ is the  space of generalized functions  ${\mathcal{D}}'( \mathbb{R}^d)$ (often called distributions).
We define a structure of measurability on ${\mathcal{D}}'( \mathbb{R}^d)$ by considering the cylindrical $\sigma$-field $\mathcal{C}$; that is, the $\sigma$-field generated by the cylinders
\begin{equation*}
	\Big\{ u \in {\mathcal{D}}'( \mathbb{R}^d), \quad \left( \langle u,\varphi_1 \rangle, \ldots, \langle u, \varphi_N\rangle \right) \in B \Big\}
\end{equation*}
with $N\geq 1$, $\varphi_1, \ldots, \varphi_N \in {\mathcal{D}}( \mathbb{R}^d)$, and $B$ a Borel  measurable set of $ \mathbb{R}^N$.

Let $(\Omega, \mathcal{F}, \mathscr{P})$ be a complete probability space. A random variable is a measurable function from $\Omega$ to $ \mathbb{R}$. The space of random variables, $L^0(\Omega)$, is a Fr\'echet space when endowed with the convergence in probability. We also define $L^p(\Omega)$, the space of random variables with finite $p$th-moment, for $0<p< \infty$. The space $L^p(\Omega)$ is a quasi-Banach space when $0<p<1$, and a Banach space otherwise.

\begin{definition} \label{def:GRP}
	A \emph{generalized random process} is a linear and continuous function $s$ from ${\mathcal{D}}( \mathbb{R}^d)$ to $L^0(\Omega)$. The linearity means that, for every $\varphi_1, \varphi_2 \in {\mathcal{D}}( \mathbb{R}^d)$ and $\lambda \in  \mathbb{R}$,
	\begin{equation*}
		s(\varphi_1 + \lambda \varphi_2) = s(\varphi_1) + \lambda s(\varphi_2) \text{ almost surely}.
	\end{equation*}
	The continuity means that if $\varphi_n \rightarrow \varphi$ in ${\mathcal{D}}( \mathbb{R}^d)$, then $s(\varphi_n)$ converges to $s(\varphi)$ in probability.
\end{definition}

Due to the nuclear structure on ${\mathcal{D}}( \mathbb{R}^d)$, a generalized random process has a version that is a measurable function from $(\Omega,\mathcal{F})$ to $({\mathcal{D}}'( \mathbb{R}^d) , \mathcal{C})$ (see \cite[Corollary 4.2]{Walsh1986introduction}).
In other words, a generalized random process is a random generalized function.
We therefore  write $s(\varphi) = \langle s,\varphi\rangle$ where $s$ is a generalized random process and $\varphi \in {\mathcal{D}}( \mathbb{R}^d)$ a test function.

\begin{definition}
The \emph{probability law} of a generalized random process $s$ is the probability measure on ${\mathcal{D}}'( \mathbb{R}^d)$ defined by
\begin{equation*}
	\mathscr{P}_s (B) = \mathscr{P} ( s \in B ) = \mathscr{P} \{\omega \in \Omega, \ \scal s {\cdot}(\omega) \in B \}
\end{equation*}
for $B$ in the cylindrical $\sigma$-field $\mathcal{C}$.
\end{definition}

Note that $\{ s\in B\}$ is a measurable set of $\Omega$ because $s :\Omega \rightarrow \mathbb{R}$ is measurable by definition.

\begin{definition}
The \emph{characteristic functional} of a generalized random process $s$ is the Fourier transform of its probability law; that is, the functional ${\widehat{\mathscr{P}}}_s : {\mathcal{D}}( \mathbb{R}^d) \rightarrow  \mathbb{C}$ defined by
\begin{equation*}
	{\widehat{\mathscr{P}}}_s(\varphi) = \int_{{\mathcal{D}}'( \mathbb{R}^d)} \mathrm{e}^{\mathrm{i} \langle u, \varphi \rangle } \mathrm{d} \mathscr{P}_s(u) = \mathbb{E} \left[ \mathrm{e}^{\mathrm{i} \langle s ,\varphi \rangle } \right].
\end{equation*}
\end{definition}

The characteristic functional characterizes the law of $s$ in the sense that two random processes are equal in law if and only if they have the same characteristic functional.

	\subsection{L\'evy White Noises, L\'evy Exponents, and L\'evy Triplets}

One of the advantages of the theory of generalized random processes is that it allows to properly define  processes that do not admit a pointwise representation. The typical example is the Gaussian white noise. Following \cite{GelVil4}, we define the L\'evy white noise from its characteristic functional.

A random variable $Y$ is said to be infinitely divisible if it can be decomposed as $Y= Y_1 + \ldots + Y_N$ with $(Y_1, \ldots , Y_N)$ i.i.d. for every $N$. The characteristic function of an infinitely divisible random variable cannot vanish and there exists a unique continuous function $\psi$ such that ${\widehat{\mathscr{P}}}_Y(\xi) = \exp ( \psi (\xi) )$ (see \cite[Theorem 8.1]{Sato1994levy}). The  log-characteristic function $\psi$ of an infinitely divisible random variable is called a \emph{L\'evy exponent} (or a characteristic exponent).
The complete family of L\'evy exponents is specified by the L\'evy-Khintchine theorem.
\begin{theorem}[Theorem 8.1, \cite{Sato1994levy}]
A function $\psi :  \mathbb{R} \rightarrow  \mathbb{C}$ is a L\'evy exponent if and only if it can  be written as
\begin{equation} \label{eq:LK}
	\psi(\xi) = \mathrm{i} \gamma \xi - \frac{\sigma^2 \xi^2}{2} + \int_{ \mathbb{R}} \left( \mathrm{e}^{\mathrm{i} x \xi} - 1 - \mathrm{i} x \xi  \One_{\abs{x}\leq 1} \right) \nu(\mathrm{d} x),
\end{equation}
for every $\xi\in  \mathbb{R}$, with $\gamma \in  \mathbb{R}$, $\sigma^2 \geq 0$, and $\nu$ a L\'evy measure, that is a measure on $ \mathbb{R}$ with $\int_{ \mathbb{R}}  (1\wedge x^2) \nu(\mathrm{d} x) < \infty$ and $\nu\{0\} = 0$.   The triplet $(\gamma,\sigma^2, \nu)$ is uniquely determined and called the \emph{L\'evy triplet} associated to $\psi$.
 \end{theorem}

\begin{definition}\label{levynoise}
	A \emph{L\'evy white noise} $\dot{X}$ is a generalized random process with characteristic functional of the form
	\begin{equation*}
		{\widehat{\mathscr{P}}}_{\dot{X}}(\varphi) = \exp\left( \int_{ \mathbb{R}^d} \psi(\varphi(t)) \mathrm{d} t \right)
	\end{equation*}
	for every $\varphi \in {\mathcal{D}}( \mathbb{R}^d)$, where $\psi$ is a L\'evy exponent.
\end{definition}
The existence of a L\'evy white noises as generalized random processes is proved in \cite{GelVil4} (see also \cite[Theorem 2]{Fageot2014}).
A L\'evy white noise $\dot{X}$ is stationary, meaning that $\dot{X}(\cdot - t_0)$ and $\dot{X}$ has the same law for every $t_0\in \mathbb{R}^d$ (here, $\dot{X}(\cdot - t_0)$ is an abuse of notation, and is defined by $\scal{\dot{X}(\cdot - t_0)}{\varphi}:=\scal{\dot X}{\varphi(\cdot+t_0)}$ for any $\varphi \in \mathcal D( \mathbb{R}^d)$). Moreover, $\dot{X}$ is independent at every point in the sense that $\langle \dot{X},\varphi_1 \rangle$ and $\langle \dot{X} ,\varphi_2 \rangle$ are independent when $\varphi_1$ and $\varphi_2$ have disjoint supports.

The random variable $\langle \dot{X},\varphi \rangle$ is a priori well-defined  for $\varphi \in {\mathcal{D}}( \mathbb{R}^d)$.
Its characteristic function $\Phi_{\langle s, \varphi \rangle} :  \mathbb{R} \rightarrow  \mathbb{C}$ is given, for every $\xi \in  \mathbb{R}$, by
\begin{equation*}
	\Phi_{\langle s, \varphi\rangle} (\xi) =\mathbb{E} \left[ \mathrm{e}^{\mathrm{i} \xi \langle \dot{X} ,\varphi \rangle } \right]
				= \exp \left( \int_{ \mathbb{R}^d} \psi( \xi \varphi(t) ) \mathrm{d} t \right).
	\end{equation*}
However, we can reasonably extend the domain of definition of the noise to broader classes of functions.

We illustrate this idea on the Gaussian white noise $ \dot{X}_{\mathrm{Gauss}}$.
Its characteristic functional is \cite{GelVil4}
$${\widehat{\mathscr{P}}}_{ \dot{X}_{\mathrm{Gauss}}}(\varphi) = \exp( - \sigma^2 \lVert \varphi \rVert^2_2 /2).$$
For each $\varphi \in {\mathcal{D}}( \mathbb{R}^d)$,   $\langle  \dot{X}_{\mathrm{Gauss}} , \varphi \rangle$ is a centered normal random variable with variance $\sigma^2 \lVert \varphi \rVert_2^2$.
One sees  easily that $\langle  \dot{X}_{\mathrm{Gauss}}, f \rangle$ can be extended to every function $f \in L^2( \mathbb{R}^d)$. To do so, consider a sequence $(\varphi_n)$ of functions in ${\mathcal{D}}( \mathbb{R}^d)$ converging to $f \in L^2( \mathbb{R}^d)$ for the usual Hilbert topology of $L^2( \mathbb{R}^d)$. Then, for every $n,m \in  \mathbb{N}$, we have
\begin{equation} \label{eq:cauchysequencegaussian}
\mathbb{E} [ (\langle  \dot{X}_{\mathrm{Gauss}}, \varphi_n \rangle - \langle  \dot{X}_{\mathrm{Gauss}}, \varphi_m \rangle)^2 ]  =\mathbb{E} [\langle  \dot{X}_{\mathrm{Gauss}}, \varphi_n - \varphi_m \rangle^2 ] = \sigma^2 \lVert \varphi_n - \varphi_m \rVert^2_2.
\end{equation}
The sequence $(\varphi_n)$ being convergent, it is  a Cauchy sequence of $L^2( \mathbb{R}^d)$. Then, \eqref{eq:cauchysequencegaussian} implies that $(\langle  \dot{X}_{\mathrm{Gauss}}, \varphi_n\rangle )$ is itself a Cauchy sequence in the complete space $L^2(\Omega)$, and hence is convergent in this space. One readily shows that the limit does not depend on the sequence $(\varphi_n)$ and we denote it by $\langle  \dot{X}_{\mathrm{Gauss}}, f \rangle$. Then, as will be made more rigorous in the sequel, the linear and continuous functional $\dot{X}_{\mathrm{Gauss}}$ initially defined from ${\mathcal{D}}( \mathbb{R}^d)$ to $L^0(\Omega)$ is actually a linear and continuous functional from $L^2( \mathbb{R}^d)$ to $L^2(\Omega)$.

Defining a Gaussian white noise only for smooth and rapidly decaying test functions appears highly conservative. As we shall see, this occurs for any L\'evy white noise. The goal of this paper is precisely to identify the domain of definition general L\'evy white noise; that is, the largest possible space of test functions that we can apply to the white noise. The identification of the domain has already been done in \cite{Rajput1989spectral} in the context of independently scattered random measures, and the unification of L\'evy white noise with this notion in Section \ref{unification} allows us to rely on this work. Moreover, we are aiming at practical criteria to identify this domain of definition for specific L\'evy white noises.

\section{Integrability Conditions with respect to L\'evy White Noise} \label{sec:domain}

In this section, a L\'evy white noise is understood as a generalized random process in the sense of Gelfand. In section \ref{unification}, we connect this construction with the framework of independent scattered random measures of Rajput and Rosinski \cite{Rajput1989spectral}. Then, in sections \ref{subsec:extension} and \ref{subsec:moments}  we use this connection and results obtained in \cite{Rajput1989spectral} to extend the domain of definition of L\'evy white noise.

	\subsection{L\'evy White Noises as Independent Scattered Random Measures}\label{unification}

A random measure is a random process whose test functions are indicator functions.
It is very popular for stochastic integration, the integral being defined for simple functions (\emph{i.e.}, linear combinations of indicator functions), and extended by a limit argument.
Essentially, a random measure is independently scattered when two indicator functions with disjoint supports define independent random variables.
For a proper definition, see \cite[Section 1]{Rajput1989spectral}.

We show in this section that a L\'evy white noise is an example of an independently scattered random measure.
We first identify integrability conditions for test functions of the form $\One_A$ with $A$ a Borel set with finite Lebesgue measure.

\begin{definition}\label{borelset}
 Let $A\subset  \mathbb{R}^d$ be a Borel measurable set. Let $\theta\in  {\mathcal{D}} ( \mathbb{R}^d)$, $\theta \geqslant 0$ and $\int_{ \mathbb{R}^d} \theta (t) \, \mathrm{d} t=1$. For $n \in  \mathbb{N}$, and $t\in  \mathbb{R}^d$, let $\theta_n(t)=n^d \theta(nt)$ be a mollified version of the Dirac impulse. Then, for all $\varphi \in \mathcal D( \mathbb{R}^d)$, we define
\begin{equation*}
 \scal{\dot X}{\varphi \mathds 1_A}:= \lim_{n\to +\infty} \scal{\dot X}{\varphi \, \left( \theta_n * \mathds 1_A\right)}\, ,
\end{equation*}
where the limit is in probability.
\end{definition}

\begin{proposition}\label{wellposed}
 Definition \ref{borelset} is well posed. In particular the limit exists and does not depend on the choice of the mollifier $\theta$. In addition, the characteristic function of the random variable $ \scal{\dot X}{\varphi \mathds 1_A}$ is given by
\begin{equation*}
\Phi_{\scal{\dot X}{\varphi \mathds 1_A}}(\xi)= \expect{ \exp \left(  i \xi \scal{\dot X}{\varphi \mathds 1_A}\right) } = \exp\left( \int_{A} \psi(\xi \varphi(t)) \, \mathrm{d} t \right)\, , \qquad \text{for all} \ \xi\in  \mathbb{R},
\end{equation*}
 where $\psi$ is the L\'evy exponent of $\dot X$. Also, for any disjoint Borel measurable sets $A,B\subset  \mathbb{R}^d$, $\scal{\dot X}{\varphi \mathds 1_{A\cup B}}=\scal{\dot X}{\varphi \mathds 1_A}+\scal{\dot X}{\varphi \mathds 1_B}$ almost surely and $\scal{\dot X}{\varphi \mathds 1_A}$ and $\scal{\dot X}{\varphi \mathds 1_B}$ are independent.
\end{proposition}

\begin{proof}
We first remark that for any $n\in  \mathbb{N}$, the function $\varphi  (\theta_n * \mathds 1_A)$ is in $\mathcal D( \mathbb{R}^d)$, therefore the random variables $Y_n= \scal{\dot X}{\varphi ( \theta_n * \mathds 1_A) }$ are well defined. It suffices to show that the sequence $(Y_n)_{n\in  \mathbb{N}}$ is Cauchy in probability. By linearity, it is enough to show that this sequence is Cauchy in law. Let $n,m\in  \mathbb{N}$. By definition, we know that for any $\xi \in  \mathbb{R}$,
$$\expect{ \exp \left( i \xi  \scal{\dot X}{\varphi \, \left( \left( \theta_n - \theta_m \right) * \mathds 1_A \right)}\right) } =
\exp\left( \int_{ \mathbb{R}^d} \psi \left( \xi \varphi(t) \left( \left( \theta_n - \theta_m \right) * \mathds 1_A(t) \right) \right)  \, \mathrm{d} t \right)\, ,$$
where
$$\psi(\xi)=i\gamma \xi-\frac{\sigma^2 \xi^2} 2  + \int_{|x|\I1}\left( e^{i \xi x} -1-i \xi x \right) \nu(\! \, \mathrm{d} x) + \int_{|x|>1}\left( e^{i\xi x} -1 \right) \nu(\! \, \mathrm{d} x) \, .$$
We treat each of the four terms of the L\'evy exponent separately. Since $\varphi \in \mathcal D( \mathbb{R}^d)$, there is a compact $K$ such that $\text{supp} \ \varphi =: K$. Then,
\begin{align*}
 \left | \int_{ \mathbb{R}^d} i \gamma  \xi \varphi(t) \left( \theta_n - \theta_m \right) * \mathds 1_A(t) \, \mathrm{d} t \right | \leqslant |\gamma \xi| \ \|\varphi \|_\infty \left \| \left( \theta_n-\theta_m\right) * \mathds 1_{A } \right \|_{L^1(K)} \, .
\end{align*}
It is well known that for $p\geqslant 1$ and $f\in L^p(K)$, $\left( \theta_n-\theta_m\right)*f\to 0$ in $L^p(K)$ as $n,m\to +\infty$. Therefore, since $\mathds 1_{A} \in L^1(K)$, $\| \left( \theta_n - \theta_m \right) * \mathds 1_{A} \|_{L^1(K)} \to 0$ as $n,m \to +\infty$. Similarly,
\begin{equation}\label{brownian}
\begin{aligned}
&  \left | \int_{ \mathbb{R}^d} \frac{ \sigma^2 \xi^2 \varphi(t)^2 \left( \theta_n - \theta_m \right) * \mathds 1_A(t)^2 } 2 \, \mathrm{d} t \right | \leqslant \frac 1 2 |\sigma^2 \xi^2 | \  \|\varphi \|_\infty^2  \left \|  \left( \theta_n - \theta_m \right) * \mathds 1_{A} \right \|_{L^2(K)}^2  \, .
\end{aligned}
\end{equation}
Since $\mathds 1_{A} \in L^2(K)$, $\| \left( \theta_n - \theta_m \right) * \mathds 1_{A} \|_{L^2(K)}^2 \to 0$ as $n,m \to +\infty$. Then, by \cite[Lemma 5.14]{Kallenberg2006foundations},
$$ \left | e^{i\xi x}-1- i \xi x \right | \leqslant \frac 1 2 |\xi x|^2\, ,$$
and
\begin{align*}
 \int_{ \mathbb{R}^d} \int_{|x|\I1}& \left | e^{i\xi \left( \theta_n - \theta_m \right) * \mathds 1_A(t) x}  -1-i \xi \left( \theta_n - \theta_m \right) * \mathds 1_A(t)x \right |  \nu(\! \, \mathrm{d} x)  \, \mathrm{d} t  \\
& \hspace{2cm} \leqslant  \frac 1 2  | \xi ^2|\|\varphi \|_\infty^2 \left( \int_{|x| \leqslant 1} x^2 \nu(\! \, \mathrm{d} x) \right)   \left \| \left( \theta_n - \theta_m \right) * \mathds 1_{A}\right \|_{L^2(K)}^2 \, \mathrm{d} t \, ,
\end{align*}
and we conclude as for \eqref{brownian}. The last term represents the compound Poisson part of the L\'evy-It\^o decomposition of the L\'evy white noise. It corresponds to the characteristic function of the random variable $M_{n,m}:=\int_{ \mathbb{R}^d} \int_ \mathbb{R} z \varphi(t) (\theta_n-\theta_m)*\mathds 1_A(t) J(\! \, \mathrm{d} t, \, \mathrm{d} z)$ where $J$ is a Poisson random measure on $ \mathbb{R}^d\times  \mathbb{R}$ with intensity measure $\! \, \mathrm{d} t \mathds 1_{|z|>1} \nu(\! \, \mathrm{d} z)$, and we know that
$$M_{n,m}= \sum_{i\geqslant 1} Z_i \varphi(T_i) \, \left( \theta_n - \theta_m \right) * \mathds 1_A(T_i)\, ,$$
for some random space-time points $(Z_i, T_i)_{i\geqslant 1}$, and the sum above has finitely many terms (independently of $m,n$) almost surely due to the compactness of the support of $\varphi$. Indeed, with $K=\text{supp} \, \varphi$, we have
$$\expect{ J\left( K\times  \mathbb{R} \right) } = \int_{K\times  \mathbb{R} } \, \mathrm{d} t \mathds 1_{|z|>1}\nu(\! \, \mathrm{d} z)=\leb d K \int_{|z|>1} \nu(\! \, \mathrm{d} z)<+\infty\, ,$$
since $\nu$ is a L\'evy measure, and $J\left( K\times  \mathbb{R} \right)$ is the random variables that counts the number of points $T_i$ that fall in the support of $\varphi$. By Lebesgue's differentiation theorem (see \cite[Chapter 7, Exercise 2]{zygmund}), $\left(\theta_n - \theta_m \right) * \mathds 1_A(t) \to 0$ as $n,m\to +\infty$ for all $t\in K\backslash H$, where $H$ is a subset of $ \mathbb{R}^d$ such that the Lebesgue measure is $\leb d H=0$. The random times $T_i$ have an absolutely continuous law. Indeed, for any Borel set $B\subset  \mathbb{R}^d$,
$$ \mathbb{P}(T_i \in B ) \leqslant \mathbb{P}(J(B \times  \mathbb{R}) \geqslant 1)\leqslant \leb d B \int_{|z| >1} \nu(\! \, \mathrm{d} z)\, .$$
Therefore,  for all $i\geqslant 1$, $\mathbb{P}(T_i\in H)=0$ and  $M_{n,m} \to 0$ as $n,m \to +\infty$ almost surely, hence also in law. Therefore $(Y_n)_{n\in  \mathbb{N}}$ is a Cauchy sequence in law, hence in probability, and there exists a random variable $Y$ such that $Y_n \to Y$ in probability as $n \to +\infty$. By checking the convergence of each term of the decomposition of the L\'evy exponent, it is easy to see that for all $\xi \in  \mathbb{R}$,
$$\int_{ \mathbb{R}^d} \psi \left( \xi \varphi(t) \theta_n  * \mathds 1_A(t)\right)  \, \mathrm{d} t \to \int_{A} \psi \left( \xi \varphi(t)\right)  \, \mathrm{d} t\, , \qquad \text{as} \ n\to +\infty\, ,$$
hence
$$\expect{ e^{i\xi Y}} =\exp \left( \int_{A} \psi \left( \xi \varphi(t)\right)  \, \mathrm{d} t\right)\, .$$
If $\tilde \theta$ is another mollifier, and $(\tilde Y_n)_{n\in  \mathbb{N}}$ and $\tilde Y$ are  the associated sequence and limit, it is easy to see by linearity of $\dot X$ that $Y_n -\tilde Y_n \to 0$ in probability as $n\to +\infty$. If $A$ and $B$ are disjoint Borel measurable sets of $ \mathbb{R}^d$, we observe that $\theta_n*\mathds 1_{A\cup B}=\theta_n*\mathds 1_{A}+ \theta_n*\mathds 1_{B}$, which proves the decomposition $\scal{\dot X}{\varphi \mathds 1_{A\cup B}}=\scal{\dot X}{\varphi \mathds 1_A}+\scal{\dot X}{\varphi \mathds 1_B}$ at the limit when $n\to +\infty$. Independence comes from the factorisation of the characteristic function of these random variables.
\end{proof}

From the previous definition, it is straightforward to define the random variables $\scal{\dot X}{\mathds 1_A}$ for any bounded borel set $A$. Indeed, it suffices to choose any $\varphi\in \mathcal D( \mathbb{R}^d)$ such that $\varphi_{|A}=1$ and set $\scal{\dot X}{\mathds 1_A}:=\scal{\dot X}{\varphi \mathds 1_A}$. This definition does not depend on the choice of $\varphi$. Indeed, if $\varphi$ and $\psi$ are two such functions, by linearity of the noise and from the expression of the characteristic function, we get that $\scal{\dot X}{\left(\varphi-\psi \right) \mathds 1_A}=0$ almost surely. In this particular case, we observe that the characteristic function of the noise takes the particular form
\begin{equation}\label{chifunction}
 \expect{ \exp \left(  i \xi \scal{\dot X}{ \mathds 1_A}\right) } = \exp\left(\leb d {A}  \psi(\xi) \right)\, .
\end{equation}
 We now extend the definition to a Borel set with finite measure (but not necessarily bounded).
\begin{definition}\label{borelset2}
 Let $A\subset  \mathbb{R}^d$ be a Borel measurable set such that $\leb d A <+\infty$. For $n\in  \mathbb{N}^*$, let $A_n=A\cap[-n,n]^d$. Then we define
 \begin{equation*}
 \scal{\dot X}{ \mathds 1_A}:= \lim_{n\to +\infty} \scal{\dot X}{ \mathds 1_{A_n}}\, ,
\end{equation*}
where the limit is in probability.
\end{definition}

\begin{proposition}\label{extension}
  Definition \ref{borelset2} is well posed. In addition, the characteristic function of the random variable $ \scal{\dot X}{ \mathds 1_A}$ is given by
\begin{equation}\label{chifunction2}
 \expect{ \exp \left(  i \xi \scal{\dot X}{ \mathds 1_A}\right) } = \exp\left( \leb d {A} \psi(\xi)  \right)\, ,  \qquad \text{for all} \ \xi \in  \mathbb{R} \, ,
\end{equation}
 where $\psi$ is the Levy exponent of $\dot X$.  Also, for any disjoint Borel measurable sets $A,B\subset  \mathbb{R}^d$, such that $\leb d A, \leb d B <+\infty$, $\scal{\dot X}{ \mathds 1_{A\cup B}}=\scal{\dot X}{ \mathds 1_A}+\scal{\dot X}{ \mathds 1_B}$ almost surely and $\scal{\dot X}{ \mathds 1_A}$ and $\scal{\dot X}{ \mathds 1_B}$ are independent.
\end{proposition}

\begin{proof}
From the expression of the characteristic function in \eqref{chifunction} and the fact that $A_n$ is a bounded Borel set, it is easy to see that the sequence $\left( \scal{\dot X}{ \mathds 1_{A_n}}\right)_{n\in  \mathbb{N}}$ is Cauchy in law, and therefore converges in probability. The expression of the characteristic function follows from the fact that $\leb d A <+\infty$ and the application of the dominated convergence theorem. The last statement comes directly from an application of Proposition \ref{wellposed} and the expression of the characteristic functions.
\end{proof}

Let $\mathcal L( \mathbb{R}^d)$ be the $\delta$-ring\footnote{A $\delta$-ring is a collection of sets that is closed under finite union, countable intersection, and relative complementation \cite[Definition 1.2.13]{Bogachev2007measure}. It appears in measure theory, especially when one wants to avoid sets with infinite measure.} of Borel-measurable subsets of $ \mathbb{R}^d$ finite finite Lebesgue measure. Since $\bigcup_{n\in  \mathbb{N}} [-n,n]^d= \mathbb{R}^d$, condition $(1.4)$ of \cite{Rajput1989spectral} is satisfied. By the two previous propositions, we have defined random set function as an extension of the L\'evy white noise $\dot X$ on $\mathcal L( \mathbb{R}^d)$. We still refer to this set function as L\'evy white noise.

\begin{theorem} \label{theo:connection}
 The extension of the L\'evy white noise $\dot X$ is an independently scattered random measure in the sense of \cite{Rajput1989spectral}.
\end{theorem}

\begin{proof}
The definition we refer to can be found in \cite[p.455]{Rajput1989spectral}. Let $(A_n)_{n\in  \mathbb{N}}$ be a sequence of disjoint sets in $\mathcal L( \mathbb{R}^d)$. Let $k\in  \mathbb{N}$ and $i_1<\dots < i_k \in  \mathbb{N}$. We show that the random variables $\scal{\dot X}{ \mathds 1_{A_{i_j}}}$,  $1\leqslant j \leqslant k$ are independent. By linearity of the noise, this fact is an immediate consequence of \eqref{chifunction2} and the $\sigma$-additivity of Lebesgue measure. This proves that $\scal{\dot X}{ \mathds 1_{A_{n}}}$,  $n\in  \mathbb{N}$ is a sequence of independent random variables. If in addition $\sum_{n\in  \mathbb{N}} \leb d {A_n} <+\infty$,  then $\bigcup_{n\in  \mathbb{N}} A_n \in \mathcal L( \mathbb{R}^d)$ and we need to show that
$$\scal{\dot X}{ \mathds 1_{\bigcup_{n\in  \mathbb{N}} A_{n}}}= \sum_{n\in  \mathbb{N}} \scal{\dot X}{ \mathds 1_{A_{n}}}\, ,$$
 where the series converges almost surely. From the second statement of Proposition \ref{extension}, it is easy to see that for any $k\in  \mathbb{N}$,
\begin{equation}\label{sum}
 \scal{\dot X}{ \mathds 1_{\bigcup_{n=1}^k A_{n}}}= \sum_{n=1}^k \scal{\dot X}{ \mathds 1_{A_{n}}}\, .
\end{equation}
From the expression of the characteristic function of the left-hand side of \eqref{sum} and by linearity, we see that $\scal{\dot X}{ \mathds 1_{\bigcup_{n=1}^k A_{n}}} \to \scal{\dot X}{ \mathds 1_{\bigcup_{n\in  \mathbb{N}} A_{n}}}$ in probability as $k\to +\infty$. Therefore the right-hand side of \eqref{sum} is a sum of independent random variables that converges in probability. By \cite[Theorem 5.3.4]{chung}, the sum converges almost surely, which concludes the proof.
\end{proof}

	\subsection{Extension of the Domain of Definition} \label{subsec:extension}

Having connected L\'evy white noises in $\mathcal D'( \mathbb{R}^d)$ with independently scattered random measures, it is then possible to construct a stochastic integral of non-random functions. This is done in \cite{Rajput1989spectral} and we simply restate the main definitions and theorems for the convenience of the reader. For any simple function $f=\sum_{i=1}^n x_i \mathds 1_{A_i}$ where $\leb d {A_i}<+\infty$ for any $1\leqslant i\leqslant n$, we can define for any Borel set $A$,
\begin{equation*}
 \scal{\dot X}{f\mathds 1_A}:=\sum_{i=1}^n x_i \scal{\dot X}{\mathds 1_{A_i\cap A}}\, .
\end{equation*}
\begin{definition}\label{defrosinski}
 We say that a Borel-measurable function $f:  \mathbb{R}^d \to  \mathbb{R}$ is \emph{$\dot X$-integrable} if there exists a sequence of simple function $(f_n)_{n\in  \mathbb{N}}$ such that $f_n\to f$ almost everywhere for the Lebesgue measure as $n\to +\infty$, and for any Borel set $A$, the sequence $\left( \scal{\dot X}{f_n \mathds 1_A}\right)_{n\in  \mathbb{N}}$ converges in probability as $n\to+\infty$. In this case we define for any Borel set $A$,
\begin{equation*}
 \scal{\dot X}{f \mathds 1_A}:=\lim_{n\to+\infty} \scal{\dot X}{f_n \mathds 1_A}\, .
\end{equation*}
We denote by $L(\dot X)$ the set of all $\dot X$-integrable functions.
\end{definition}

Definition \ref{defrosinski} identifies the class of measurable test functions $f$ such that $\langle \dot{X}, f \rangle$ is well-defined.
Then, we have a characterization of $\dot X$-integrable functions.

\begin{proposition}\label{integrable}\cite[Theorem 2.7]{Rajput1989spectral}
 Let $\dot X$ be a L\'evy white noise with characteristic triplet $(\gamma, \sigma, \nu)$, and $f: \mathbb{R}^d \to  \mathbb{R}$ be a Borel-measurable function. Then $f \in L(\dot X)$ if and only if the following conditions are satisfied:
\begin{itemize}
 \item[(i)] $\int_{ \mathbb{R}^d} \left | \gamma  f(s)+ \left( \int_{ \mathbb{R}} x f(s) \left( \mathds 1_{|xf(s)| \leqslant 1}- \mathds 1_{|x|\leqslant 1} \right) \nu(\! \, \mathrm{d} x) \right) \right | \, \mathrm{d} s<+\infty$,
 \item[(ii)] $\sigma f\in L^2( \mathbb{R}^d)$,
 \item[(iii)] $ \int_{ \mathbb{R}^d \times  \mathbb{R}} \left( |x f(s)|^2 \wedge 1 \, \mathrm{d} s \right) \nu(\! \, \mathrm{d} x)<+\infty$.
\end{itemize}
\end{proposition}

\begin{definition}
 We set
\begin{align*} \label{eq:Psi}
\Psi (\xi) =   \Big\lvert \gamma \xi + \int_{ \mathbb{R}} x \xi \left( \One_{\abs{x \xi}\leq 1} - \One_{\abs{x}\leq 1} \right) \nu(\mathrm{d} x) \Big\rvert
				+ \sigma^2 \xi^2
				+ \int_{ \mathbb{R}}  \left( 1\wedge (x^2\xi^2) \right) \nu(\mathrm{d} x).
\end{align*}
We call $\Psi$ the \emph{Rajput-Rosinski exponent} of $\dot{X}$.
\end{definition}

Then, $f\in L(\dot{X})$ if and only if $\Psi(f) := \int_{ \mathbb{R}^d} \Psi(f(t)) \mathrm{d} t < \infty$.
It turns out that the space of $\dot X$-integrable functions has a rich structure of generalized Orlicz spaces.
The definition and first properties of those spaces are recall in Appendix \ref{app:orlicz}.
We refer to \cite{Rao1991theory} for an in-depth exposition, with a special emphasis on the Chapter X.

\begin{proposition} \label{prop:LdotXOrlicz}
	The Rajput-Rosinski exponent $\Psi$ of a L\'evy white noise $\dot{X}$ is a $\Delta_2$-regular $\varphi$-function in the sense of Definition \ref{def:phifun} (see Appendix \ref{app:orlicz}).
	Therefore, $L(\dot{X}) = L^\Psi( \mathbb{R}^d)$ is a generalized Orlicz space. It is therefore a complete linear metric space equipped with the F-norm
	\begin{equation*}
		\lVert f \rVert_{\Psi} := \inf \{ \lambda > 0, \Psi( f / \lambda ) \leq \lambda \}.
	\end{equation*}
	The space ${\mathcal{D}}( \mathbb{R}^d)$ is dense in $L(\dot{X})$ and the convergence of a sequence of functions $f_k$ in $L(\dot{X})$ to $0$ is equivalent to $\Psi(f_n) \rightarrow 0$ as $n \rightarrow \infty$.
\end{proposition}

\begin{proof}
Rajput and Rosinski have shown that $\Psi$ is a $\varphi$-function in  \cite[Lemma 3.1]{Rajput1989spectral}.
By definition, $L(\dot{X}) = L^{\Psi}( \mathbb{R}^d)$ is therefore a generalized Orlicz space in the sense of Definition \ref{def:orlicz}.
Except for the density, the  properties of the space $L(\dot{X})$ are then derived from Proposition \ref{prop:orlicz}.
For the density, Proposition \ref{prop:orlicz} implies that simple functions are dense in $L(\dot{X})$. It suffices therefore to remark that a simple function can be easily approximated by a function in ${\mathcal{D}}( \mathbb{R}^d)$ for the topology of $L(\dot{X})$ by taking a regularized version the simple function.
\end{proof}

 The connection between L\'evy white noises in ${\mathcal{D}}'( \mathbb{R}^d)$ and independently scattered random measures established in Theorem \ref{theo:connection} allows us to apply \cite[Theorem 3.3]{Rajput1989spectral} to the L\'evy white noise and leads to the following result.

 \begin{theorem}  \label{prop:continuity}
Let $\dot{X}$  be a L\'evy white noise. Then, the functional
\begin{align*}
	\dot{X} :     L(\dot{X}) &\rightarrow L^0 (\Omega) \\
			 f &\mapsto \langle \dot{X}, f  \rangle
\end{align*}
is linear and continuous. In other words, $\dot{X}$ is a random linear functional on $L(\dot{X})$.
\end{theorem}

Theorem \ref{prop:continuity} gives general conditions  on test functions for integrability with respect to a given L\'evy white noise. It therefore specifies the domain of definition of $\dot{X}$; that is, the broadest class test functions on which $\dot X$ is a random linear functional. Once the random variable $\langle \dot{X}, f \rangle$ is well-defined, it is important to identify its characteristic function. The following result is the last part of \cite[Theorem 2.7]{Rajput1989spectral}.

\begin{proposition}  \label{prop:CharacteristicFunction}
	Consider a white noise $\dot{X}$ with L\'evy exponent $\psi$.
	For a fixed $f \in L(\dot{X})$, the characteristic function of $\langle \dot{X} ,f \rangle$ is
	\begin{equation*}
		\Phi_{\langle \dot{X} ,f \rangle}(\xi) = \exp\left( \int_{ \mathbb{R}^d} \psi(\xi f (t)) \mathrm{d} t \right).
	\end{equation*}
\end{proposition}

	More generally, one uses the previous results to extend the domain of continuity and positive-definiteness of the characteristic functional of a L\'evy white noise.

	\begin{proposition} \label{prop:continuityCF}
	For any L\'evy white noise $\dot{X}$, the characteristic functional ${\widehat{\mathscr{P}}}_{\dot{X}}$ is well-defined, continuous, and positive-definite over $L(\dot{X})$, and is given by
	\begin{equation*}
		{\widehat{\mathscr{P}}}_{\dot{X}}(f) = \exp\left( \int_{ \mathbb{R}^d} \psi(  f (t)) \mathrm{d} t \right).
	\end{equation*}
	\end{proposition}

	\begin{proof}
	The characteristic functional $\varphi \mapsto {\widehat{\mathscr{P}}}_{\dot{X}}(\varphi) = \mathbb{E}[\mathrm{e}^{\mathrm{i} \langle \dot{X},\varphi\rangle} ]$ is  continuous over ${\mathcal{D}}( \mathbb{R}^d)$.
	For any $f \in L(\dot{X})$, we know that $\langle \dot{X},f \rangle$ is a well-defined random variable according to Proposition \ref{prop:CharacteristicFunction}, and that its characteristic function is $\xi \mapsto  \mathbb{E}[\mathrm{e}^{\mathrm{i} \xi \langle \dot{X},f\rangle}] = \exp\left( \int_{ \mathbb{R}^d} \psi( \xi f(t)) \mathrm{d} t \right)$.  We can therefore extend ${\widehat{\mathscr{P}}}_{\dot{X}}$ to $L(\dot{X})$ by setting
	\begin{equation*}
		{\widehat{\mathscr{P}}}_{\dot{X}} (f) =  \mathbb{E}[\mathrm{e}^{\mathrm{i}   \langle \dot{X},f\rangle}] =\exp\left( \int_{ \mathbb{R}^d} \psi(   f(t)) \mathrm{d} t \right).
	\end{equation*}

	\textit{Positive-definiteness.} Let $N \geq 1$, $a_n \in  \mathbb{C}$, $f_n \in L(\dot{X})$, $n=1,\cdots, N$.
	The space ${\mathcal{D}}( \mathbb{R}^d)$ is dense in $L(\dot{X})$ (Proposition \ref{prop:LdotXOrlicz}), hence there exists for $N$ sequences $(\varphi_k^n)_{k \in  \mathbb{N}}$ such that $\varphi_k^n \rightarrow f_n$ in $L(\dot{X})$ for $n= 1, \cdot , N$.
	From Theorem \ref{theo:continuity}, we know that $f \mapsto \langle \dot{X}, f \rangle$ is continuous from $L(\dot{X})$ to $L^0 (\Omega)$. In particular, we have $\mathbb{E}[\mathrm{e}^{\mathrm{i} \langle \dot{X}, \varphi_k^i -  \varphi_k^j \rangle}] \underset{k \rightarrow \infty}{\longrightarrow} \mathbb{E}[\mathrm{e}^{\mathrm{i} \langle \dot{X}, f_i - f_j \rangle}]$ for every $1\leq i,j \leq N$. Finally, we have that
	\begin{align*}
		\sum_{1\leq i,j\leq N} a_i \bar{a}_j {\widehat{\mathscr{P}}}_{\dot{X}} (f_i - f_j)
		&=
		\sum_{1\leq i,j\leq N} a_i \bar{a}_j\mathbb{E}[\mathrm{e}^{\mathrm{i} \langle \dot{X}, f_i - f_j \rangle}] \nonumber \\
		&=
		\lim_{k\rightarrow \infty}  \sum_{1\leq i,j\leq N} a_i \bar{a}_j \mathbb{E}[\mathrm{e}^{\mathrm{i} \langle \dot{X}, \varphi_k^i -  \varphi_k^j \rangle}] \nonumber \\
		&=
			\lim_{k\rightarrow \infty}  \sum_{1\leq i,j\leq N} a_i \bar{a}_j {\widehat{\mathscr{P}}}_{\dot{X}} (\varphi_i -\varphi_j) \nonumber \\
		&\geq 0
	\end{align*}
	since ${\widehat{\mathscr{P}}}_{\dot{X}}$ is positive-definite over ${\mathcal{D}}( \mathbb{R}^d)$.

	\textit{Continuity.} Using the L\'evy-Khintchine representation \eqref{eq:LK} of $\psi$ with L\'evy triplet $(\gamma, \sigma^2, \nu)$, we have
	\begin{align} \label{eq:boundpsiPsi}
		\lvert \psi (\xi) \rvert &= \Big\lvert   \mathrm{i} \gamma \xi  + \mathrm{i} \int_{ \mathbb{R}}  x \xi \left( \One_{\abs{x \xi }\leq 1}- \One_{\abs{x}\leq 1} \right) \nu(\mathrm{d} x) + \sigma^2 \xi^2 + \int_{ \mathbb{R}} (\mathrm{e}^{\mathrm{i} x \xi} - 1 - \mathrm{i} x \xi \One_{\abs{ x \xi } \leq 1} ) \nu (\mathrm{d} x) \Big\rvert \nonumber \\
		&\leq \Big\lvert \gamma \xi + \int_{ \mathbb{R}} x \xi \left( \One_{\abs{x \xi }\leq 1}- \One_{\abs{x}\leq 1} \right) \nu(\mathrm{d} x) \Big\rvert
				+ \sigma^2 \xi^2
				+ 2 \int_{ \mathbb{R}} 1\wedge (x^2\xi^2) \nu(\mathrm{d} x) \nonumber \\
		&\leq 2 \Psi(\xi),
	\end{align}
	where we used the triangular inequality and the relation $\lvert \mathrm{e}^{\mathrm{i} y} -1 - \mathrm{i} y \One_{\abs{y} \leq 1} \rvert \leq 2(1 \wedge y^2)$ applied to $y= x \xi$. Applying \eqref{eq:boundpsiPsi} to $\xi = f(t)$ and integrating over $ \mathbb{R}^d$, we have for every $f \in L(\dot{X})$,
	\begin{equation*}
		\lvert \log {\widehat{\mathscr{P}}}_{\dot{X}}(f) \rvert \leq \int_{ \mathbb{R}^d} \lvert \psi ( f(t) )\rvert \mathrm{d} t \leq 2 \rVert f \lVert_{\Psi}.
	\end{equation*}
	Hence ${\widehat{\mathscr{P}}}_{\dot{X}}$ is continuous at $0$. The functional ${\widehat{\mathscr{P}}}_{\dot{X}}$ is positive-definite and continuous at $0$, and therefore continuous \cite[Section IV.1.2, Proposition 1.1]{ProbaBanach1987}.
	\end{proof}

	\subsection{Moments of $\langle \dot{X},f\rangle$}\label{subsec:moments}

We say that a generalized random process $s$ has \emph{finite $p$th moments} when $\mathbb{E} [\lvert {\langle \dot{X}, \varphi \rangle} \rvert^p] < \infty$ for any test function $\varphi \in {\mathcal{D}}( \mathbb{R}^d)$.
Consider the $d$-dimensional L\'evy white noise $\dot{X}$ and the infinitely divisible random variable $Y$ with common L\'evy exponent.
For $p > 0$, we have the equivalence \cite{Unser2014sparse}
\begin{equation*}
	\mathbb{E} [\abs{Y}^p] < \infty \Longleftrightarrow \forall \varphi \in {\mathcal{D}}( \mathbb{R}^d), \mathbb{E} [\lvert {\langle \dot{X}, \varphi \rangle}\rvert^p]  < \infty.
\end{equation*}
In particular, if $\dot{X}$ has finite $p$-th moments, the random variables $\langle \dot{X}, \varphi \rangle$ have all a finite $p$th moment.
This is not anymore the case in general for $\langle \dot{X}, f \rangle$ when $f \in L(\dot{X})$.

\begin{definition}
	If $\dot{X}$ as finite $p$-th-moments for some $p>0$, we set
	\begin{equation*}
		L^p (\dot{X}) = \{ f \in L(\dot{X}), \ \mathbb{E}[\lvert \langle \dot{X}, f \rangle \rvert^p] < \infty.
	\end{equation*}
\end{definition}

\begin{proposition}[Existence of moments, Theorem 3.3 in  \cite{Rajput1989spectral}]
	Let $0\leq p$.
	For a white noise $\dot{X}$ and $f \in L(\dot{X})$, we have
	\begin{equation*}
		f \in L^p(\dot{X}) \Longleftrightarrow \int_{ \mathbb{R}^d}\int_{ \mathbb{R}} \lvert x  \varphi(t ) \rvert^p \One_{\abs{ x \varphi(t) } > 1}  \nu(\mathrm{d} x) \mathrm{d} t < \infty.
	\end{equation*}
\end{proposition}

\begin{definition}
	For $p > 0$, the \emph{$p$th-order Rajput-Rosinski exponent} of the L\'evy white noise $\dot{X}$ is defined as
\begin{align} \label{eq:Psip}
\Psi _p (\xi) = &   \Big\lvert \gamma \xi + \int_{ \mathbb{R}} x \xi \left( \One_{\abs{x \xi }\leq 1} - \One_{\abs{x}\leq 1} \right) \nu(\mathrm{d} x) \Big\rvert
				+ \sigma^2 \xi^2 \nonumber \\
				& + \int_{ \mathbb{R}} \left( \lvert x  \varphi(t ) \rvert^p \One_{\abs{ x \varphi(t) } > 1} +  \lvert x \varphi(t ) \rvert^2 \One_{\abs{ x \varphi(t) } \leq 1}  \right) \nu(\mathrm{d} x).
\end{align}
\end{definition}
Then, $f\in L^p(\dot{X})$ if and only if $ \Psi_p(f) := \int_{ \mathbb{R}^d} \Psi_p(f(t)) \mathrm{d} t < \infty$.
Again, according to \cite[Lemma 3.1]{Rajput1989spectral}, $\Psi_p$ is a $\varphi$-function for every $p>0$. Its means that $L^p(\dot{X})$ is a generalized Orlicz space associated to $\Psi_p$. Proposition \ref{prop:LdotXOrlicz} is therefore valid for $L^p(\dot{X})$ instead of $L(\dot{X})$.
The space $L^p(\dot{X})$ is the largest space of test functions such that $\langle \dot{X} , f \rangle$ is well-defined and has a finite $p$th-moment.
Of course, $L^p(\dot{X}) \subseteq L(\dot{X})$, and we know from Theorem \ref{prop:continuity} that $\dot{X}$ is a random linear map from $L^p(\dot{X})$ to $L^0(\Omega)$. We have actually a stronger result, still taken from \cite{Rajput1989spectral}.

\begin{theorem}[Theorem 3.3 in  \cite{Rajput1989spectral}] \label{theo:continuity}
Let $\dot{X}$  be a L\'evy white noise. Then, the functional
\begin{align*}
	\dot{X} :     L^p(\dot{X}) &\rightarrow L^p (\Omega) \\
			 f &\mapsto \langle \dot{X}, f  \rangle
\end{align*}
is linear and continuous. In other words, $\dot{X}$ is an $L^p(\Omega)$-valued random linear functional on $L^p(\dot{X})$.
\end{theorem}

\section{Representation of the L\'evy White Noise  as a Stochastic Integral} \label{sec:integral}

The definition of L\'evy white noise we gave is in terms of its characteristic functional. It is an abstract definition that relies on the Minlos-Bochner theorem, which is not constructive. We give here an example of a construction of L\'evy white noise as the derivative (in the sense of Schwartz's theory of generalized functions) of a L\'evy process (or a L\'evy field in the case of dimension $d \geq 2$). We recall the  definition of a L\'evy field. A general presentation of the theory of \emph{multiparameter L\'evy fields} can be found in \cite{adler}; see also \cite{Dalang1992}.

\subsection{From L\'evy Sheets to L\'evy White Noises} \label{subsec:sheetnoise}

Let $(X_t)_{t\in  \mathbb{R}_+^d}$ be a $d$-parameter random field. We define the $2^d$ relations $\mathcal R=(\mathcal R_1, ..., \mathcal R_d)$, where $\mathcal R_i$ is either $\leqslant$ or $>$, and $a \,\mathcal R \, b$ if and only if $a_i\mathcal R_i b_i$ for all $1\leqslant i \leqslant d$. For $a\leqslant b \in  \mathbb{R}^d_+$, we define the box $(a,b]=\left\{t\in  \mathbb{R}_+^d : a<t \leqslant b \right \}$, and the increment $ \Delta_a^b X$ of $X$ over the box $(a,b]$ by
\begin{equation*}
 \Delta_a^b X=\sum_{\varepsilon \in \{0,1\}^d}(-1)^{|\varepsilon |}X_{c_\varepsilon (a,b)} \ ,
\end{equation*}
where for any $\varepsilon \in \{0,1\}^d$, we write $|\varepsilon|=\sum_{i=1}^d \varepsilon_i$ and $c_\varepsilon(a,b) \in  \mathbb{R}^d_+$ is defined by $ c_\varepsilon(a,b)_i=a_i \mathds 1 _{\{\varepsilon_i=1\}}+b_i \mathds 1 _{\{\varepsilon_i=0\}}$, for all $1\leqslant i \leqslant d$.
The next definition is a generalization of the \textit{càdlàg} property to processes indexed by $ \mathbb{R}_+^d$.
\begin{definition}
 Using the terminology in \cite{adler} and \cite{straf}, we say that $X$ is \emph{lamp} (for limit along monotone paths) if we have the following:
\begin{itemize}
 \item[(i)] For all $2^d$ relations $\mathcal R$, $\displaystyle \lim_{u\to t, \, t\mathcal R u} \ X_u$ exists.
 \item[(ii)] If $\mathcal R=\left( \leqslant, ..., \leqslant \right)$ then $\displaystyle X_t=\lim_{u\to t, \, t\mathcal R u} \ X_u$.
 \item[(iii)] $X_t=0$ if $t_i=0$ for some $1\leqslant i \leqslant d$.
\end{itemize}
\end{definition}

\noindent
We are now ready to give the definition of a L\'evy field in $ \mathbb{R}_+^d$.

\begin{definition}
$X=(X_t)_{t\in  \mathbb{R}_+^d}$ is a $d$-parameter L\'evy field if it has the following properties:
\begin{itemize}
 \item[(i)]$X$ is continuous in probability.
 \item[(ii)]$X$ is \textit{lamp} almost surely.
 \item[(iii)]For any sequence of disjoint boxes $(a_k,b_k], \ 1\leqslant k \leqslant n$, the random variables $\Delta_{a_k}^{b_k} X$ are independent.
 \item[(iv)]Given two boxes $(a,b]$ and $(c,d]$ in $ \mathbb{R}_+^d$ such that $(a,b]+t=(c,d]$ for some $t\in  \mathbb{R}^d$, the increments $\Delta_a^bX$ and $\Delta_c^dX$ are identically distributed.
\end{itemize}
The jump $\Delta_t X$ of $X$ at time $t$ is defined by $ \displaystyle \Delta_tX=\lim_{u\to t, \, u<t} \Delta_u^t X$.
\end{definition}

This definition coincides with the notion of L\'evy process when $d=1$. In addition, for all $t=(t_1,... , t_d)\in  \mathbb{R}_+^d$, and for all $1\leqslant i\leqslant d$, the process $X^{i,t}_\cdot=X_{(t_1,...,t_{i-1},\, \cdot \, ,t_{i+1},...,t_d)}$ is a L\'evy process (the notation here means that it is the process in one parameter obtained by fixing all the coordinates of $t$ except the $i$-th).

The Brownian sheet is an example of such a $d$-parameter L\'evy field. It is the analog in this framework of Brownian motion and further properties of this field are detailed in \cite{dalang_levelset}, \cite{walshdalang_browniansheet}, \cite{multi} or \cite{Walsh1984}.

We can now state the multidimensional analog of the L\'evy-It\^o decomposition, taken from \cite[Theorem 4.6]{adler} particularized to the case of stationary increments (see also \cite{Dalang1992}).

\begin{theorem}\label{lidd}
 Let $X$ be a $d$-parameter L\'evy field with characteristic triplet $(\gamma, \sigma, \nu)$. The following holds:
\begin{itemize}
 \item[(i)] The jump measure $J_X$ defined on $ \mathbb{R}_+^d \times \left(  \mathbb{R}\backslash \{0\} \right) $ by $J_X(B)=\# \left \{ (t, \Delta_t X) \in B   \right \}$, for $B$ in the Borel $\sigma$-algebra of $ \mathbb{R}_+^d \times \left(  \mathbb{R}\backslash \{0\} \right)$, is a Poisson random measure with intensity $ \text{Leb}_d\times \nu$, and $\nu$ is a L\'evy measure. In particular,
 $$\int_ \mathbb{R} \left( x^2 \wedge 1 \right)  \nu(\! \, \mathrm{d} x)<+\infty\, .$$
 \item[(ii)] For all $t\in  \mathbb{R}_+^d$, we have the decomposition
 $$X_t= \gamma \text{Leb}_d ([0,t])+ \sigma W_t + \int_{[0,t]}\int_{|x|>1}xJ_X( \! \, \mathrm{d} s, \, \mathrm{d} x)+ \int_{[0,t]}\int_{|x|\leqslant 1}x\tilde J_X( \! \, \mathrm{d} s, \, \mathrm{d} x)\, ,$$
 where $W$ is a Brownian sheet, $\tilde J_X=J_X- \text{Leb}_d \times \nu$ is the compensated jump measure, and the equality holds almost surely. In addition, the terms of the decomposition are independent random fields.
\end{itemize}
\end{theorem}

If $X$ is a $d$-parameter L\'evy field, by the \textit{lamp} property of its sample paths, it is locally bounded and defines almost surely an element of $\mathcal D'( \mathbb{R}^d)$ via the $L^2$-inner product. We have restricted our study to the case where the parameter space is $ \mathbb{R}_+^d$, but we can easily generalize this notion of L\'evy field on $ \mathbb{R}^d$ by taking independent copies of $X$ on each $2^d$ orthants of $ \mathbb{R}^d$. We say this random field is a \emph{L\'evy field on $ \mathbb{R}^d$}.

\begin{proposition} \label{prop:derivativesheet}
Let $X$ be a $d$-parameter L\'evy field on $ \mathbb{R}^d$ with characteristic triplet $(\gamma, \sigma, \nu)$. The $d^{\text {th}}$ cross-derivative of $X$ in the sense of Schwartz distributions is a L\'evy white noise: for $\omega \in \Omega$ and $\varphi \in \mathcal D( \mathbb{R}^d)$,
 $$\scal{\dot X} \varphi (\omega):= (-1)^d \scal{X}{\varphi^{(\mathbf 1_d)}} (\omega) :=(-1)^d\int_{ \mathbb{R}^d}X_s (\omega) \varphi^{(\mathbf 1_d)}(s) \, \mathrm{d} s \, ,$$
 where $\varphi^{(\mathbf 1_d)}=\frac{\partial ^d}{\partial t_1 \cdots \partial t_d} \varphi$.
\end{proposition}

\subsection{Comparison with It\^o-type Integration}

According to \cite[Proposition 3.17]{Dalang2015Levy}, Proposition \ref{prop:derivativesheet} is coherent with the definition of L\'evy white noise given in Definition \ref{levynoise}. In particular, it can integrate not only functions in $\mathcal D( \mathbb{R}^d)$, but also functions in $L(\dot X)$. This particular version of L\'evy white noise allows us to derive a stochastic integral representation of the noise. More precisely, we define two stochastic integral operators $I$ and $\tilde I$, and we show that, on their domain of definition, they agree with the L\'evy white noise $\dot X$ defined above.

\begin{definition}
 We say a Borel measurable function $f: \mathbb{R}^d\to  \mathbb{R}$ is \emph{It\^o $\dot X$-integrable} if the following exists:
\begin{align*}
   I(f) &:=\int_{ \mathbb{R}^d} \gamma f(s) \, \mathrm{d} s + \int_{ \mathbb{R}^d} \sigma f(s) \, \mathrm{d} W_s \\
   &\qquad+ \int_{ \mathbb{R}^d} \int_{|x|\leqslant 1} xf(s) \tilde J_X(\! \, \mathrm{d} s ,\, \mathrm{d} x) +  \int_{ \mathbb{R}^d} \int_{|x|> 1} xf(s)  J_X(\! \, \mathrm{d} s ,\, \mathrm{d} x)  \, ,
\end{align*}
 where the first integral is in the $L^1( \mathbb{R}^d)$ sense, the second is a Wiener integral, and the others are Poisson integrals in the sense of \cite[Lemma 12.13]{Kallenberg2006foundations}. We denote by $ L( I)$ the space of It\^o $\dot X$-integrable functions.

\end{definition}

We will see in the following that this stochastic integral representation is not always well defined for $f\in L (\dot X )$. That is why we introduce an other operator $\tilde I$.

\begin{definition}
 We say a Borel measurable function $f: \mathbb{R}^d\to  \mathbb{R}$ is \emph{Poisson $\dot X$-integrable} if the following integrals exist:
\begin{equation}\label{poissondecomposition}
  \begin{aligned}
 \tilde I(f) &:= \int_{ \mathbb{R}^d}  \left( \gamma  f(s)+  \int_{ \mathbb{R}} x f(s) \left( \mathds 1_{|xf(s)| \leqslant 1}- \mathds 1_{|x|\leqslant 1} \right) \nu(\! \, \mathrm{d} x) \right) \, \mathrm{d} s  + \int_{ \mathbb{R}^d} \sigma f(s) \, \mathrm{d} W_s  \\
  & \hspace{1cm}+ \int_{ \mathbb{R}^d \times  \mathbb{R}} x f(s) \mathds 1_{|xf(s)| \leqslant 1} \tilde J_X( \! \, \mathrm{d} s , \, \mathrm{d} x) + \int_{ \mathbb{R}^d \times  \mathbb{R}} x f(s) \mathds 1_{|xf(s)| > 1}  J_X( \! \, \mathrm{d} s , \, \mathrm{d} x) \, ,
\end{aligned}
\end{equation}
where the first integral is an $L^1( \mathbb{R}^d)$ integral, the second is a Wiener integral, and the last two are compensated Poisson and Poisson stochastic integrals as defined in \cite[Chapter 12]{Kallenberg2006foundations}. We denote by $ L( \tilde I)$ the space of Poisson $\dot X$-integrable functions.
\end{definition}

We can in fact characterize the domains of the operators $I$ and $\tilde I$.

\begin{proposition}
 The following holds:
\begin{itemize}
\item[ (i) ] $f \in  L(I)$ if and only if $\gamma f \in L^1( \mathbb{R}^d)$, $\sigma f \in L^2( \mathbb{R}^d)$, and
\begin{equation}\label{conditionpoisson}
\begin{aligned}
 &\int_{ \mathbb{R}^d} \int_{|x|>1} \left( |xf (s)| \wedge 1\right) \, \mathrm{d} s \, \nu(\! \, \mathrm{d} x)\\
 &\qquad+ \int_{ \mathbb{R}^d} \int_{|x|\I1} \left( |xf (s)|^2 \wedge |xf (s)| \right) \, \mathrm{d} s \, \nu(\! \, \mathrm{d} x)< +\infty\, .
\end{aligned}
\end{equation}
\item[(ii)]$ L(I) \subset L(\tilde I)$, and for all $f \in  L(I)$, $ I(f)=\tilde I(f)$ almost surely.
\item[(iii)] $L(\tilde I)=L(\dot X)$ and for all $f\in L(\tilde I)$, $\tilde I(f)= \scal{\dot X} f$ almost surely.
\end{itemize}
\end{proposition}

\begin{proof}
 We first prove \textit{(i)}. The deterministic and Wiener integral exist under the well known conditions $\gamma \varphi \in L^1( \mathbb{R}^d)$, $\sigma \varphi \in L^2( \mathbb{R}^d)$. By \cite[Lemma 12.13]{Kallenberg2006foundations}, the compensated Poisson and Poisson integrals exist if and only if \eqref{conditionpoisson} is satisfied. To verify the equality $L(\dot X)=L(\tilde I)$, we can use the existence criterions of the different terms in \eqref{poissondecomposition} (see \cite[Lemma 12.13]{Kallenberg2006foundations} for the Poisson and compensated Poisson integrals) to see that a function $f$ is $\dot X$ integrable if and only if it is Poisson $\dot X$-integrable. Then, let $f \in  L(I)$. Condition \textit{(ii)} of Proposition \ref{integrable} is satisfied. Moreover, we can see that
\begin{align*}
  \int_{ \mathbb{R}^d} \int_{|x|>1} \left( |xf (s)| \wedge 1\right) \, \mathrm{d} s \, \nu(\! \, \mathrm{d} x) &+ \int_{ \mathbb{R}^d} \int_{|x|\I1} \left( |xf (s)|^2 \wedge |xf (s)| \right) \, \mathrm{d} s \, \nu(\! \, \mathrm{d} x) \\
  &\geqslant  \int_{ \mathbb{R}^d \times  \mathbb{R}} \left( |x f(s)|^2 \wedge 1\right) \, \mathrm{d} s\, \nu(\! \, \mathrm{d} x)\, ,
\end{align*}
 therefore condition \textit{(iii)} of Proposition \ref{integrable} is satisfied. Then,
\begin{equation}\label{majoration}
\begin{aligned}
 &\int_{ \mathbb{R}^d} \int_{ \mathbb{R}} \left |x f(s)\right |  \left | \mathds 1_{|xf(s)| \leqslant 1}- \mathds 1_{|x|\leqslant 1}  \right |  \nu(\! \, \mathrm{d} x)\, \mathrm{d} s \\
 & = \int_{ \mathbb{R}^d} \int_{|x|>1} \left |x f(s)\right |   \mathds 1_{|xf(s)| \leqslant 1}  \nu(\! \, \mathrm{d} x)\, \mathrm{d} s+ \int_{ \mathbb{R}^d} \int_{|x|\I1} \left |x f(s)\right |   \mathds 1_{|xf(s)| > 1}  \nu(\! \, \mathrm{d} x)\, \mathrm{d} s <+\infty\, ,
\end{aligned}
\end{equation}
therefore \textit{(i)} of Proposition \ref{integrable} is satisfied, and the inclusion $ L(I) \subset L(\dot X)=L(\tilde I)$ is satisfied. Then, we can assume without loss of generality that $\gamma=\sigma=0$.
\begin{equation}\label{zero}
\begin{aligned}
  I(f) &= \int_{ \mathbb{R}^d}\int_{|x|\I1} xf(s) \tilde J_X(\! \, \mathrm{d} x, \, \mathrm{d} s)+ \int_{ \mathbb{R}^d}\int_{|x|>1} xf(s)  J_X(\! \, \mathrm{d} x, \, \mathrm{d} s)\\
 &=\int_{ \mathbb{R}^d}\int_{|x|\I1} xf(s)\mathds 1_{|xf(s) |\leqslant 1} \tilde J_X(\! \, \mathrm{d} x, \, \mathrm{d} s)+\int_{ \mathbb{R}^d}\int_{|x|\I1} xf(s) \mathds 1_{|xf(s) |>1} \tilde J_X(\! \, \mathrm{d} x, \, \mathrm{d} s)\\
 &+  \int_{ \mathbb{R}^d}\int_{|x|>1} xf(s) 1_{|xf(s) |\leqslant 1} J_X(\! \, \mathrm{d} x, \, \mathrm{d} s)+  \int_{ \mathbb{R}^d}\int_{|x|>1} xf(s) 1_{|xf(s) |> 1} J_X(\! \, \mathrm{d} x, \, \mathrm{d} s)\, .
\end{aligned}
\end{equation}
By \eqref{majoration}, we can write
\begin{equation}\label{first}
\begin{aligned}
 \int_{ \mathbb{R}^d}\int_{|x|\I1} xf(s) \mathds 1_{|xf(s) |>1} \tilde J_X(\! \, \mathrm{d} x, \, \mathrm{d} s)&=\int_{ \mathbb{R}^d}\int_{|x|\I1} xf(s) \mathds 1_{|xf(s) |>1}  J_X(\! \, \mathrm{d} x, \, \mathrm{d} s) \\
 &\hspace{1cm}-\int_{ \mathbb{R}^d}\int_{|x|\I1} xf(s) \mathds 1_{|xf(s) |>1}  \nu(\! \, \mathrm{d} x) \, \mathrm{d} s\, ,
\end{aligned}
\end{equation}
and
\begin{equation}\label{second}
\begin{aligned}
  \int_{ \mathbb{R}^d}\int_{|x|>1} xf(s) 1_{|xf(s) |\leqslant 1} J_X(\! \, \mathrm{d} x, \, \mathrm{d} s)&= \int_{ \mathbb{R}^d}\int_{|x|>1} xf(s) 1_{|xf(s) |\leqslant 1} \tilde J_X(\! \, \mathrm{d} x, \, \mathrm{d} s) \\
 &\hspace{1cm}+\int_{ \mathbb{R}^d}\int_{|x|>1} xf(s) \mathds 1_{|xf(s) |\I1}  \nu(\! \, \mathrm{d} x) \, \mathrm{d} s\, .
\end{aligned}
\end{equation}
 Recombining \eqref{first} and \eqref{second} in \eqref{zero}, we get
\begin{align*}
  I(f)&=\int_{ \mathbb{R}^d}\int_{ \mathbb{R}} xf(s) 1_{|xf(s) |\leqslant 1} \tilde J_X(\! \, \mathrm{d} x, \, \mathrm{d} s)+\int_{ \mathbb{R}^d}\int_{ \mathbb{R}} xf(s) 1_{|xf(s) |> 1}  J_X(\! \, \mathrm{d} x, \, \mathrm{d} s) \\
 &\hspace{1cm}+ \int_{ \mathbb{R}^d} \int_ \mathbb{R} xf(s) \left( \mathds 1_{|xf(s) | \leqslant 1} -\mathds 1_{|x| \leqslant 1} \right) \nu(\! \, \mathrm{d} x) \, \mathrm{d} s=\tilde I(f)\, .
\end{align*}

We finally show that for $f\in L(\tilde I)$, $\scal{\dot X} f=\tilde I(f)$. For $f \in \mathcal D( \mathbb{R}^d)$, we can use \cite[Lemma 3.6]{Dalang2015Levy} to deduce that $\scal{\dot X} f= I(f)$, and since $\mathcal D( \mathbb{R}^d) \subset L(I)$, we get $\scal{\dot X} f=\tilde I(f)$. Then let $A\subset  \mathbb{R}^d$ be a Borel set such that $\leb d A <+\infty$. Let $(\theta_n)_{n\geqslant 1}$ be a sequence of mollifier as in Definition \ref{borelset}. Since for any $n\in  \mathbb{N}$, $\scal{\dot X}{f (\theta_n * \mathds 1_A)}=\tilde I(f( \theta_n * \mathds 1_A))$, and since $ \scal{\dot X}{f (\theta_n * \mathds 1_A)}\to  \scal{\dot X}{f  \mathds 1_A}$ in probability as $n\to +\infty$, it suffices to show that $\tilde I(f (\theta_n * \mathds 1_A))\to \tilde I(f   \mathds 1_A)$ in probability as $n\to+\infty$. In fact, one easily checks that $f (\theta_n * \mathds 1_A)$ and $f  \mathds 1_A \in L(I)$. Therefore it is enough to show that $ I(f (\theta_n * \mathds 1_A))\to  I(f   \mathds 1_A)$ in probability as $n\to+\infty$, and this is obtained using the linearity of $I$ and the convergence in law of each part of the decomposition of the L\'evy exponent as in the proof of Proposition \ref{wellposed}.
The same reasoning works to show $\scal{\dot X}{\mathds 1_A}=\tilde I(\mathds 1_A)$. To extend the result to simple functions, the problem we have is that each term of the decomposition of $\tilde I$ is not linear (although we will see that $\tilde I$ is linear). Let $\alpha >0$. Then,
\begin{equation*}
\begin{aligned}
 & \int_{ \mathbb{R}^d\times  \mathbb{R}} x\mathds 1_{s\in A} \mathds 1_{|x| \leqslant 1} \tilde J_X(\! \, \mathrm{d} s , \, \mathrm{d} x) +  \int_{ \mathbb{R}^d\times  \mathbb{R}} x\mathds 1_{s \in A} \mathds 1_{|x| > 1}  J_X(\! \, \mathrm{d} s , \, \mathrm{d} x) \\
  & \hspace{2cm} =  \int_{ \mathbb{R}^d\times  \mathbb{R}} x\mathds 1_{s\in A} \mathds 1_{|x| \leqslant \alpha} \tilde J_X(\! \, \mathrm{d} s , \, \mathrm{d} x) +  \int_{ \mathbb{R}^d\times  \mathbb{R}} x\mathds 1_{s \in A} \mathds 1_{|x| > \alpha}  J_X(\! \, \mathrm{d} s , \, \mathrm{d} x) \\
  & \hspace{3cm} + \int_{ \mathbb{R}^d\times  \mathbb{R}} x \mathds 1_{s\in A} \left( \mathds 1_{|x| \leqslant \alpha} - \mathds 1_{|x| \leqslant 1} \right) \, \mathrm{d} s \nu(\! \, \mathrm{d} x) \\
  & \hspace{2cm} := I_M^\alpha(A)+ I_P^\alpha(A)+ D^\alpha(A)\, .
\end{aligned}
\end{equation*}
 Let $f=\sum_{i=1}^n y_i \mathds 1_{A_i}$ be a simple function, where for all $1 \leqslant i \leqslant n$, $\leb d {A_i} <+\infty$ and $|y_i|>0$. Then,
\begin{align*}
 \scal{\dot X}{f}&=\sum_{i=1}^n y_i \scal{\dot X}{\mathds 1_{A_i}}\\
 &=\sum_{i=1}^n y_i  \int_{A_i\cap  \mathbb{R}^d}  \gamma  \, \mathrm{d} s  + \int_{ \mathbb{R}^d} \sigma \mathds 1_{A_i} (s) \, \mathrm{d} W_s+ I_M^1(A_i)+I_P^1(A_i)\\
 &=\int_{ \mathbb{R}^d} \gamma f(s) \, \mathrm{d} s+  \int_{ \mathbb{R}^d} \sigma f(s)  \, \mathrm{d} W_s + \sum_{i=1}^n y_i \left( I_M^{|y_i|^{-1}}(A_i)+I_P^{|y_i|^{-1}}(A_i)+D^{|y_i|^{-1}}(A_i)\right) \\
 &=\tilde I(f) \, .
\end{align*}

 Let $f \in L(\dot X)$. By definition, there is a sequence of simple functions $f_n$ such that $f_n \to f$ almost everywhere as $n\to +\infty$, and $\scal{\dot X}{f_n} \to \scal{\dot X}{f}$ in probability as $n\to +\infty$. The proof of \cite[Theorem 2.7]{Rajput1989spectral} shows that the sequence $(f_n)_{n\geqslant 1}$ can be chosen such that for any $n \in  \mathbb{N}$, $|f_n | \leqslant |f|$. We only need to show that $\tilde I(f_n) \to \tilde I(f)$ in probability as $n\to +\infty$. We show the convergence in probability of each part of the decomposition of the stochastic integral. First we deal with the Gaussian part. By classical properties of Wiener stochastic integration, we get
\begin{equation*}
 \mathbb{E} \left [ \left( \int_{ \mathbb{R}^d} \sigma f_n(s) \, \mathrm{d} W_s -\int_{ \mathbb{R}^d} \sigma f(s) \, \mathrm{d} W_s\right) ^2 \right ]= \int_{ \mathbb{R}^d} \sigma^2 \left( f_n(s)-f(s) \right)^2 \, \mathrm{d} s \to 0 \qquad \text{as} \ n\to +\infty\, ,
\end{equation*}
 by the dominated convergence theorem. We deduce that $ \int_{ \mathbb{R}^d} \sigma f_n(s) \, \mathrm{d} W_s\to  \int_{ \mathbb{R}^d} \sigma f(s) \, \mathrm{d} W_s$ in $L^2(\Omega)$ as $n\to+\infty$, which implies the convergence in probability. Then, we show the convergence of the compensated Poisson term.
\begin{align*}
I_1:&= \int_{ \mathbb{R}^d \times  \mathbb{R}} x f_n(s) \mathds 1_{|xf_n(s)| \leqslant 1} \tilde J_X( \! \, \mathrm{d} s , \, \mathrm{d} x) - \int_{ \mathbb{R}^d \times  \mathbb{R}} x f(s) \mathds 1_{|xf(s)| \leqslant 1} \tilde J_X( \! \, \mathrm{d} s , \, \mathrm{d} x) \\
 &=  \int_{ \mathbb{R}^d \times  \mathbb{R}} x \left( f_n(s) - f(s)\right)  \mathds 1_{|xf(s)| \leqslant 1} \tilde J_X( \! \, \mathrm{d} s , \, \mathrm{d} x)\\
 &\qquad  +  \int_{ \mathbb{R}^d \times  \mathbb{R}} x f_n(s)\mathds 1_{|xf_n(s)| \leqslant 1\, , |xf(s)|>1}  \tilde J_X( \! \, \mathrm{d} s , \, \mathrm{d} x) \, .
\end{align*}
 Each of these two integrals exist since
\begin{align*}
 \int_{ \mathbb{R}^d \times  \mathbb{R}} \left | x \left( f_n(s) - f(s)\right)  \mathds 1_{|xf(s)| \leqslant 1}  \right |^2 &\wedge  \left | x \left( f_n(s) - f(s)\right)  \mathds 1_{|xf(s)| \leqslant 1} \right | \, \mathrm{d} s \nu(\! \, \mathrm{d} x)\\
 & \leqslant 4  \int_{ \mathbb{R}^d \times  \mathbb{R}} \left | x f(s) \mathds 1_{|xf(s)| \leqslant 1} \right |^2 \, \mathrm{d} s \nu(\! \, \mathrm{d} x) < +\infty\, ,
\end{align*}
and
\begin{align*}
 \int_{ \mathbb{R}^d \times  \mathbb{R}} \left | x  f_n(s)   \mathds 1_{|xf_n(s)| \leqslant 1\, , |xf(s)|>1}  \right |^2 &\wedge \left | x  f_n(s)   \mathds 1_{|xf_n(s)| \leqslant 1\, , |xf(s)|>1}  \right | \, \mathrm{d} s \nu(\! \, \mathrm{d} x) \\
 &\leqslant  \int_{ \mathbb{R}^d \times  \mathbb{R}} \left | x  f_n(s)   \mathds 1_{|xf_n(s)| \leqslant 1}  \right |^2 \, \mathrm{d} s \nu(\! \, \mathrm{d} x)<+\infty\, .
\end{align*}
Furthermore, since these two integrals are compensated Poisson integrals over disjoint subsets of $ \mathbb{R}^d \times  \mathbb{R}$, they are independent and their mean is zero. Then,
\begin{align*}
 \mathbb{E} \left( I_1^2\right)= \int_{ \mathbb{R}^d \times  \mathbb{R}} |x (f_n(s)-f(s)|^2 \mathds 1_{|xf(s)| \leqslant 1} \, \mathrm{d} s \nu(\! \, \mathrm{d} x) + \int_{ \mathbb{R}^d \times  \mathbb{R}} |x f_n(s)|^2   \mathds 1_{|xf_n(s)| \leqslant 1\, , |xf(s)|>1} \, \mathrm{d} s \nu(\! \, \mathrm{d} x) \, .
\end{align*}
Then, $|x (f_n(s)-f(s)|^2 \mathds 1_{|xf(s)| \leqslant 1}\leqslant 4 |x f(s) |^2 \mathds 1_{|xf(s) \leqslant 1}$, so by dominated convergence theorem we get the convergence to zero of the first integral. Similarly,
$$ |x f_n(s)|^2   \mathds 1_{|xf_n(s)| \leqslant 1\, , |xf(s)|>1} \leqslant  \mathds 1_{ |xf(s)|>1}\, ,$$
so by dominated convergence again, the second integral converges to zero. We deduce that
$$\int_{ \mathbb{R}^d \times  \mathbb{R}} x f_n(s) \mathds 1_{|xf_n(s)| \leqslant 1} \tilde J_X( \! \, \mathrm{d} s , \, \mathrm{d} x)\to \int_{ \mathbb{R}^d \times  \mathbb{R}} x f(s) \mathds 1_{|xf(s)| \leqslant 1} \tilde J_X( \! \, \mathrm{d} s , \, \mathrm{d} x)\, ,$$ in $L^2(\Omega)$ as $n\to +\infty$, which implies the convergence in probability. The treatment of the compound Poisson term goes as follows:
\begin{align*}
  \int_{ \mathbb{R}^d \times  \mathbb{R}} x f_n(s) \mathds 1_{|xf_n(s)| > 1}  J_X( \! \, \mathrm{d} s , \, \mathrm{d} x)=\sum_{i\geqslant 1} X_i f_n(T_i)  \mathds 1_{|X_i f_n(T_i)| > 1}\, ,
\end{align*}
where the sum is finite almost surely. Then, $1_{|xf_n(s)| > 1} \leqslant 1_{|xf(s)| > 1}$, therefore $J_X(\{ |xf_n(s)| > 1 \})\leqslant J_X(\{ |xf(s)| > 1 \})=: N$. $N$ is an almost surely finite random variable, and does not depends on $n$. Therefore, since $f_n\to f$ almost everywhere, and  since the law of $T_i$ is absolutely continuous with respect to Lebesgue measure, we deduce that
\begin{align*}
 \sum_{i\geqslant 1} X_i f_n(T_i)  \mathds 1_{|X_i f_n(T_i)| > 1}\to \sum_{i\geqslant 1} X_i f (T_i)  \mathds 1_{|X_i f(T_i)| > 1}= \int_{ \mathbb{R}^d \times  \mathbb{R}} x f(s) \mathds 1_{|xf(s)| > 1}  J_X( \! \, \mathrm{d} s , \, \mathrm{d} x)\, ,
\end{align*}
almost surely as $n\to +\infty$, which implies the convergence in probability. We then have the following:
$$\scal{\dot X}{f_n} =\tilde  I(f_n)=: U(f_n)+\tilde I_W(f_n)+\tilde I_M(f_n)+\tilde I_P(f_n) \qquad \text{a.s.} $$
Also, we proved that $\tilde I_W(f_n)+\tilde I_M(f_n)+\tilde I_P(f_n)\to \tilde I_W(f)+\tilde I_M(f)+\tilde I_P(f)$ in probability as $n\to +\infty$. Also, $\scal{\dot X}{f_n}$ converges in probability to $\scal{\dot X}{f}$ as $n\to +\infty$, hence also in law. From these facts, we deduce that the deterministic part of the decomposition $U(f_n)$ converges as $n\to +\infty$, and from the expression of the characteristic function,
\begin{align*}
U(f_n) \to U(f):= \int_{ \mathbb{R}^d}  \gamma  f(s)+ \left( \int_{ \mathbb{R}} x f(s) \left( \mathds 1_{|xf(s)| \leqslant 1}- \mathds 1_{|x|\leqslant 1} \right) \nu(\! \, \mathrm{d} x) \right) \, \mathrm{d} s \qquad \text{as} \ n\to+\infty \, .
\end{align*}
This concludes the proof.
 \end{proof}

In general the inclusion $ L( I) \subset L(\dot X)$ is strict. For example, we can consider the case of an $\alpha$-stable white noise $\dot X_\alpha$ on $ \mathbb{R}^d$, $\alpha\in (0,2)$, that is a L\'evy white noise on $ \mathbb{R}^d$ with characteristic triplet $\left( 0,0,\nu_\alpha(\! \, \mathrm{d} x)\right)$, where $\nu_\alpha(\! \, \mathrm{d} x)=\frac{\! \, \mathrm{d} x}{|x|^{\alpha+1}}$. Then, we have that $L(\dot{X}_\alpha) = L^\alpha ( \mathbb{R}^d)$ (see Proposition \ref{prop:salphas}).
On the other hand, $\varphi \in \tilde L(I)$ if and only if
\begin{align}\label{alpha}
 \int_{ \mathbb{R}^d}\int_{|x|>1}  |x \varphi(s)| \wedge 1 \, \mathrm{d} s\, \nu_\alpha(\! \, \mathrm{d} x)+\int_{ \mathbb{R}^d}\int_{|x|\I1}  |x \varphi(s)|^2 \wedge |x \varphi(s)| \, \mathrm{d} s\, \nu_\alpha(\! \, \mathrm{d} x)<+\infty\, .
\end{align}
Replacing $ \nu_\alpha(\! \, \mathrm{d} x)$ by $\frac 1 {|x|^{\alpha+1}} \, \mathrm{d} x$ yields when $\alpha\neq 1$
\begin{align*}
 \frac 2 \alpha \leb d {\{s\in  \mathbb{R}^d : |\varphi(s)|>1 \}} &+ \int_{ \mathbb{R}^d} |\varphi(s)|^\alpha \left( \frac 2 \alpha \mathds 1_{|\varphi(s)|\leqslant 1}+ \frac 2 {2-\alpha} \mathds 1_{|\varphi(s)|>1} \right) \, \mathrm{d} s \\
 &+\frac{2}{|1-\alpha|} \int_{ \mathbb{R}^d} \left | |\varphi(s)|^\alpha -|\varphi(s)|\right | \, \mathrm{d} s \\
 &+ \frac 2 {2-\alpha} \int_{ \mathbb{R}^d} |\varphi(x)|^2 \mathds 1_{|\varphi(s)|\leqslant 1} \, \mathrm{d} s<+\infty\, .
 \end{align*}
 For $\alpha=1$, \eqref{alpha} is equivalent to
\begin{align*}
 \int_{ \mathbb{R}^d}|\varphi(s)| \, \mathrm{d} s +\int_{ \mathbb{R}^d}|\varphi(s)| \ln\left( |\varphi(s)| \right) \, \mathrm{d} s<+\infty\, .
\end{align*}

The integral representation via the operator $\tilde I$ is adaptive in the sense that the way we split the parts in the decomposition depends on the function $f$. Even if $\tilde{I}$ is linear, the different parts of the decomposition are not, which make the representation quite unusual. However, the traditional L\'evy-It\^o decomposition appears to be insufficient, since it requires an $L^1$-type constraint on $f$ that is artificial. This is better illustrated by the case of symmetric compound Poisson white noise. We shall see in Section \ref{subsubsec:poisson} that $L(\dot{X}_{\mathrm{Poisson}}) = L^{0,2}( \mathbb{R}^d)$ (the space of measurable functions that are asymptotically in $L^2$; see Definition \ref{def:Lpq}). However, due to \cite[Lemma 12.13]{Kallenberg2006foundations}, we easily see that $L(I) = L^{0,2}( \mathbb{R}^d) \bigcap L^{1}( \mathbb{R}^d) = L^{1,2}( \mathbb{R}^d)$.

\section{Practical Determination of the Domain of Definition} \label{sec:criteria}

We provide here several criteria for the practical identification of the domain of definition of a L\'evy white noise.
We apply our result to the Gaussian, S$\alpha$S, compound Poisson, and generalized Laplace noises.
The results presented here are new for the two latter classes of noise to the best of our knowledge.
Similar considerations are given for the domain of finite $p$th moments for $0< p \leq 2$.
In the rest of the paper, we shall consider the spaces $L^p(\dot{X})$ for $0 \leq p \leq 2$, with the convention that $L^0(\dot{X}) = L(\dot{X})$ is the domain of definition of the L\'evy white noise $\dot{X}$.

	\subsection{Basic Properties}

\begin{proposition} \label{prop:affineinvariance}
	Let $\dot{X}$ be a L\'evy white noise  and $p \geq 0$.
	\begin{itemize}
		\item \emph{Linearity:} for $f,g \in L^p(\dot{X})$ and $\lambda \in  \mathbb{R}$, $f + \lambda g \in L^p(\dot{X})$.
		\item \emph{Invariances:} for $f \in L^p(\dot{X})$ and $H :  \mathbb{R}^d \rightarrow  \mathbb{R}^d$, a $C_1$-diffeomorphisme, we have $$t \mapsto f(H (t)) \in L^p(\dot{X}).$$
		In particular,   the translations $f(\cdot - t_0)$, rescalings $f(b \cdot)$, and rotations $f(\mathrm{R} \cdot)$ of $f$, with $t_0 \in  \mathbb{R}^d$, $b \neq 0$, and $\mathrm{R} \in \mathrm{SO}(d)$ an rotation matrix, are in $L^p(\dot{X})$.
	\end{itemize}
\end{proposition}

\begin{proof}
	The linearity is already known since $L^p(\dot{X})$ is a vectorial space by Proposition \ref{prop:LdotXOrlicz}.
	For the invariance, we simply remark that, by the substitution $u =H(t)$, $$\int_{ \mathbb{R}^d} \Psi_p( f( H(t) )) \mathrm{d} t = \int_{ \mathbb{R}^d}   \lvert \det  J_{H^{-1}}(u) \rvert  \Psi_p (f(u))   \mathrm{d} u$$ with $J_H$ the invertible Jacobian matrix of $H$. By assumption on $H$, $ \lvert \det  J_{H^{-1}}(u) \rvert $ is bounded above and below by finite strictly positive constants, implying the result.
\end{proof}

For $\dot{X}$ a L\'evy   noise, the rescaling $\dot{X}(\cdot / b)$ of a factor $b\neq 0$ is the generalized random process defined by $\langle \dot{X}(\cdot / b) , \varphi \rangle = \langle \dot{X}, b^d \varphi(b \cdot) \rangle$, for $\varphi \in {\mathcal{D}}( \mathbb{R}^d)$. We see easily that $\dot{X}(\cdot / b)$ is itself a L\'evy white noise. Similarly, for $a \neq 0$, the generalized random process $a \dot{X}$ is still a L\'evy white noise.
We say that two generalized random processes $s_1$ and $s_2$   \emph{independent} if their finite-dimensional marginals  are independent.
By linearity, this is equivalent to the relation
\begin{equation*}
{\widehat{\mathscr{P}}}_{s_1 + s_2}(\varphi) = {\widehat{\mathscr{P}}}_{s_1}(\varphi) {\widehat{\mathscr{P}}}_{s_2}(\varphi), \qquad \text{for all } \varphi \in {\mathcal{D}}( \mathbb{R}^d).
\end{equation*}
If $\dot{X}_1$ and $\dot{X}_2$ are two independent L\'evy white noises, then $\dot{X}_1 + \dot{X}_2$ is also a L\'evy white noise.

\begin{proposition} \label{prop:scalingindependence}
	Let $\dot{X}$ be a L\'evy white noise and $p \geq 0$. Then we have, for $a$ and $b$ nonzero, and $t_0 \in  \mathbb{R}^d$,
	\begin{equation*}
		L^p(\dot{X}) = L^p(a \dot{X}) = L^p(  \dot{X} (\cdot / b))=L^p(\dot X(\cdot-t_0) )
	\end{equation*}
	If $\dot{X}_1$ and $\dot{X}_2$ are two independent L\'evy white noises, then
	\begin{equation} \label{eq:domainsumnoises}
		L^p(\dot{X}_1) \cap L^p(\dot{X}_2) \subseteq L^p(\dot{X}_1+\dot{X}_2).
	\end{equation}
	Moreover, if at least one of the two L\'evy white noises is symmetric, then \eqref{eq:domainsumnoises} is an equality.
\end{proposition}

\begin{proof}
	We have $\langle \dot{X}(\cdot / b) , f \rangle = \langle \dot{X}, b^d  f (b \cdot) \rangle$, so that $f \in L^p(\dot{X}(\cdot / b)) $ if and only if $ b^d f(b \cdot) \in L^p(\dot{X})$. Then, $L^p(\dot{X})$ being a linear space that is invariant by rescaling (Proposition \ref{prop:affineinvariance}), the latter condition is equivalent to $f\in L^p(\dot{X})$, hence $L^p(  \dot{X} (\cdot / b)) = L^p(\dot{X})$. We proceed similarly for $L^p( a \dot{X})$ and $L^p(\dot X(\cdot-t_0) )$.

	For $i=1,2$, the L\'evy triplet of $\dot{X}_i$ ($\dot{X})$, respectively) is denoted by $(\gamma_i, \sigma^2_i, \nu_i)$ ($(\gamma,\sigma^2,\nu)$, respectively), and the corresponding exponent is $\Psi_{p,i}$ ($\Psi_{p}$, respectively).
	If $\dot{X}_1$ and $\dot{X}_2$ are independent, we have the relations $$\gamma = \gamma_1 + \gamma_2, \ \sigma^2 = \sigma_1^2 + \sigma_2^2, \ \nu = \nu_1 + \nu_2.$$
	Therefore, by the triangular inequality, we have
	\begin{align*}
		\Psi_p (\xi) & =   \Big\lvert (\gamma_1 + \gamma_2) \xi + \int_{ \mathbb{R}} x \xi \left( \One_{\abs{x \xi }\leq 1}- \One_{\abs{x}\leq 1} \right)( \nu_1 + \nu_2) (\mathrm{d} x) \Big\rvert
				 \nonumber \\
				& \quad + (\sigma^2_1 + \sigma^2_2) \xi^2  + \int_{ \mathbb{R}} (\lvert \xi x \rvert^p  \wedge \lvert \xi x \rvert^2) ( \nu_1 + \nu_2)(\mathrm{d} x)  \nonumber \\
				& \leq \Psi_{p,1}(\xi) + \Psi_{p,2}(\xi),
	\end{align*}
	which proves \eqref{eq:domainsumnoises}. When one of the noise is symmetric, for instance $\dot{X}_1$, the latter inequality is an equality since $  \gamma_1 \xi + \int_{ \mathbb{R}}  x \xi \left( \One_{\abs{x \xi }\leq 1}- \One_{\abs{x}\leq 1} \right) \nu_1(\mathrm{d} x)   =0$ and \eqref{eq:domainsumnoises} is an equality.
\end{proof}

In general, \eqref{eq:domainsumnoises} is only an inclusion. Consider for instance the case where $\dot{X}_1$ and $\dot{X}_2$ have L\'evy triplet $(1,1,0)$ and $(-1,0,0)$ respectively, meaning that $\dot{X}_1$ is a Gaussian white noise with drift $\gamma =1$ and $\dot{X}_2$ a pure drift $\gamma = -1$. Then, $\dot{X}_1$ and $\dot{X}_2$ are clearly independent, and $\dot{X}_1 + \dot{X}_2$ is a Gaussian white noise with no drift. Therefore, $L^p(\dot{X}_1 + \dot{X}_2) = L^2( \mathbb{R}^d)$ but $L^p(\dot{X}_1) \cap L^p(\dot{X}_2) = L^1( \mathbb{R}^d) \cap L^2( \mathbb{R}^d)$ (see Section \ref{sec:GaussAndDrift} for more details on the determination of those domains).

For $\gamma \in  \mathbb{R}$ and $\nu$ a L\'evy measure, we set
 \begin{equation*}
 m_{\gamma,\nu}(\xi) = \left\vert \gamma \xi + \int_{ \mathbb{R}}  x \xi \left( \One_{\abs{x \xi }\leq 1}- \One_{\abs{x}\leq 1} \right)  \nu(\mathrm{d} x) \right\rvert.
 \end{equation*}
The next  result is taken from \cite{Rajput1989spectral}.

\begin{proposition}[Reduction to the symmetric case without Gaussian part] \label{prop:reduction}
	Let $(\gamma, \sigma^2, \nu)$ be a L\'evy triplet. We also denote by $\nu_{\mathrm{sym}}$ the symmetrization of $\nu$. We consider the following L\'evy white noises:
	\begin{itemize}
		\item $\dot{X}$ with L\'evy triplet $(\gamma, \sigma^2, \nu)$,
		\item $\dot{X}_2$ with L\'evy triplet $(\gamma, 0, \nu)$,
		\item $\dot{X}_{\mathrm{sym}}$ with L\'evy triplet $(0,\sigma^2,\nu_{\mathrm{sym}})$.
	\end{itemize}
Then, we have the following relations for $p \geq 0$:
	\begin{itemize}
		\item If $\sigma^2 \neq 0$, then
		\begin{align} \label{eq:LpwithoutGaussian}
			L^p(\dot{X}) &= L^2( \mathbb{R}^d) \cap L^p(\dot{X}_2).
		\end{align}
		\item In any case,
		\begin{align} \label{eq:Lpsymmetrized}
			L^p(\dot{X}) &= L^p(\dot{X}_{\mathrm{sym}}) \cap \{f \in L(\dot{X}), \ \int_{ \mathbb{R}^d} m_{\gamma,\nu}(f(t)) \mathrm{d} t < \infty \}.
		\end{align}
	\end{itemize}
\end{proposition}
\begin{proof}
We can decompose $\dot{X} = \dot{X}_2 + \dot{X}_{\mathrm{Gauss}}$, where $\dot{X}_2$ and $ \dot{X}_{\mathrm{Gauss}}$ are independent with respective L\'evy triplets $(\gamma, 0 ,\nu)$ and $(0,\sigma^2,0)$. Then, $ \dot{X}_{\mathrm{Gauss}}$ is a Gaussian white noise, for which $L^p( \dot{X}_{\mathrm{Gauss}}) = L^2( \mathbb{R}^d)$. We apply \eqref{eq:domainsumnoises} with equality ($ \dot{X}_{\mathrm{Gauss}}$ being symmetric) to obtain  \eqref{eq:LpwithoutGaussian}.
Finally, \eqref{eq:Lpsymmetrized} is a reformulation of \cite[Proposition 2.9]{Rajput1989spectral}.
\end{proof}

Based on Proposition \ref{prop:reduction}, we restrict our attention to symmetric L\'evy white noises without Gaussian part.
We first reduce to the case $ \sigma^2=0$ thanks to   \eqref{eq:LpwithoutGaussian}.
The only remaining part to deduce the general case from the symmetric one is the identification of functions $f$ satisfying   $ \int_{ \mathbb{R}^d} m_{\gamma,\nu}(f(t)) \mathrm{d} t < \infty$.
Primarily, for non-symmetric noise, this usually relies on $L^1$-type conditions, but we leave this topic open for further investigations.

	\subsection{The spaces $L^{p_0,p_\infty}( \mathbb{R}^d)$}

        We introduce the family of function spaces that generalize the $L^p$-spaces for $0<p<\infty$.
They will be identified later on as the domain of definition of important classes of L\'evy white noises.
	We first give some notations. For $0\leq p_0 , p_\infty < \infty$, we set
        \begin{align*}
        		\rho_{p_0 , p_\infty}(\xi) &:=  \lvert \xi \vert^{p_0} \One_{\lvert \xi \rvert > 1} +  \lvert \xi \vert^{p_\infty} \One_{\lvert \xi \rvert \leq  1}, \\
        		\rho_{\log,p_\infty}(\xi) &:= (1+ \log  \lvert \xi \rvert) \One_{\lvert \xi \rvert > 1} +  \lvert \xi \vert^{p_\infty} \One_{\lvert \xi \rvert \leq  1} .
	\end{align*}
	with the convention that $0^0=1$.

        \begin{definition} \label{def:Lpq}
        		For $0\leq p_0, p_\infty < \infty$, we define
		\begin{align*}
			L^{p_0, p_\infty}( \mathbb{R}^d) &= \left\{ f   \text{ measurable}, \  \rho_{p_0,p_\infty} (f)  := \int_{ \mathbb{R}^d} \rho_{p_0,p_\infty} (f(t)) \mathrm{d} t  < \infty  \right\}, \\
			L^{\log, p_\infty}( \mathbb{R}^d) &= \left\{ f  \text{ measurable},  \ \rho_{\log,p_\infty} (f)   :=  \int_{ \mathbb{R}^d}  \rho_{\log,p_\infty} (f(t)) \mathrm{d} t < \infty  \right\}.
		\end{align*}
        \end{definition}
    For $p>0$, we have $L^{p,p} ( \mathbb{R}^d) = L^p ( \mathbb{R}^d)$.
    Roughly speaking, $p_0$ measures the local integrability of a function, while $p_\infty$ indicates the asymptotic one.
This is illustrated by the following example. For $\alpha,\beta > 0$, the function $f(t) = \abs{t}^{-\alpha} \One_{\abs{t} < 1} +  \abs{t}^{-\beta} \One_{\abs{t} \geq 1}$ is such that
    \begin{align*}
    	\rho_{p_0,p_\infty}(f) &= \int_{ \mathbb{R}^d} \left( \lvert f(t) \vert^{p_0} \One_{\lvert f(t) \rvert > 1} +  \lvert f(t) \vert^{p_\infty} \One_{\lvert f(t) \rvert \leq  1} \right)  \mathrm{d} t  \\
				&= \int_{\abs{t} < 1} \abs{t}^{- p_0 \alpha} \mathrm{d} t +  \int_{\abs{t}  \geq  1} \abs{t}^{- p_\infty\beta} \mathrm{d} t.
    \end{align*}
    Therefore, $f$ is in $L^{p_0,p_\infty}( \mathbb{R}^d)$ if and only if
   \begin{equation*}
  \alpha < \frac{d}{p_0} \text{ and } \beta > \frac{d}{p_\infty}.
   \end{equation*}
   The first inequality effectively refers to the integrability of $f$ at the origin (or local integrability), while the second covers its asymptotic integrability.

       As we did in Sections~\ref{subsec:extension} and~\ref{subsec:moments} with the spaces $L(\dot{X})$ and  $L^p(\dot{X})$, we rely on generalized Orlicz spaces~\cite[Chapter X]{Rao1991theory} to identify the structure of the spaces $L^{p_0,p_\infty}( \mathbb{R}^d)$.

	\begin{proposition}  \label{prop:rhopqisnice}
		We fix  $p_0 \geq 0$ and $p_\infty > 0$.
 		The functions $\rho_{p_0,p_\infty}$ and $\rho_{\log,p_\infty}$ are  $\Delta_2$-regular $\varphi$-functions.
	\end{proposition}

	\begin{proof}
		To simplify the notation, we write $\rho = \rho_{p_0,p_\infty}$ in this proof.
		The function $\rho$ is continuous , non-decreasing, symmetric, and vanishes at the origin (since $p_\infty \neq 0$).
		It is therefore a $\varphi$-function.

		Then, we have the following decomposition
		\begin{equation*} \label{eq:boundDelta2_second}
			\rho( 2 \xi )
			= 2^{p_0} \abs{\xi}^{p_0} \One_{\abs{\xi} > 1}
			+ 2^{p_0} \abs{\xi}^{p_0} \One_{1/2 < \abs{\xi} \leq 1}
			+ 2^{p_\infty} \abs{\xi}^{p_\infty} \One_{\abs{\xi} \leq 1/2}.
		\end{equation*}
		For $1/2 \leq \abs{\xi} \leq 1$, we have that $\abs{\xi}^{p_0 - p_\infty} \leq \max (2^{p_\infty, p_0}, 1) $. Therefore, we have that
		\begin{align*}
			\rho( 2 \xi )
			& \leq 2^{p_0} \abs{\xi}^{p_0} \One_{\abs{\xi} > 1}
			+ 2^{p_0}  \max (2^{p_\infty, p_0}, 1)  \abs{\xi}^{p_\infty} \One_{1/2 < \abs{\xi} \leq 1}
			+ 2^{p_\infty} \abs{\xi}^{p_\infty} \One_{\abs{\xi} \leq 1/2} \\
			& \leq \max(2^{p_0}, 2^{p_\infty}) \rho(\xi).
		\end{align*}
		Therefore, $\rho$ is $\Delta_2$-regular.
		The proof for $\rho_{\log,p_\infty}$   is very similar.
	\end{proof}

	Proposition \ref{prop:rhopqisnice} coupled with Proposition \ref{prop:orlicz} allow us to identify the structure of the spaces $L^{p_0,p_\infty}( \mathbb{R}^d)$ and $L^{\log,p_\infty}( \mathbb{R}^d)$

    \begin{proposition}  \label{prop:structureLpq}
    	We fix  $ p_0 \geq 0$ and $p_\infty  > 0$.
	Then, $L^{p_0,p_\infty}( \mathbb{R}^d) = L^{\rho_{p_0,p_\infty}}( \mathbb{R}^d)$ is a generalized Orlicz space associated to the $\varphi$-function $\rho_{p_0,p_\infty}$. It is in particular a complete linear metric space for the F-norm
	\begin{equation*}
		\lVert f \rVert_{\rho_{p_0,p_\infty}} := \inf \{\lambda > 0 , \ \rho_{p_0,p_\infty}(f / \lambda) \leq \lambda \}.
	\end{equation*}
	Finally, simple functions are dense in $L^{p_0,p_\infty}( \mathbb{R}^d)$.
	The same conclusions occur for $L^{\log,p_\infty}( \mathbb{R}^d)$.
	\end{proposition}

	The following embeddings are easily deduced by bounding the F-norm of the considered function spaces.

\begin{proposition}	 \label{prop:embeddingsLpq}
	We fix  $ p_0 \geq 0$ and $p_\infty  > 0$.
	\begin{enumerate}
		\item If  $0\leq p_1 \leq p_2<\infty$, we have the   embedding
			\begin{equation*}  \label{eq:embedding1}
			 L^{p_2,p_\infty}( \mathbb{R}^d) \subseteq L^{p_1,p_\infty}( \mathbb{R}^d).
			\end{equation*}
		\item If $0< p_1 \leq p_2<\infty$, we have the   embedding
		\begin{equation*}  \label{eq:embedding2}
			L^{p_0,p_1}( \mathbb{R}^d) \subseteq L^{p_0,p_2}( \mathbb{R}^d).
		\end{equation*}
		\item Conditions 1., 2., and 3. remain true by changing $p_0$ by $\log$ and we have the   embeddings, for any $0< p_0 \leq 2$ and $0\leq p_\infty \leq 2$.
				\begin{equation*} \label{eq:embedding3}
			L^{p_0,p_\infty}( \mathbb{R}^d) \subseteq L^{\log,p_\infty}( \mathbb{R}^d) \subseteq L^{0,p_\infty}( \mathbb{R}^d) .
			\end{equation*}
	\end{enumerate}
    \end{proposition}

In Propositions \ref{prop:structureLpq} and \ref{prop:embeddingsLpq},
we restricted ourselves to the case when $p_\infty \neq 0$.
The reason is that $\rho_{p_0,0}(0) \neq 0$,
so that $\rho_{p_0,0}$ is not a $\varphi$-function.
Therefore, we do not define a generalized Orlicz space in the sense of Rao and Ren \cite{Rao1991theory}.
The space $L^{p_0,0}( \mathbb{R}^d)$ can be described as follows:
It is the space of functions in $L^{p_0} ( \mathbb{R}^d)$ whose support has a finite Lebesgue measure.
We do not specify any topological structure on those spaces, since they will not appear as the domain of definition of any L\'evy   noise.
However, the space $L^{2,0}( \mathbb{R}^d)$ will play a role as a common subspace to all the domains of definition of the L\'evy white noises (see Proposition \ref{prop:limitspaces}).

	\subsection{Criteria for the Determination of the Domain of Definition}\label{subsec:criteria}

In this section, we consider a symmetric white noise $\dot{X}$ without Gaussian part and with symmetric L\'evy measure $\nu$.
In particular, for $p \geq 0$, the function  $\Psi_p$ exponent in \eqref{eq:Psip}  simply becomes
\begin{align}
  	 \Psi_p(\xi)    =  \lvert \xi \rvert^2 \int_{|x| \leq 1/|\xi|} \lvert x \rvert^2 \nu (\mathrm{d} x) +  \lvert \xi \rvert^p  \int_{|x|> 1/ \lvert \xi \rvert} \lvert x \rvert^p \nu (\mathrm{d} x). \label{eq:psimmetric}
\end{align}
The first criterion is applicable as soon as we are able to estimate the behavior of the function $\Psi_p$ at the origin and/or at infinity.

\begin{proposition}[Criteria for the determination of the domain of definition] \label{prop:criteria}
	Let $\dot{X}$ be a symmetric L\'evy white noise without Gaussian part and $0 \leq p \leq 2$.
	\begin{enumerate}
	\item Assume that $\Psi_p(\xi) \leq C \rho_{p_0,p_\infty}(\xi)$ for some constant $C>0$ and every $\xi$, then we have the   embedding
	\begin{equation} \label{eq:practicalX1}
		L^{p_0,p_\infty}( \mathbb{R}^d) \subseteq L^p(\dot{X}).
	\end{equation}
	\item Assume that $\rho_{p_0,p_\infty}(\xi) \leq C  \Psi_p(\xi)  $ for some constant $C>0$ and every $\xi$, then we have the   embedding
	\begin{equation} \label{eq:practicalX2}
		L^p(\dot{X}) \subseteq L^{p_0,p_\infty}( \mathbb{R}^d)  .
	\end{equation}
	\item Assume that $\Psi_p(\xi) \underset{0}{\sim} A \lvert \xi \rvert^{p_\infty}$ and $\Psi_p(\xi) \underset{\infty}{\sim} B \lvert \xi \rvert^{p_0}$,
	then \begin{equation}  \label{eq:practicalX3}  L^p(\dot{X}) = L^{p_0,p_\infty}( \mathbb{R}^d) . \end{equation}
	\item  The same holds with $L^{\log,p_\infty}( \mathbb{R}^d)$ instead of $L^{p_0,p_\infty}( \mathbb{R}^d)$  if we replace $\lvert \xi \rvert^{p_0}$ by $ \log \lvert \xi\rvert $.

	\end{enumerate}

\end{proposition}

\begin{proof}
	The condition $\Psi_p (\xi) \leq C \rho_{p_0,p_\infty} (\xi)$ implies that, for any function $f \in L^{p_0,p_\infty}( \mathbb{R}^d) $, we have
	\begin{equation*}
		\lVert f \rVert_{\Psi_p} = \int_{ \mathbb{R}^d} \Psi_p(f(t)) \mathrm{d} t \leq C \int_{ \mathbb{R}^d} \rho_{p_0,p_\infty} (f(t) )\mathrm{d} t = C  \lVert f \rVert_{p_0,p_\infty}.
	\end{equation*}
	Therefore, the identity map is continuous from    $L^{p_0,p_\infty}( \mathbb{R}^d)$ to $L^p(\dot{X})$ proving \eqref{eq:practicalX1}. The proof of \eqref{eq:practicalX2} is very similar.
	For the last point, we remark that the two functions $\Psi_p$ and $\rho_{p_0,p_\infty}$ do not vanish for $\xi \neq 0$,  are continuous, and are equivalent at $0$ and infinity. Hence, there exists two constants  such that
	\begin{equation*}
		C_1  \rho_{p_0,p_\infty}(\xi) \leq  \Psi_p(\xi) \leq C_2 \rho_{p_0,p_\infty}(\xi).
	\end{equation*}
	We then apply \eqref{eq:practicalX1}  and \eqref{eq:practicalX2}  to obtain \eqref{eq:practicalX3}
\end{proof}

Note that the local integrability of test functions (parameter $p_0$) is linked with the asymptotic behavior of $\Psi_p$, while the asymptotic integrability (parameter $p_\infty$) is linked to the behavior of $\Psi_p$ at $0$.

If we know that the L\'evy measure has some finite moments, then we obtain new information on the domain of definition of the L\'evy white noise.   For  $p,q \geq 0$, we set
\begin{equation} \label{eq:genmoments}
	m_{p,q} (\nu) := \int_{ \mathbb{R}} \rho_{p,q} (x) \nu(\mathrm{d} x) = \int_{\abs{t}> 1} \abs{t}^p \nu (\mathrm{d} t) + \int_{\abs{t} \leq 1} \abs{t}^q \nu (\mathrm{d} t),
\end{equation}
called  the \emph{generalized moments} of $\nu$.
Then, $\nu$ being a L\'evy measure, we have that $m_{0,2} (\nu) < \infty$.

Consider the L\'evy process $X$, together with its corresponding L\'evy white noise $\dot{X}$, with L\'evy triplet $(0,0,\nu)$.
Then, $\dot{X}$  has finite $p$th moments if and only if $m_{p,2} (\nu) < \infty$ \cite[Theorem 25.3]{Sato1994levy}.
The quantity
\begin{equation*}
	\beta_0 := \sup\{ 0\leq p \leq 2, \ m_{p,2}(\nu) < \infty \}
\end{equation*}
is called the \emph{Pruitt index} and was introduced in \cite{Pruitt1981growth} to study the asymptotic behavior of L\'evy processes.
It measures the growth rate of $X$ at infinity \cite[Section 5.3]{Bottcher2014levy} and is therefore strongly related with the required rate of decay for the control of the Besov regularity of the L\'evy white noise $\dot{X}$ \cite[Theorem 3]{Fageot2017multidimensional}.

In contrast, the \emph{Blumenthal-Getoor index}, defined as
\begin{equation*}
	\beta_\infty := \inf\{ 0\leq q \leq 2, \ m_{0,q}(\nu) < \infty \},
\end{equation*}
relies on the local regularity of $X$ and $\dot{X}$.
This can be formulated in terms of the strong variation of $X$ \cite[Section 5.4]{Bottcher2014levy} or the local Besov regularity of $X$  (see \cite{Schilling1997Feller}) and of  $\dot{X}$ (see \cite[Corollary 3]{Fageot2017multidimensional}).

In accordance with the previous remarks, the generalized moments of a L\'evy measure $\nu$ have important interpretations for the local and asymptotic behaviors of the L\'evy white noise and the corresponding L\'evy process.

\begin{proposition} \label{lemma:controlPsip}
	Let $w$ be a symmetric L\'evy  noise without Gaussian part and with L\'evy measure $\nu$.
	\begin{itemize}
	\item We assume that $m_{p,2} (\nu) < \infty$ for some $0 \leq p \leq 2$.
	Then, we have, for any $\xi \in  \mathbb{R}$, that
	\begin{equation} \label{eq:PsipAlphap}
		m_{p,2}(\nu)  \rho_{p,2}(\xi)  \leq \Psi_p(\xi) \leq  m_{p,2}(\nu)   \rho_{2,p} (\xi).
	\end{equation}
	\item We assume that $m_{p,2} (\nu) < \infty$ for some $p \geq 2$.
	Then, we have, for any $\xi \in  \mathbb{R}$, that
	\begin{equation} \label{eq:PsipAlphapbiggerthan2}
		m_{p,2}(\nu)  \rho_{2,p}(\xi)  \leq \Psi_p(\xi) \leq  m_{p,2}(\nu)   \rho_{p,2} (\xi).
	\end{equation}
	\item For $p>2$, we condense \eqref{eq:PsipAlphap} and \eqref{eq:PsipAlphapbiggerthan2} as
\begin{equation*}
	m_{p,2}(\nu) \rho_{\min(p,2), \max(p,2)} (\xi) \leq \Psi_p(\xi) \leq m_{p,2}(\nu) \rho_{\max(p,2), \min(p,2)} (\xi).
\end{equation*}
	\item If   $m_{p_\infty, p_0} (\nu) < \infty$ for some $0\leq p_0 \leq 2$, $0 < p_\infty < \infty$ and if $p\leq p_0, p_\infty$, then
	\begin{equation} \label{eq:PsipAlphap0pinf}
		 \Psi_p(\xi) \leq  m_{\min(p_\infty,2), p_0}(\nu)   \rho_{p_0,\min(p_\infty,2)} (\xi).
	\end{equation}
	\end{itemize}
\end{proposition}

\begin{proof}
	All the inequalities will be obtained by exploiting the position of $\abs{x}$, $\abs{\xi}$, or $\abs{x \xi}$ with respect to $1$.
	We first show \eqref{eq:PsipAlphap}, the proof for \eqref{eq:PsipAlphapbiggerthan2} being very similar.
	We start by proving the upper bound of \eqref{eq:PsipAlphap}. We first assume that $\abs{\xi} \leq 1$. Then, using \eqref{eq:psimmetric}, we  decompose $\Psi_p$ as
	\begin{equation} \label{eq:developPsi}
		\Psi_p(\xi) =
		\int_{\lvert x \rvert \leq 1} \lvert x \xi  \rvert^2 \nu(\mathrm{d} x)
		+
		\int_{1< \lvert x \rvert \leq \frac{1}{\lvert \xi \rvert }} \lvert x \xi  \rvert^2 \nu(\mathrm{d} x)
		+
		 \int_{\lvert x \rvert > \frac{1}{\lvert \xi \rvert }} \lvert x   \xi \rvert^p\nu(\mathrm{d} x) .
	\end{equation}
	Since $p \leq 2$, we have that
	\begin{align}
	\Psi_p(\xi)
		& \leq
		\int_{\lvert x \rvert \leq 1} \lvert x \rvert^2   \lvert \xi \rvert^p \nu(\mathrm{d} x)
		+
		\int_{1< \lvert x \rvert \leq \frac{1}{\lvert \xi \rvert }} \lvert  x \xi  \rvert^p \nu(\mathrm{d} x)
		+
		\int_{\lvert x \rvert > \frac{1}{\lvert \xi \rvert }} \lvert x \xi \rvert^p \nu(\mathrm{d} x)  	\nonumber \\
		& = \left( \int_{\abs{x} \leq 1} \abs{x}^2 \nu(\mathrm{d} x ) + \int_{1 < \abs{x}} \abs{x}^p \nu(\mathrm{d} x)\right)  \abs{\xi}^p \nonumber \\
		& = m_{p,2}(\nu) \abs{\xi}^p. \label{eq:upperboundsmaller}
	\end{align}
	Assume now that $\abs{\xi} > 1$. Then, we use the decomposition
	\begin{equation} \label{eq:developPsi2}
		\Psi_p(\xi) =
		\int_{\lvert x \rvert \leq \frac{1}{\abs{\xi}}} \lvert x \xi  \rvert^2 \nu(\mathrm{d} x)
		+
		\int_{\frac{1}{\abs{\xi}}< \abs{x} \leq 1} \lvert x \xi  \rvert^p \nu(\mathrm{d} x)
		+
		 \int_{\lvert x \rvert > 1} \lvert x   \xi \rvert^p\nu(\mathrm{d} x) .
	\end{equation}
	Again, due to $p\leq 2$, we have that
	\begin{align}
	\Psi_p(\xi)
		& \leq
		\int_{\lvert x \rvert \leq \frac{1}{\abs{\xi}}} \lvert x \xi  \rvert^2 \nu(\mathrm{d} x)
		+
		\int_{\frac{1}{\abs{\xi}}< \abs{x} \leq 1} \lvert x \xi  \rvert^2 \nu(\mathrm{d} x)
		+
		 \int_{\lvert x \rvert > 1} \lvert x \rvert^p \abs{\xi}^2 \nu(\mathrm{d} x)
	\nonumber \\
		& = \left( \int_{\abs{x} \leq 1} \abs{x}^2 \nu(\mathrm{d} x ) + \int_{1 < \abs{x}} \abs{x}^p \nu(\mathrm{d} x)\right)  \abs{\xi}^2 \nonumber \\
		& = m_{p,2}(\nu) \abs{\xi}^2. \label{eq:upperboundbigger}
	\end{align}
	Combining \eqref{eq:upperboundsmaller} and \eqref{eq:upperboundbigger}, we deduce that $\Psi_p(\xi) \leq m_{p,2} (\nu) \rho_{2,p} (\xi)$.

	For the lower bound in \eqref{eq:PsipAlphap}, we first assume that $\abs{\xi} \leq 1$. Then, starting from \eqref{eq:developPsi}, we have that
	\begin{align}
	\Psi_p(\xi)
		& \geq
		\int_{\lvert x \rvert \leq 1} \lvert x \rvert^2   \lvert \xi \rvert^2 \nu(\mathrm{d} x)
		+
		\int_{1< \lvert x \rvert \leq \frac{1}{\lvert \xi \rvert }} \abs{x}^p \abs{\xi}^2 \nu(\mathrm{d} x)
		+
		\int_{\lvert x \rvert > \frac{1}{\lvert \xi \rvert }} \lvert x \rvert^p \lvert \xi \rvert^2 \nu(\mathrm{d} x)  	\nonumber \\
		& = m_{p,2}(\nu) \abs{\xi}^2. \label{eq:lowerboundsmaller}
	\end{align}
	And finally, when $\abs{\xi}>1$, we have, using \eqref{eq:developPsi2}, that
	\begin{align}
	\Psi_p(\xi)
		& \geq
		\int_{\lvert x \rvert \leq \frac{1}{\abs{\xi}}} \lvert x   \rvert^2 \abs{\xi}^p \nu(\mathrm{d} x)
		+
		\int_{\frac{1}{\abs{\xi}}< \abs{x} \leq 1} \lvert x    \rvert^2 \abs{\xi}^p \nu(\mathrm{d} x)
		+
		 \int_{\lvert x \rvert > 1} \abs{ x \xi }^p \nu(\mathrm{d} x)
	\nonumber \\
		& = m_{p,2}(\nu) \abs{\xi}^p. \label{eq:lowerboundbigger}
	\end{align}
	With \eqref{eq:lowerboundsmaller} and \eqref{eq:lowerboundbigger}, we deduce that $\Psi_p(\xi) \geq  m_{p,2}(\nu)  \rho_{p,2}(\xi)$ and \eqref{eq:PsipAlphap} is proved.  \\

	Equation \eqref{eq:PsipAlphap0pinf} is proved using the same principle.
	Assume that $\abs{\xi} \leq 1$ and $p \leq p_\infty \leq 2$. Then, using \eqref{eq:developPsi}, we deduce that
	\begin{align*}
	\Psi_p(\xi)
		& \leq
		\int_{\lvert x \rvert \leq 1} \lvert x \rvert^2   \lvert \xi \rvert^{p_\infty} \nu(\mathrm{d} x)
		+
		\int_{1< \lvert x \rvert \leq \frac{1}{\lvert \xi \rvert }} \lvert  x \xi  \rvert^{p_\infty} \nu(\mathrm{d} x)
		+
		\int_{\lvert x \rvert > \frac{1}{\lvert \xi \rvert }} \lvert x \xi \rvert^{p_\infty} \nu(\mathrm{d} x)  	\nonumber \\
		& = m_{p_\infty,2}(\nu) \abs{\xi}^p.
	\end{align*}
	If now $p_\infty > 2$, we have, still for $\abs{\xi} \leq 1$, that
	\begin{align*}
	\Psi_p(\xi)
		& \leq
		\int_{\lvert x \rvert \leq 1} \lvert x \rvert^2   \lvert \xi \rvert^{2} \nu(\mathrm{d} x)
		+
		\int_{1< \lvert x \rvert \leq \frac{1}{\lvert \xi \rvert }} \abs{x}^{p_\infty} \abs{\xi}^{2} \nu(\mathrm{d} x)
		+
		\int_{\lvert x \rvert > \frac{1}{\lvert \xi \rvert }} \abs{   \xi }^{2} \nu(\mathrm{d} x)
		\\
		& =  m_{2,2}(\nu) \abs{\xi}^2.
	\end{align*}
	We deduce that  $\Psi_p(\xi) \leq  m_{\min(p_\infty,2),2}(\nu)  \abs{\xi}^{\min(p_\infty , 2)}$.

	When $\abs{\xi} > 1$, $p \leq p_0 \leq 2$, and $p < p_\infty$, we have using \eqref{eq:developPsi2} that
	\begin{align*}
	\Psi_p(\xi)
		& \geq
		\int_{\lvert x \rvert \leq \frac{1}{\abs{\xi}}} \abs{ x \xi}^{p_0} \nu(\mathrm{d} x)
		+
		\int_{\frac{1}{\abs{\xi}}< \abs{x} \leq 1} \abs{ x\xi }^{p_0} \nu(\mathrm{d} x)
		+
		 \int_{\lvert x \rvert > 1} \abs{ x }^{\min(p_\infty,2)} \abs{ \xi }^{p_0} \nu(\mathrm{d} x)
	\nonumber \\
		& = m_{\min(p_\infty,2), p_0}(\nu) \abs{\xi}^{p_0}
	\end{align*}
	Remarking that $m_{\min(p_\infty,2) , 2}(\nu) \leq m_{\min(p_\infty,2) , p_0}(\nu)$ and combining the bounds for $\abs{\xi} \leq 1$ and $\abs{\xi} > 1$,  we deduce \eqref{eq:PsipAlphap0pinf}.
\end{proof}

\begin{proposition} \label{prop:limitspaces}
  For any L\'evy  noise, we have
	\begin{align}\label{eq:limitL0}
		L^{2,0}( \mathbb{R}^d) \subseteq L(\dot{X}) \subseteq L^{0,2}( \mathbb{R}^d),
	\end{align}
Let $0< p \leq 2$. For any symmetric L\'evy white noise such that $m_{p,2}(\nu) < \infty$, we have
\begin{equation} \label{eq:limitLp}
	L^{2,p}( \mathbb{R}^d) \subseteq L^{p}(\dot{X}) \subseteq L^{p,2}( \mathbb{R}^d).
\end{equation}
Let $p \geq 2$. For any symmetric L\'evy white noise such that $m_{p,2}(\nu) < \infty$, we have
\begin{equation} \label{eq:limitLpbigp}
	L^{p,2}( \mathbb{R}^d) \subseteq L^p(\dot{X}) \subseteq L^{2,p}( \mathbb{R}^d).
\end{equation}
For $p > 0$, assuming that $m_{p,2}(\nu) < \infty$, we condense \eqref{eq:limitLp} and \eqref{eq:limitLpbigp} as
\begin{equation} \label{eq:limitLpcondensed}
	L^{\max(p,2),\min(p,2)}( \mathbb{R}^d) \subseteq L^p(\dot{X}) \subseteq L^{\min(p,2),\max(p,2)}( \mathbb{R}^d) .
\end{equation}
In particular,   for any symmetric   finite-variance L\'evy   noise
	\begin{equation}\label{eq:limitL2}
		L^{2}(\dot{X}) = L^2( \mathbb{R}^d).
	\end{equation}
For any symmetric L\'evy white noise without Gaussian part such that $m_{p_\infty, p_0}(\nu) < \infty$, with $0\leq p \leq p_0, p_\infty \leq 2$, we have
	\begin{equation} \label{eq:limitmoreprecise}
	 L^{p_0,p_\infty}( \mathbb{R}^d) \subseteq L^p(\dot{X}) .
	\end{equation}
\end{proposition}

\begin{proof}
	When $w$ is symmetric without Gaussian part, \eqref{eq:limitL0}, \eqref{eq:limitLp}, and \eqref{eq:limitL2} are directly deduced from \eqref{eq:PsipAlphap} by taking $p=0$, $p$ general, and $p=2$, respectively.
	Adding a Gaussian part  does not change the conclusions since $L^{2,p} ( \mathbb{R}^d) \subseteq L^{p}(\dot{X}_{\mathrm{Gauss}}) = L^2( \mathbb{R}^d) \subseteq L^{p,2}( \mathbb{R}^d)$ for all $0 \leq p \leq 2$ and thanks to \eqref{eq:LpwithoutGaussian}.

	We now consider a general L\'evy  noise  $\dot{X}$ with L\'evy triplet  $(\gamma,\sigma^2,\nu)$  and $w_{\mathrm{sym}}$ its symmetric version with triplet  $(0,\sigma^2,\nu_{\mathrm{sym}})$.
We already now that
$L^{2,0}( \mathbb{R}^d) \subseteq  L(\dot{X}_{\mathrm{sym}}) \subseteq L^{0,2}( \mathbb{R}^d)$.
Moreover, from \eqref{eq:Lpsymmetrized}, we know that
\begin{equation} \label{eq:rappel}
	L(\dot{X}) = L(\dot{X}_{\mathrm{sym}}) \cap \{f \in L(\dot{X}), \ \int_{ \mathbb{R}^d} m_{\gamma,\nu}(f(t)) \mathrm{d} t < \infty \}.
\end{equation}
First, we have that $L(\dot{X}) \subseteq L(\dot{X}_{\mathrm{sym}}) \subseteq L^{0,2}( \mathbb{R}^d)$. Second, due to \eqref{eq:rappel},  it is sufficient to prove that
$$L^{2,0} ( \mathbb{R}^d) \subseteq  \{f \in L(\dot{X}), \ \int_{ \mathbb{R}^d} m_{\gamma,\nu}(f(t)) \mathrm{d} t < \infty \}$$
to deduce that $L^{2,0}( \mathbb{R}^d) \subseteq L(\dot{X})$. We remark that, for $\lvert \xi \rvert \leq 1$,
\begin{align*}
  m_{\gamma,\nu}(\xi) &= \left\lvert \gamma \xi + \int_{1\leq \lvert x \rvert \leq \frac{1}{\lvert \xi\rvert}}  \xi t  \nu({\mathrm{d} x}) \right\rvert
   \leq  \abs{\gamma \xi} + \int_{1\leq \lvert x \rvert \leq \frac{1}{\lvert \xi\rvert}}  \nu({\mathrm{d} x})  \\
  	& \leq \rvert  \gamma \lvert + \int_{1 \leq \lvert x \rvert} \nu(\mathrm{d} x),
\end{align*}
and that, for $\lvert \xi \rvert > 1$,
\begin{align*}
 m_{\gamma,\nu}(\xi) &=  \left\lvert \gamma \xi + \int_{\frac{1}{\lvert \xi\rvert} \leq \lvert x \rvert \leq 1}   \xi t   \nu(\mathrm{d} x) \right\rvert
 \leq \abs{ \gamma \xi } + \int_{\frac{1}{\lvert \xi\rvert} \leq \lvert x \rvert \leq 1} \lvert \xi t \rvert^2 \nu(\mathrm{d} x)  \\
  	& \leq \left( \rvert \gamma \lvert + \int_{ \lvert t\rvert \leq 1} t^2  \nu(\mathrm{d} x)\right) \xi^2.
\end{align*}
Therefore, we have $  m_{\gamma,\nu}(\xi)   \leq C \rho_{2,0}(\xi)$ for some constant $C$, which implies that $L^{2,0}( \mathbb{R}^d)$ is included into  $\{f \in L(\dot{X}), \ \int_{ \mathbb{R}^d} m_{\gamma,\nu}(f(t)) \mathrm{d} t < \infty \}$, as expected.

Finally, \eqref{eq:limitmoreprecise}   is a direct consequence of \eqref{eq:PsipAlphap0pinf}.
\end{proof}

\textit{Remarks.}
\begin{itemize}
\item The embeddings \eqref{eq:limitL0} inform on  the extreme cases.
In particular, a function in $L^{2,0}( \mathbb{R}^d)$---the space of  functions  in $L^2( \mathbb{R}^d)$ whose support has a finite Lebesgue measure---can be applied to any   L\'evy  noise.
This includes in particular all the indicator functions $\One_{B}$ with $B$ a Borel set with finite Lebesgue measure, or the Daubechies wavelets that are compactly supported and in $L^2( \mathbb{R}^d)$.
Remarkably, finite-variance compound Poisson  noises reach the largest possible domain of definition $L^{0,2}( \mathbb{R}^d)$ (see below).

\item Moreover, \eqref{eq:limitmoreprecise} is particularly important as it gives the implication of having finite moments of the form $\int_{\abs{t} > 1} \abs{t}^{p_\infty} \nu(\mathrm{d} x) < \infty$ and $\int_{\abs{t}\leq 1} \abs{t}^{p_0} \nu(\mathrm{d} x) < \infty$. This result will play a crucial role when identifying compatibility condition between a whitening operator and a L\'evy white noise in Section \ref{sec:application}.

\item The embeddings \eqref{eq:limitLpcondensed} are useful to understand the finiteness of the moments of $\langle w, f \rangle$ for a L\'evy white noise with finite $p$th-moments. In particular, a test function $f$ that is bounded with compact support is in the domain of definition of any noise and $\langle w , f \rangle$ has a finite $p$th-moment as soon as $w$ has.
\end{itemize}

	\subsection{Examples} \label{subsec:examples}

        In this section, we consider subfamilies of infinitely divisible laws that define important classes of L\'evy white noises.
        For these different classes, we specify the domain of definition $L(\dot{X})$ and the domains $L^p(\dot{X})$ of the considered L\'evy white noises.

        The function $\One_{[0,1]^d} \in L^{2,0}( \mathbb{R}^d) $ is in the domain of definition of every L\'evy white noise. Moreover, according to Proposition \ref{prop:CharacteristicFunction}, a L\'evy white noise with L\'evy exponent $\psi$ is such that
        \begin{equation*}
        		\Phi_{\langle \dot{X}, \One_{[0,1] } \rangle }(\xi) = \exp\left( \int_{ \mathbb{R}^d} \psi ( \xi \One_{[0,1]^d} (t) ) \mathrm{d} t\right) = \exp (\psi(\xi))
        \end{equation*}
        since $\psi(0) = 0$. Therefore, the L\'evy exponent of the L\'evy white noise is also the L\'evy exponent of the random variable $\langle \dot{X}, \One_{[0,1]^d} \rangle$. We take the convention that the terminology for the  law of this random variable is inherited by the   L\'evy white noise. For instance, a  white noise is said to be Gaussian if the random variable $\langle \dot{X}, \One_{[0,1]^d} \rangle$ is Gaussian.

        We   illustrate how to deduce the domain of definitions of Gaussian, S$\alpha$S, generalized Laplace, and compound Poisson white noises.

        		\subsubsection{Gaussian White Noises and Pure Drift White Noises} \label{sec:GaussAndDrift}

		The Gaussian white noise of variance $\sigma^2$ is characterized by the L\'evy triplet $(0,\sigma^2,0)$. With Proposition \ref{integrable}, we directly obtain that, for every $0\leq p \leq 2$,
		\begin{equation*}
		L^p(\dot{X}_{\mathrm{Gauss}}) = L^2( \mathbb{R}^d).
		\end{equation*}
		Based on these   considerations and Proposition \ref{prop:reduction}, we shall consider L\'evy triplets with $\sigma^2 = 0$ from now.

		Similarly, the pure drift white noise $\dot{X}_{\mathrm{drift}}$ with mean $\gamma$ is defined from its triplet $(\gamma, 0,0)$. We have in that case that $\dot{X}_{\mathrm{drift}} = \gamma$ almost surely and $\dot{X}_{\mathrm{drift}}$ is a constant---therefore non stochastic---process. Then, we have
		\begin{equation*}
		L^p(\dot{X}_{\mathrm{drift}}) = L^1( \mathbb{R}^d)
		\end{equation*}
		 for every $0\leq p \leq 2$.

		If now $\dot{X} = \dot{X}_{\mathrm{Gauss}} + \dot{X}_{\mathrm{drift}}$ has L\'evy triplet $(\gamma, \sigma^2,0)$ with $\gamma$ and $\sigma^2 \neq 0$, then the domain is, due to \eqref{eq:domainsumnoises} with equality,
		\begin{equation*}
		 	L^p(\dot{X}) =   L^1( \mathbb{R}^d)\cap  L^2( \mathbb{R}^d).
		\end{equation*}

		\subsubsection{Non-Gaussian S$\alpha$S White Noises}

		Stable random variables are an important subclass of infinitely divisible random variables. Extensive details on S$\alpha$S random variables and random processes can be found in \cite{Taqqu1994stable}.

	The extension of the symmetric $\alpha$-stable noise to an independently scattered random measure is in fact an $\alpha$-stable random measure in the sense of \cite{Taqqu1994stable}. The identification of the space of deterministic integrable functions has already been carried out in this context in \cite{Taqqu1994stable}, and we merely re-state the result and prove it within our framework.

		We fix $0< \alpha < 2$.
		A random variable is symmetric-$\alpha$-stable (S$\alpha$S) if its characteristic function can be written as  $\mathrm{e}^{- \gamma \lvert \xi \rvert^\alpha}$ for some $\gamma >0$. For simplicity, we should only consider $\gamma = 1$ thereafter, since a different $\gamma$ will not change the domain of definition according to Proposition \ref{prop:scalingindependence}. A S$\alpha$S  white noise $\dot{X}_\alpha$ is a L\'evy white noise such that $\langle \dot{X}_\alpha , \One_{[0,1]^d} \rangle$ is a S$\alpha$S random variable. Its characteristic functional is given for $\varphi \in {\mathcal{D}}( \mathbb{R}^d)$ by \cite[Section 4.2.2]{Unser2014sparse}
		\begin{equation*}
			{\widehat{\mathscr{P}}}_{\dot{X}_\alpha}(\varphi) = \exp \left( - \lVert \varphi \rVert^{\alpha}_\alpha \right).
		\end{equation*}

		\begin{proposition}\label{prop:salphas}
			Let $0 < \alpha < 2$. Then, for every $0\leq p < \alpha$, we have
			\begin{equation*}
				L^p(\dot{X}_{\alpha}) = L^{\alpha}( \mathbb{R}^d).
			\end{equation*}
			For $p \geq \alpha$, we have $L^p(\dot{X}_\alpha) = \{0\}$.
		\end{proposition}

\begin{proof
	The L\'evy measure of $\dot{X}_\alpha$ is  $\nu(\mathrm{d} x) = \frac{C_\alpha}{\lvert x \rvert^{\alpha + 1}} \mathrm{d} x$ with $C_\alpha$ a constant. A non-trivial S$\alpha$S random variable has an infinite $p$th-moment for $p\geq \alpha$, and for every $f \in L(\dot{X}_\alpha)$, $\langle \dot{X}, f \rangle$ is a S$\alpha$S random variable. Hence $L^p (\dot{X}) = \{0\}$ for $p\geq \alpha$. The case of interest is  therefore  $0\leq p < \alpha$. Then, from \eqref{eq:psimmetric},
	\begin{align*}
		\Psi_{p}(\xi) &= 2C_\alpha \int_{0}^{1/\lvert \xi \rvert} \frac{\xi^2}{x^{\alpha + 1}} \mathrm{d} x + 2C_\alpha \int_{1 / \lvert \xi \rvert} \frac{\lvert \xi \rvert^p}{x^{\alpha + 1 - p}} \mathrm{d} x \\
				&= 2C_\alpha \lvert \xi \rvert^\alpha \left( \int_0^1 \frac{\mathrm{d} y}{y^{\alpha -1}} + \int_{1}^\infty \frac{\mathrm{d} y }{y^{\alpha + 1 - p}} \right) \\
				&= \left( \frac{2(2-p)C_\alpha}{(2-\alpha)(\alpha-p)} \right) \lvert \xi \rvert^\alpha.
	\end{align*}
 Finally, the result follows from Proposition \ref{prop:criteria}.
\end{proof}

		\subsubsection{Generalized Laplace White Noises}

		Our goal is to study the   Laplace white noise, for which $\langle \dot{X} , \One_{[0,1]^d} \rangle$ follows a Laplace law. It requires to introduce the family of generalized Laplace laws. We follow here the terminology of \cite[Section 4.1.1]{Koltz2001laplace} and consider only the symmetric case.

		A random variable $Y$ is called a \emph{generalized Laplace random variable} if its characteristic function can be written as
		\begin{equation*}
			\Phi(\xi) = \frac{1}{(1 + \frac{1}{2} \sigma^2 \xi^2)^\tau} = \exp \left( - \tau \log (1 + \frac{1}{2} \sigma^2 \xi^2) \right),
		\end{equation*}
		with $\tau > 0$ the \emph{shape parameter} and $\sigma^2$ the \emph{scaling parameter}. We denote this situation by $Y \sim \mathcal{GL}(\sigma,\tau)$. Note that the variance of $Y$ is $\tau \sigma^2$. When $\tau = 1$, we recover the traditional Laplace law. The generalized Laplace laws are infinitely divisible \cite[Section 2.4.1]{Koltz2001laplace} and associated with the L\'evy triplet $(0,0, \nu_{\tau,\sigma^2})$ with \cite[Proposition 2.4.2]{Koltz2001laplace}

	\begin{equation*} \label{eq:levymeasurelaplace}
	\nu_{\tau,\sigma^2}(\mathrm{d} x) = \tau \frac{1}{\lvert x \rvert} \mathrm{e}^{- 2 \lvert x \rvert / \sigma^2} \mathrm{d} x.
	\end{equation*}

		\begin{definition}
			We say that a L\'evy white noise $\dot{X}_{\mathrm{Laplace}}$ is a \emph{generalized Laplace white noise} if $\langle \dot{X}_{\mathrm{Laplace}} , \One_{[0,1]^d} \rangle \sim \mathcal{GL}(\sigma,\tau)$ for some $\tau, \sigma^2>0$. We call $\tau$ and $\sigma^2$ respectively the shape parameter and the scaling parameter of $\dot{X}_{\mathrm{Laplace}}$. When $\tau = 1$, we simply say that $\dot{X}_{\mathrm{Laplace}}$ is a \emph{Laplace white noise}.
		\end{definition}

	For the best of our knowledge, general integrability conditions for (generalized) Lapalce noise has not been investigated in the literature. Proposition \ref{prop:laplacecase} provides such conditions.

	\begin{proposition} \label{prop:laplacecase}
		For every generalized Laplace white noise $\dot{X}_{\mathrm{Laplace}}$, we have
		\begin{equation} \label{prop:domainlaplace}
			L(\dot{X}_{\mathrm{Laplace}}) = L^{\log, 2}( \mathbb{R}^d).
		\end{equation}
		Moreover, for $0<p\leq 2$, we have
		\begin{equation} \label{prop:domainlaplacep}
			L^p(\dot{X}_{\mathrm{Laplace}}) = L^{p, 2}( \mathbb{R}^d).
		\end{equation}
	\end{proposition}

	\begin{proof}
		Let $0 \leq p \leq 2$.
		We start from \eqref{eq:psimmetric} and write
		\begin{equation*}
		 \Psi_p(\xi)     =  \xi^2 \int_{|x| \leq 1/|\xi|} x^2 \nu_{\tau,\sigma^2} (\mathrm{d} x) +  \lvert \xi \rvert^p  \int_{|x|> 1/ \lvert \xi \rvert} \lvert x \rvert^p \nu_{\tau,\sigma^2} (\mathrm{d} x) := \Psi_{p,1}(\xi) + \Psi_{p,2}(\xi).
		\end{equation*}
		Without loss of generality, we   consider the case $\sigma^2 = 2$ and $\tau = 1$, in which case $\nu_{1,2}(\mathrm{d} x) = \frac{\mathrm{e}^{-\lvert x \rvert}}{\lvert x \rvert} \mathrm{d} x$. Then, by integration by parts, we have
		\begin{align*}
			\Psi_{p,1}(\xi) &= 2 \lvert \xi \rvert^2 \int_{0}^{1 / \lvert \xi\rvert} x \mathrm{e}^{- x} \mathrm{d} x \nonumber \\
					&= 2 \lvert \xi \rvert^2 \left( 1 - \mathrm{e}^{- 1 / \lvert \xi \rvert} (1 + \frac{1}{\lvert \xi \rvert} ) \right)
		\end{align*}
		Hence, we have $\Psi_{p,1} (\xi) \underset{\xi \rightarrow \infty}{\longrightarrow} 2$ and $\Psi_{p,1} (\xi) \underset{\xi \rightarrow 0}{\sim} 2\lvert \xi\vert^2$.

		For $\Psi_{p,2}(\xi) = \lvert \xi \rvert^p  \int_{|x|> 1/ \lvert \xi \rvert} \lvert x \rvert^p \nu_{\tau,\sigma^2} (\mathrm{d} x)$, we shall distinguish between $p=0$ and $p>0$. For $p>0$, the function $x^{p-1}\mathrm{e}^{-x}$ is integrable over $ \mathbb{R}$, so that $\Psi_{p,2}(\xi)  \underset{\xi \rightarrow \infty}{\sim} \left( \int_{ \mathbb{R}} x^{p-1}\mathrm{e}^{-x} \mathrm{d} x\right) \lvert \xi\vert^p$. For $p=0$, the function $x^{-1} \mathrm{e}^{-x} $ is not anymore integrable around $0$. Using the equivalence $ x^{-1} \mathrm{e}^{-x}  \underset{x \rightarrow 0}{\sim} x^{-1}$, we deduce that
		$$ \Psi_{p,2}(\xi) = 2 \int_{   \frac{ 1}{\lvert \xi \rvert}}^{\infty}  x^{-1}\mathrm{e}^{-x} \mathrm{d} x  \underset{\xi \rightarrow \infty}{\sim}   2 \int_{   \frac{ 1}{\lvert \xi \rvert}}^{1}  x^{-1}\mathrm{e}^{-x} \mathrm{d} x\underset{\xi \rightarrow \infty}{\sim} 2 \int_{\frac{1}{\lvert \xi \rvert}}^1 x^{-1} \mathrm{d} x = 2 \log \abs{\xi}.$$
		Moreover, since $p\leq 2$, we have, again by integration by parts,
		\begin{align*}
		\Psi_{p,2}(\xi) &= 2 \int_{x \lvert \xi \rvert>1} (x \lvert \xi \rvert)^p \mathrm{e}^{-x} \frac{ \mathrm{d} x}{x} \leq 2 \int_{x \lvert \xi \rvert>1} (x \lvert \xi \rvert)^2 \mathrm{e}^{-x} \frac{ \mathrm{d} x}{x}  = 2 \lvert \xi \rvert (1 + \lvert \xi \rvert) \mathrm{e}^{- 1 / \lvert \xi \rvert} ,
		\end{align*}
		implying that $\Psi_{p,2}(\xi) \underset{\xi \rightarrow 0}{=} o(\lvert \xi \rvert^2)$. By combining the results on $\Psi_{p,1}$ and $\Psi_{p,2}$, we obtain that
		\begin{itemize}
			\item for $0\leq p\leq2$, $\Psi_{p}(\xi) \underset{\xi \rightarrow 0}{\sim} 2 \lvert \xi \rvert^2$;
			\item for $0< p\leq 2$, $\Psi_{p}(\xi) \underset{\xi \rightarrow \infty}{\sim}  \left( \int_{ \mathbb{R}} x^{p-1}\mathrm{e}^{-x} \mathrm{d} x\right) \lvert \xi \rvert^p$;
			\item for $p=0$, $\Psi_{0}(\xi) = \Psi(\xi) \underset{\xi \rightarrow \infty}{\sim} 2 \log \lvert \xi\rvert$.
		\end{itemize}
		We apply now Proposition \ref{prop:criteria} to deduce \eqref{prop:domainlaplace} and \eqref{prop:domainlaplacep}.
	\end{proof}

		\subsubsection{Compound Poisson White Noises}\label{subsubsec:poisson}

\begin{definition}\label{def:poisson}
 A L\'evy white noise is a \emph{compound Poisson noise} if its L\'evy triplet has the form $(0,0,\nu)$ and if its L\'evy measure satisfies
 \begin{equation*}
 \lambda := \int_{ \mathbb{R}} \nu(\mathrm{d} x) < \infty.
 \end{equation*}
In that case, $\nu = \lambda \mathbb{P}$ with $\mathbb{P}$ a probability measure.
\end{definition}

One can represent a compound Poisson noise $\dot{X}_{\mathrm{Poisson}}$ as \cite[Theorem 1]{Unser2011stochastic}
		\begin{equation*}
			\dot{X}_{\mathrm{Poisson}} = \sum_{n\geq 0} a_n \delta ( \cdot - t_n)
		\end{equation*}
		with $\delta$ the Dirac distribution, $(a_n)$ i.i.d. random variables with probability law $\mathbb{P}$, and $(t_n)$ independent of $(a_n)$ such that, for every Borel set $B \in  \mathbb{R}^d$, the random number of elements $t_n$ in $B$ follows a Poisson law with rate $\lambda \mathrm{Leb}(B)$.
	Then, $\dot{X}_{\mathrm{Poisson}}$ has a finite variance if and only if $\mathbb{P}$ has. In that case, it has a zero mean if and only if $\mathbb{P}$ has.

	The integration with respect to compound Poisson noise is treated for instance in \cite[Chapter 12]{Kallenberg2006foundations}. However, it relies on  $L^1$-type conditions, inherited from the fact that the stochastic integration follows the L\'evy-It\^o decomposition (see Section \ref{subsec:sheetnoise} for more details). We provide more general integratibility conditions in Proposiiton \ref{prop:poissoncase}, that are not sensitive to $L^1$-type integrabitiliy.

	\begin{proposition}\label{prop:poissoncase}
	If $\dot{X}_{\mathrm{Poisson}}$ is a symmetric compound Poisson white noise  with finite variance, then
	\begin{equation*}
				L^p (\dot{X}_{\mathrm{Poisson}}) = L^{p,2} ( \mathbb{R}^d).
	\end{equation*}
for every $0 \leq p \leq 2$.
	\end{proposition}

	\begin{proof}
		First, $L^p (\dot{X}_{\mathrm{Poisson}}) \subseteq L^{p,2} ( \mathbb{R}^d)$ as for any symmetric L\'evy white noise, according to \eqref{eq:limitLp}.  Moreover, for a compound Poisson noise with finite variance, we have for every $q \in [0,2]$ that $ \int_{ \mathbb{R} } \lvert x  \rvert^q   \mathbb{P}(\mathrm{d} x) < \infty$. Therefore, we have
		\begin{align*}
			\Psi_p(\xi) &= \lambda \int_{ \mathbb{R} } (  \lvert x \xi \rvert^p \wedge   \lvert x \xi \rvert^2) \mathbb{P}(\mathrm{d} x) \nonumber \\
			& \leq \lambda \min\left( \lvert \xi \rvert^p   \int_{ \mathbb{R} } \lvert x  \rvert^p   \mathbb{P}(\mathrm{d} x), \lvert \xi \rvert^2   \int_{ \mathbb{R} } \lvert x  \rvert^2   \mathbb{P}(\mathrm{d} x) \right) \nonumber \\
			& \leq C ( \lvert \xi \rvert^p\wedge   \lvert \xi \rvert^2) = \rho_{p,2}(\xi),
		\end{align*}
		so that $\lVert f \rVert_{\Psi_p} \leq C \lVert f \rVert_{p,2}$. This means that $ L^{p,2} ( \mathbb{R}^d) \subseteq L^p (\dot{X}_{\mathrm{Poisson}}) $, finishing the proof.
	\end{proof}

	        We summarize the results of this section in Table \ref{table:domains}.

\begin{table*}[t!]
\centering
\caption{Definition Domains of some L\'evy White Noises}

\begin{tabular}{ccccc}
\hline
\hline\\[-2ex]
White noise & Parameters  & $	\Phi_{\langle \dot{X}, \One_{[0,1]}\rangle}(\xi) $ &  $L(\dot{X})$ & $L^p(\dot{X})$ \\
& & & & $0<p \leq2$
\\
\hline\\[-1ex]
Gaussian & $\sigma^2 >  0$ & $\mathrm{e}^{- \sigma^2 \xi^2}$ & $L^2( \mathbb{R}^d)$ & $L^2( \mathbb{R}^d)$ \\[+1ex]
Pure drift & $\gamma \in  \mathbb{R}$ & $\mathrm{e}^{\mathrm{i}\gamma \xi}$ & $L^1( \mathbb{R}^d)$ & $L^1( \mathbb{R}^d)$ \\[+1ex]
S$\alpha$S & $0<\alpha<2$ & $\mathrm{e}^{-\lvert \xi \rvert^\alpha}$ & $L^\alpha( \mathbb{R}^d)$ & $\begin{cases} L^\alpha( \mathbb{R}^d) &\mbox{if } p<\alpha \\
\{0\} & \mbox{if } p\geq \alpha \end{cases}$  \\[+1ex]
generalized & $\sigma^2 > 0$ & $\frac{1}{(1 + \sigma^2\xi/2)^\tau}$ & $L^{\log,2}( \mathbb{R}^d)$ & $L^{p,2}( \mathbb{R}^d)$ \\
Laplace & $\tau >0$  & & & \\[+1ex]
symmetric finite-variance & $\lambda >0$ & $\mathrm{e}^{\lambda (\widehat{\mathbb{P}} (\xi) -1 )}$ & $L^{0,2}( \mathbb{R}^d)$ & $L^{p,2}( \mathbb{R}^d)$ \\
compound Poisson & $\mathbb{P}$ & & & \\
\hline
\hline
\end{tabular} \label{table:domains}
\end{table*}

\section{Application to  Linear Stochastic Partial Differential Equations} \label{sec:application}

Until now, we restricted ourself to the study of L\'evy white noise. In this section, we   see how to apply our results to solve linear stochastic differential equations of the form
\begin{equation*} \label{eq:SDE}
	\Lop s  = \dot{X}
\end{equation*}
with $\dot{X}$ a L\'evy white noise and $\Lop$ a differential operator.
We give new conditions of compatibility between the operator $\Lop$ and the white noise $\dot{X}$ such that the process $s$ exists as a generalized random process.
Our results extend our previous works  \cite{Fageot2014,Unser2014sparse,Unser2014unifiedContinuous}.

\begin{theorem}\label{theo:generalcriterion}
	We consider a L\'evy white noise $\dot{X}$.
	We assume that $\Top$ is a continuous and linear operator from ${\mathcal{D}}( \mathbb{R}^d)$ to $L(\dot{X})$.
	Then, the mapping
	\begin{eqnarray}
	s :  & {\mathcal{D}}( \mathbb{R}^d) &\rightarrow L^0 (\Omega) \nonumber \\
			& \varphi  & \mapsto \langle s ,\varphi \rangle := \langle \dot{X}, \Top \{\varphi \}\rangle
	\end{eqnarray}
	specifies a generalized random process in the sense of Definition \ref{def:GRP}.

	If moreover there exists an operator $\Lop$ such that $\Top \Lop^* \varphi = \varphi$ for every $\varphi \in {\mathcal{D}}( \mathbb{R}^d)$ (left-inverse property), then we have that
	\begin{equation}
		\Lop s = w.
	\end{equation}
\end{theorem}

\begin{proof}
	The mapping $\varphi \mapsto \langle \dot{X}, \Top \{\varphi\}$ is well-defined for any  $\varphi \in {\mathcal{D}}( \mathbb{R}^d)$ because $\Top \{\varphi\} \in L(\dot{X})$ by assumption.
	It is actually the composition of the operator $\Top$ with the random linear functional $\dot{X}$. This two mappings being linear, the composition is linear. Moreover, $\Top$ is continuous from ${\mathcal{D}}( \mathbb{R}^d)$ to $L(\dot{X})$ by assumption and $\dot{X}$ is continuous from $\L(\dot{X})$ to $L^0(\Omega)$ according to Theorem \ref{prop:continuity}. Therefore, $s$ is linear and continuous mapping from ${\mathcal{D}}( \mathbb{R}^d)$ to $L(\dot{X})$ and therefore a valid generalized random process in ${\mathcal{D}}'( \mathbb{R}^d)$.

	If now $\Top \Lop^*\{\varphi\} = \varphi$ for every $\varphi \in {\mathcal{D}}( \mathbb{R}^d)$, then the process $\Lop s$, defined as
	\begin{equation}
		\Lop s \ : \varphi \mapsto \langle s, \Lop^* \varphi \rangle,
	\end{equation}
	satisfies the relation
	\begin{equation} \label{eq:whyonlyleftinverse}
		\langle \Lop s , \varphi \rangle = \langle \dot{X} , \Top \Lop^* \varphi \rangle = \langle \dot{X},\varphi\rangle
	\end{equation}
	for every $\varphi \in {\mathcal{D}}( \mathbb{R}^d)$. Equivalently, we have shown that $\Lop s = \dot{X}$ as elements of ${\mathcal{D}}'( \mathbb{R}^d)$, as expected.
\end{proof}

We justify shortly the assumptions of Theorem \ref{theo:generalcriterion}.
Many differential operators admit a natural inverse $\Lop^{-1}$, that is typically defined using the Green's function of $\Lop$.
A solution of \eqref{eq:SDE} can therefore be formally written as $s = \Lop^{-1} w$; that is,
\begin{equation} \label{eq:formalrelation}
	\langle s, \varphi \rangle = \langle \Lop^{-1} \dot{X}, \varphi \rangle = \langle \dot{X}, (\Lop^*)^{-1} \{\varphi\} \rangle.
\end{equation}
In order to be valid, \eqref{eq:formalrelation} should at least be meaningful for any $\varphi \in {\mathcal{D}}( \mathbb{R}^d)$. It means in particular that $(\Lop^*)^{-1} \{\varphi\}$ should be in domain of definition of the L\'evy white noise $\dot{X}$.
However, for many differential operators, including the derivative, the natural inverse operator to $\Lop^*$ exists but is not stable in the sense that it is not continuous from ${\mathcal{D}}( \mathbb{R}^d)$ to any domain of definition $L(\dot{X})$.
In that case, it is required to correct $(\Lop^*)^{-1}$ in order to make it stable.
The role of the corrected version of $(\Lop^*)^{-1}$ is played by $\Top$.  Thanks to \eqref{eq:whyonlyleftinverse}, we moreover see that we only require that $\Top$ is a \textit{left}-inverse of $\Lop^*$.
For the specification of stable left-inverses of important classes of differential operators, including the derivative of any order, fractional derivatives, fractional Laplacian, we refer the reader to \cite{Unser2014sparse}.

\begin{definition}\label{def:GLP}
A generalized random process constructed according to Theorem \ref{theo:generalcriterion} is called a \emph{generalized L\'evy process} (or \emph{generalized L\'evy field} when $d\geq 2$). The operator $\Lop$ is the \emph{whitening operator} of $s$ and $\dot{X}$ the underlying L\'evy white noise.
\end{definition}

The following result links the stability properties of the corrected left-inverse operator $\Top$ with the finiteness of the generalized moments of the L\'evy measure of $\dot{X}$.

 \begin{proposition} \label{prop:constructioncriterion}
 	We consider a symmetric L\'evy white noise without Gaussian  part $\dot{X}$ and a linear, continuous, and   shift-invariant operator $\Lop$.
	We assume that, for $0\leq p_0, p_\infty \leq 2$, we have
	\begin{itemize}
		\item $m_{p_\infty, p_0}(\nu) = \int_{ \mathbb{R}} \rho_{p_\infty, p_0} (t) \nu(\mathrm{d} t) < \infty$, and
		\item the adjoint operator $\Lop^*$ admits a left-inverse $\Top$ that maps continuously ${\mathcal{D}}( \mathbb{R}^d)$ to $L^{p_0,p_\infty}( \mathbb{R}^d)$.
	\end{itemize}
	Then, there exists a generalized L\'evy process $s$ such that $\Lop s = \dot{X}$.
 \end{proposition}

 \begin{proof}
 	Applying \eqref{eq:limitmoreprecise} with $p=0$, the condition $\int_{ \mathbb{R}} \rho_{p_\infty, p_0} (t) \nu(\mathrm{d} t) < \infty$ ensures that $L^{p_0,p_\infty}( \mathbb{R}^d) \subset L(\dot{X})$. This embedding and the assumption on $\Top$ imply that $\Top$ maps continuously ${\mathcal{D}}( \mathbb{R}^d)$ to $L(\dot{X})$, and Theorem \ref{theo:generalcriterion} applies.
 \end{proof}

We recall that the condition $m_{p_\infty,p_0}(\nu) < \infty$ introduced in \eqref{eq:genmoments} is connected with the local properties (parameter $p_0$) and the asymptotic properties (parameter $p_\infty$) of the L\'evy white noise.

 \paragraph{Comparison with previous works.}
 Theorem \ref{theo:generalcriterion} and Proposition \ref{prop:constructioncriterion} can be compared with other conditions of compatibility between the whitening operator $\Lop$ and the L\'evy white noise $\dot{X}$. The results are reformulated with our notation.

\begin{itemize}

	\item First of all, we differentiate between two types of solutions of the linear SDE $\Lop s = w$. We say that $s$ is a \emph{generalized solution} if $\Lop s = w$ almost surely. In contrast, a generalized random process $s$ such that $\Lop s = w$ in law (that is, $\mathscr{P}_{\Lop s} = \mathscr{P}_w$) is called a \emph{solution in law}. In our previous works, we constructed solutions in law of \eqref{eq:SDE} essentially relying on the Minlos-Bochner theorem for the construction of $s$. One important contribution of this paper is to construct generalized solutions, what requires the identification of the domain of definition of the L\'evy white noise $\dot{X}$ to be as general as possible.

	\item Throughout the paper, we have considered L\'evy white noise as random elements in ${\mathcal{D}}'( \mathbb{R}^d)$. This is in line with the original work of Gelfand and Vilenkin \cite{GelVil4}.
It can be of interest, however, to restrict to the class of \emph{tempered} L\'evy white noise, that is, to consider L\'evy white noise, and by extension generalized L\'evy processes, as random elements in the space ${\mathcal{S}}'( \mathbb{R}^d)$.
The construction of generalized L\'evy processes in ${\mathcal{S}}'( \mathbb{R}^d)$ instead of ${\mathcal{D}}'( \mathbb{R}^d)$ can be found in \cite{Unser2014sparse}; see also \cite{bierme2017generalized} for a recent exposition of the main results in this framework. For a comparison between the two constructions, we refer to \cite[Chapter 3]{Fageot2014}.

Not every L\'evy white noise is tempered. Actually, a L\'evy white noise is tempered if and only if it has finite $p$th moments for some $p>0$ (see \cite{Dalang2015Levy}).
The main point is that the construction is really analogous, since the spaces ${\mathcal{S}}( \mathbb{R}^d)$ and ${\mathcal{S}}'( \mathbb{R}^d)$ are nuclear.
Among the consequences, we mention that Theorem \ref{theo:generalcriterion} and Proposition \ref{prop:constructioncriterion} remain valid when replacing ${\mathcal{D}}( \mathbb{R}^d)$ by ${\mathcal{S}}( \mathbb{R}^d)$. In that case, the processes $\dot{X}$ and $s$ are both located in ${\mathcal{S}}'( \mathbb{R}^d)$. In what follows, when comparing these two results with previous contributions, we consider the version of the result for tempered generalized random processes.

	\item  For $1\leq p \leq 2$, the L\'evy exponent $\psi$ is \emph{$p$-admissibile} if $\abs{\psi(\xi)} + \abs{\xi} \abs{\psi'(\xi)} \leq C \abs{\xi}^p$. Note that the derivative $\psi'(\xi)$ is well-defined as soon as the first moment of the underlying infinitely divisible random variable is finite, what we assume now. This notion was introduced in \cite{Unser2014sparse} together with the following compatibility condition: if $\psi$ is $p$-admissible and $\Top$ continuously map ${\mathcal{S}}( \mathbb{R}^d)$ to $L^p( \mathbb{R}^d)$, then there exists a solution in law of $\Lop s = w$ with characteristic functional
	\begin{equation} \label{eq:CFTop}
	{\widehat{\mathscr{P}}}_s : \varphi \mapsto {\widehat{\mathscr{P}}}_{\dot{X}} ( \Top \{\varphi\}).
	\end{equation}
	 A sufficient condition for the $p$-admissible is that $\int_{ \mathbb{R}} \abs{t}^p \nu(\mathrm{d} t) < \infty$. Therefore, \eqref{eq:CFTop} is a valid characteristic functional as soon as $\int_{ \mathbb{R}} \abs{t}^p \nu(\mathrm{d} t) < \infty$ and $\Top$ maps continuously ${\mathcal{S}}( \mathbb{R}^d)$ to $L^p( \mathbb{R}^d)$ for some $1 \leq p \leq 2$.
	We recover this result by selecting $p_0 = p_\infty = p$ in Proposition \ref{prop:constructioncriterion}. Actually, Proposition \ref{prop:constructioncriterion} extends this criterion in three ways. First, we can distinguish between the behavior of $\nu$ around $0$ and at $\infty$. Second, we do not restrict to the case $p \geq 1$ (this second improvement was already achieved in our work \cite{Fageot2016scaling} thanks to a relaxation of the $p$-admissibility). Finally, as we have said already, we specify generalized solutions, and not only solutions in law, of $\Lop s = \dot{X}$.

	\item  In our work with A. Amini, we have shown that the characteristic functional \eqref{eq:CFTop} specifies a generalized L\'evy process if $\int_{ \mathbb{R}} \rho_{p_\infty,p_0} (t) \nu(\mathrm{d} t)$ and $\Top$ maps continuously ${\mathcal{S}}( \mathbb{R}^d)$ to $L^{p_0,p_\infty}( \mathbb{R}^d)$ for $0 < p_\infty \leq p_0 \leq 2$ \cite[Theorem 5]{Fageot2014}. When $p_\infty \leq p_0$, we have that
	\begin{equation*}
		\max ( \abs{\xi}^{p_0} , \abs{\xi}^{p_\infty} ) \leq \rho_{p_0,p_\infty}(\xi) \leq  \abs{\xi}^{p_0} + \abs{\xi}^{p_\infty}.
	\end{equation*}
	Therefore, $L^{p_0,p_\infty}( \mathbb{R}^d) = L^{p_0}( \mathbb{R}^d) \cap L^{p_\infty}( \mathbb{R}^d)$ and we recover our previous result (at least for symmetric L\'evy white noise without Gaussian part).
	Moreover, Proposition \ref{prop:constructioncriterion} is a improvement, since one can consider $p_\infty > p_0$. In that case, $L^{p_0,p_\infty}( \mathbb{R}^d)$ contains but is strictly bigger than $L^{p_0}( \mathbb{R}^d) \cap L^{p_\infty}( \mathbb{R}^d)$ and the requirement on $\Top$ is less restrictive.

	\item Combining \eqref{eq:limitmoreprecise} and Proposition \ref{prop:continuityCF}, we generalize \cite[Theorem 2]{Amini2014sparsity} again   by considering the case $p_\infty > p_0$: we are able to specify a larger domain of definition and of continuity than $L^{p_0}( \mathbb{R}^d) \cap L^{p_\infty}( \mathbb{R}^d)$ in that case.

\end{itemize}

redpajama_set_name: RedPajamaArXiv