Existence of a density of the 2Dim Stochastic Navier Stokes Equation driven by Levy processes or fractional Brownian motion

TL;DR

This paper investigates the regularity of the probability distribution of solutions to the 2D stochastic Navier-Stokes equation driven by Levy processes or fractional Brownian motion, focusing on conditions for absolute continuity of finite-dimensional projections.

Contribution

It establishes conditions under which the law of the solution's projection is absolutely continuous, advancing understanding of stochastic Navier-Stokes equations with Levy and fractional noise.

Findings

Conditions for absolute continuity of the solution's law

Regularity properties of the probability measure

Impact of Levy measure and Hurst parameter

Abstract

In this article we are interested in the regularity properties of the probability measure induced by the solution process of the L\'evy noise or a fractional Brownian motion driven Navier Stokes Equation on the two dimensional torus $\mathbb{T}$. We mainly investigate under which conditions on the characteristic measure of the L\'evy process or the Hurst parameter of the fractal Brownian motion the law of the projection of $u(t)$ onto any finite dimensional $F\subset L^2(\mathbb{T})$ is absolutely continuous with respect to the Lebesgue measure on $F$.

Equations

$$\{\begin{array}{l}{\partial }_{t}u(t)-\nu \Delta u(t)+u(t)\cdot \nabla u(t)+\nabla \mathfrak{p}(t)=\dot{\Xi }(t),\\ \nabla \cdot u(t)=0,\\ u(0)=u_0,\end{array}$$

$$\Xi$$

$$A_{(E_0,E_1)}(y)=\{x\in E_0:x+y\in A\}.$$

$${\mu }_{E_0}:\mathcal{B}(E_0)\ni A\mapsto \mu (A+E_1)\in [0,1].$$

$${\mu }_{({\tilde{E}}_0,E_1)}:\mathcal{B}({\tilde{E}}_0)\ni A\mapsto \mu (A+E_1)\in [0,1].$$

$$\text{mbox}{Leb}_{E_0}:\mathcal{B}(E_0)\ni U\mapsto {\mu }_{E_0}(U):=\text{mbox}{Leb}_{{\mathbb{R}}^n}({\iota }^{-1}(U)),$$

$$l_n:G^n\times \mathcal{B}(F_1\oplus \cdots \oplus F_n)\to {\mathbb{R}}_0^{+},$$

$$\mu (A)={\int }_{G^n}{\int }_{A_n(y)}{l}_n(\mathbf{y},dx){\mu }_{G^n}(d\mathbf{y}),$$

$$F_n=\{\lambda e_n:\lambda \in \mathbb{R}\}.$$

$$G_n=F_0\oplus F_1\oplus \dots \oplus F_n\text{ and }{G}^n=G_n^{\perp },$$

$$l_n:G^n\times \mathcal{B}(G_n)\to [0,1].$$

$$l_n(\mathbf{y},U)={\int }_U{\rho }_n(\mathbf{y},x)dx.$$

$$\{F_n,G^n,l_n{\}}_{n=0}^{\infty }$$

$$Y=\sum _{j=1}^{\infty }{y}_j{e}_j\in E,$$

$${\mu }_{G_n}(B_{G_n}({\pi }_{G_n}Y,\delta ))>0.$$

$$\mu \left(\left\{x\in E\mid |x-x_0{|}_E\le ϵ\right\}\right)\ge \delta .$$

$${\mu }_{G^n}\left(\left\{y\in G^n:{\mu }_n(A_n(y))=0\text{mbox}or1\right\}\right)=1,$$

$${\mathcal{N}}_f:=\{x\in E:f(x)=0\}.$$

$$\mu ({\mathcal{N}}_f)=0\text{mbox}or1.$$

$$\mbox{Leb}_{G_{n}}\biggl{(}G_{n}\setminus\left\{x\in G_{n}:f(x)\neq 0\right\}\biggr{)}=0.$$

$$\mbox{Leb}_{G_{n}}\biggl{(}\left\{x\in G_{n}:f(x)=0\right\}\biggr{)}=0,$$

$$\displaystyle\left\{\begin{array}[]{rcl}du(t)+{\mathcal{L}}u(t)\,dt+B(u(t),u(t))&=&\dot{\Xi}(t),\quad t>0,\\ u(0)&=&u_{0}\in\mathscr{H}.\end{array}\right.$$

$$\hskip 56.9055pt\left\{\begin{array}[]{rcl}du(t)+\mathcal{L}u(t)\,dt+B(u(t),u(t))&=&v(t),\quad t>0,\\ u(0)&=&u_{0}\in\mathscr{H},\end{array}\right.$$

$${\mathcal{R}}_T:\mathcal{H}\times L^r(0,T;U)$$

$${\pi }_F{\mathcal{R}}_T(u_0,L^r(0,T;U))\supset F$$

$$\underset{x\in K_{ϵ}}{\sup }|\Phi (x)-{\pi }_F{\mathcal{R}}_T(u_0,x){|}_F\le ϵ,$$

$$\Phi (K_{ϵ})\supset B_F(R).$$

$$\mathbb{E}{1}_{\mathcal{O}}({\pi }_{F}u(T,u_0))={\int }_{\mathcal{O}}\rho (x)\text{mbox}{Leb}_F(dx).$$

$$f:\mathcal{H}\times X\ni (u_0,\xi )\mapsto {\pi }_F{\mathcal{R}}_T(u_0,\xi )\in F,$$

$$f(u_0,B_0)\supset B_F.$$

Full text

1. Introduction

We consider the Navier-Stokes equations (NSEs) subjected to the periodic boundary condition on the torus

[TABLE]

where $u$ and $\mathfrak{p}$ are unknown vector field and scalar periodic functions in the space variable representing the fluid velocity and the pressure, respectively. We assume that we are given an initial velocity $u_{0}$. The perturbation $\dot{\Xi}$ denotes, roughly speaking, the Radon-Nikodym derivative of a Lévy process $\Xi=L$ or a fractional Brownian motion $\Xi=B^{H}$. In the case when $\Xi$ is a Wiener noise the above system has been the subject of intensive mathematical studies since the pioneering work of Bensoussan and Temam. The analysis of the qualitative properties and long time behaviour of its solutions has generated several important results, see for instance [5, 8, 14, 16, 20], to cite a few results. Particularly, when

[TABLE]

where $(b_{j})_{j\in\mathbb{N}}$ is a sequence of non-negative numbers, $(\beta_{j})_{j\in\mathbb{N}}$ is a sequence of independent, identically distributed real-valued Brownian motions and $(e_{j})_{j\in\mathbb{N}}$ is an orthonormal basis of the space of square integrable, periodic and divergence free functions with mean zero, the authors in [9], [1] and [22] proved the existence of densities for the laws of finite dimensional functionals of its solutions. In these papers different methods are used to prove the existence of such densities, for instance in [9] a method based on Girsanov theorem is used and the Malliavin calculus is used in [22]. In [1] a method based on controllability of (1) in finite-dimensional projections and an abstract result on image of decomposable measure under analytic mappings is used. This method does not use the Gaussian structure of the noise as the methods in [9] and [22]. In this paper we are mainly interested in proving the existence of densities for the laws of finite-dimensional analytic functionals of the solution of (1) when the driving noise $\Xi$ is a Lévy noise or a fractional Brownian motion. For this purpose we extend the results in [1] to our framework. Although we closely follow the approach in [1] the extension of the result therein to our setting is not trivial. In fact, the proof in [1] relies very much on the natural decomposability of the driving noise law in a Hilbert space $\mathscr{H}$ which is not naturally satisfied by a Lévy process or a fractional Brownian. In fact, even if the Lévy noise (or fractional Brownian motion) $\Xi$ has a decomposition as in (2), which is one of the main assumptions in [1], it is not known whether there exists a Hilbert space $\mathscr{H}$ on which the law of $\Xi$ on $\mathscr{H}$ is decomposable. In order to overcome this difficulty we prove, by using wavelet analysis and the decomposability of measure on Banach space introduced in [11], that there exists a Banach space $\mathscr{H}$ with Schauder basis on which the law of $\Xi$ is decomposable. With this result at hand and using the solid controllability of (1) we can prove the existence of densities for the laws of finite-dimensional projection of the solutions of (1).

In the next section we will fix the notation and present some preliminary results. Section 3 is devoted to the statement and the proof of our main result which will be applied to the stochastic 2D Navier-Stokes equations in the torus. In Appendix A and Appendix B we present and prove several results related to the wavelet expansion of Lévy noise and fractional Brownian motion, respectively. In Appendix C we establish a zero one law result, which is crucial for the proof of the main result, for decomposable measures.

2. Notations, Hypotheses and preliminary results

For a separable Banach space $E$ we denote by ${{\mathcal{B}}}(E)$ its Borel $\sigma$–algebra. For a subspace $E_{0}$ of $E$ we denote by $E_{1}$ the subspace of $E$ such that $E=E_{0}\oplus E_{1}$, i.e., $E_{1}=E_{0}^{\perp}$. Furthermore, for $A\subset E$ and $y\in E_{1}$ we set

[TABLE]

Let $\mu$ be a probability measure on $(E,{{\mathcal{B}}}(E))$ and $E_{0}$ and $E_{1}$ as above. We define a probability measure $\mu_{E_{0}}$ on $(E_{0},\mathcal{B}(E_{0}))$ by

[TABLE]

For a subspace $\tilde{E}_{0}\subset E_{1}$ we set

[TABLE]

If $E_{0}$ is finite dimensional, then we denote by $\mbox{Leb}_{E_{0}}$ the measure defined by

[TABLE]

where $\iota$ is the isomorphism $\iota:E_{0}\to{\mathbb{R}}^{n}$, $n=\dim(E_{0})$.

We can now introduce the following definition.

Definition 2.1**.**

Let $\{F_{n}:n\in{\rm I\kern-1.99997ptN}\}$ be a family of mutually disjoint closed subspaces of $E$, i.e. $F_{j}\cap F_{k}=\{0\}$, $j\not=k$. We set $G_{n}:=F_{1}\oplus\cdots\oplus F_{n}$ and $G^{n}:=(F_{1}\oplus\cdots\oplus F_{n})^{\perp}$. If for any $n\in\mathbb{N}$ there exists a kernel

[TABLE]

such that

[TABLE]

where $A_{n}({{\bf y}})=A_{(F_{1}\oplus\cdots\oplus F_{n},G^{n})}({{\bf y}})$, then we say that the measure $\mu$ is decomposable with decomposition $\{F_{n},G^{n},l_{n}\}_{n=1}^{\infty}$.

Hereafter we fix a separable Banach space $E$ with Schauder basis $\{e_{n}:n\in{\rm I\kern-1.99997ptN}\}$ and we set

[TABLE]

We also set

[TABLE]

along which we consider a probability kernel

[TABLE]

The projection onto any nontrivial subspace $F\subset E$ is denoted by $\pi_{F}$. Having fixed these notations we now proceed to the statement of our standing assumptions.

Analysing Theorem 2.2 of [1], one can easily verify that following assumption is essential.

Assumption 2.1**.**

Let $\mu\in{{\mathcal{P}}}(E)$ be a decomposable measure with decomposition $\{F_{n},G^{n},l_{n}\}_{n=0}^{\infty}$. We assume that for any $n\in{\rm I\kern-1.99997ptN}$ there exists a positive function $\rho_{n}:G^{n}\times G_{n}\to{\mathbb{R}}^{+}_{0}$ such that $\mu_{{G^{n}}}$–a.s. we have for all $U\in{{\mathcal{B}}}(G_{n})$

[TABLE]

Assumption 2.1 is often difficult to verify. Hence we formulate the next assumption which is more stronger but easier to check than the above. In fact, we prove in Lemma C.1 that the following assumption, i.e. Assumption 2.2, implies Assumption 2.1.

Assumption 2.2**.**

Let $\mu\in{{\mathcal{P}}}(E)$ be a decomposable measure with decomposition

[TABLE]

such that $\mu_{G_{n}}$ is absolutely continuous with respect to the Lebesgue measure $\mbox{Leb}_{G_{n}}$.

Our third standing set of conditions is given in the following next lines.

Assumption 2.3**.**

Let $\mu\in{{\mathcal{P}}}(E)$ and let $\{F_{n},G^{n},l_{n}\}_{n=0}^{\infty}$ be a decomposition of $\mu$. There exists a point

[TABLE]

such that

1. (1)

for any $n\in{\rm I\kern-1.99997ptN}$ and $\delta>0$

[TABLE] 2. (2)

for all numbers $N\in{\rm I\kern-1.99997ptN}$ there exists a $R_{N}>0$ such that for all $x_{0}\in B_{G_{N}}(\pi_{G_{N}}Y,R)$, and all $\epsilon>0$, there exists a $\delta>0$ such that

[TABLE]

In order to clarify the role of the above assumption we shall introduce the following definition.

Definition 2.2**.**

We call a set $A\in{{\mathcal{B}}}^{\mu}(E)$ a finite zero one $\mu$–set if and only if for all $n\in{\rm I\kern-1.99997ptN}$

[TABLE]

where $A_{n}(y)=A_{(F_{1}\oplus\cdots\oplus F_{n},G^{n})}(y)$.

Let $F_{\infty}=\cup_{n\in{\rm I\kern-1.39998ptN}}\{F_{0}+F_{1}+\cdots+F_{n}\}$. Now, let us present the generalization of Theorem 4 in [11], respective [1, Theorem 1.6]], whose proof requires that the measure $\mu$ is decomposable and has a finite second moment (see [1, Property (P), page 402] for the precise statement).

Theorem 2.1**.**

Let $f:X\to{\mathbb{R}}$ be an analytic function and let $\mu\in{{\mathcal{P}}}(X)$ be a decomposable measure with density satisfying Assumption 2.2 and Assumption 2.3. Let ${{\mathcal{N}}}_{f}\subset E$ be defined by

[TABLE]

Then, we have

[TABLE]

Furthermore, if $f$ is not identical zero, then $\mu({{\mathcal{N}}}_{f})=0$.

Proof of Theorem 2.1:.

Let ${{\mathcal{N}}}_{f}:=\{x\in G:f(x)=0\}$. Since $f$ is analytic, for all $n\in{\rm I\kern-1.99997ptN}$ for any $y\in G^{n}$ the function $f_{y}(x):=f(y+x)$ is also analytic. Therefore, either $\mbox{Leb}_{G_{n}}({{\mathcal{N}}}_{f}^{n}(y))=0$ or $\mbox{Leb}_{G_{n}}(E\setminus{{\mathcal{N}}}_{f}^{n}(y))=0$, where ${{\mathcal{N}}}_{f}^{n}(y)=\{x\in G_{n}:x+y\in{{\mathcal{N}}}_{f}\}$. Thus, ${{\mathcal{N}}}_{f}$ is a finite zero–one $\mu$ set, and there exists a set $\tilde{{{\mathcal{N}}}}_{f}\in{{\mathcal{B}}}(E)$ such that $\tilde{{{\mathcal{N}}}}_{f}+F_{(\infty)}=\tilde{{{\mathcal{N}}}}_{f}$ and $\mu(\tilde{{{\mathcal{N}}}}_{f})=\mu({{\mathcal{N}}}_{f})$.

To prove the second part we assume that $f\not\equiv 0$ and we shall show that $\mu({{\mathcal{N}}}_{f}^{c})>0$. For this purpose let $n\in\mathbb{N}$ be fixed and set $f_{n}:=f_{\lvert_{G_{n}}}$ and $Y_{n}:=\pi_{G_{n}}Y$ where $Y$ is the point from Assumption 2.3. Observe that $f_{n}$ is analytic and thus

[TABLE]

We shall now distinguish two cases: $f_{n}(Y_{n})\neq 0$ and $f_{n}(Y_{n})=0$. For the first case, i.e., $f_{n}(Y_{n})\neq 0$ we observe that by the continuity of $f_{n}$ there exists a number $\delta>0$ such that $f(x)\neq 0$ for all $x\in B_{E}(Y_{n},\delta)$ from which along with item (2) of Assumption 2.3 we easily conclude that $\mu({{\mathcal{N}}}_{f}^{c})>0$. To treat the second case, i.e, $f_{n}(Y_{n})=0$, we first notice that, since $f_{n}$ is analytic, we have

[TABLE]

which implies that for any $\epsilon>0$ one can find $x_{0}\in G_{n}$ such that $\lvert x_{0}-Y_{n}\rvert_{E}\leq\epsilon$ and $f(x_{0})\neq 0$. Since $f$ is continuous we can find a number $\delta>0$ such that $f(x)\neq 0$ for all $x\in B_{E}(x_{0},\delta)$. Item (2) of Assumption 2.3 with $\epsilon=\tfrac{R}{2}$ yields that $\mu(B_{E}(x_{0},\delta)>0)$ from which it easily follows that $\mu({{\mathcal{N}}}_{f}^{c})>0$.

The above theorem will, as in [1, Theorem 2.2], be used to prove the existence of the density of law of the finite projection on finite dimensional space of the solution of a stochastic evolution equation driven by Lévy noise and fractional Brownian motion.

3. The main result

In this section we consider an abstract stochastic evolution equation in a separable Banach space $E$

[TABLE]

where the driving noise $\Xi$ is either a Lévy process or a fractional Brownian motion, $\mathcal{L}:D(\mathcal{L})\to\mathscr{H}$ and $B:\mathscr{H}\times\mathscr{H}\to\mathscr{H}$ is a densely defined bilinear operator taking values in $\mathscr{H}$. We assume that the above equation is uniquely solvable in $\mathscr{H}$ and we denote the solution starting from $u_{0}\in\mathscr{H}$ at time $t=0$ by $\{u(t,u_{0}):t\geq 0\}$.

In order to formulate the main result of this section we need to introduce few concepts from the control theory. For this aim, let $U\subset\mathscr{H}$ be a separable Banach space $r\geq 1$ be fixed number and let us consider the following control problem

[TABLE]

where $v\in L^{r}(0,T;U)$ is the control and $U$ is the control space (the trajectories of our noise will be basically belong to $L^{r}(0,T;U)$ ). For a fixed time $T>0$ we denote by

[TABLE]

the so called solution operator that takes each function $g\in L^{r}(0,T;U)$ and initial condition $u_{0}\in\mathscr{H}$ to the solution $u(T,u_{0})$ of the system (9).

Definition 3.1**.**

A system is controllable in time $T>0$ for a finite dimensional subspace $F\subset\mathscr{H}$ if and only if

[TABLE]

for any $u_{0}\in\mathscr{H}$.

Definition 3.2**.**

A system is solidly controllable in time $T>0$ for a finite dimensional subspace $F\subset\mathscr{H}$, if and only if for any $R>0$ and any $u_{0}\in\mathscr{H}$, there exists an $\epsilon>0$ and a compact set $K_{\epsilon}\subset L^{r}(0,T;U)$ such that for any function $\Phi:K_{\epsilon}\to F$ satisfying

[TABLE]

we have

[TABLE]

With this preliminary works the following general result can be shown.

Theorem 3.1**.**

Let $E$ be a separable Banach space with Schauder basis $\{e_{n}:n\in\mathbb{N}\}$. Let $F$ be a finite dimensional subspace of $\mathscr{H}$. We assume that the embedding $L^{r}(0,T;U)\hookrightarrow E$ is continuous, $\{e_{n}:n\in\mathbb{N}\}$ is also a Schauder basis in $L^{r}(0,T;U)$, and the law $\mu$ of the noise $\dot{\Xi}$ on $E$ is decomposable on $E$ with the decomposition $\{F_{n},G^{n},l_{n}\}_{n=0}^{\infty}$, where notation used in (3) and (4) is enforced, satisfying Assumptions 2.2 and 2.3. For a fixed number $T>0$ we also assume that

1. (A1)

the solution operator ${\mathcal{R}}_{T}$ defined in (10) which is generated by the system (9) is analytic, 2. (A2)

and for any finite dimensional space $F\subset\mathscr{H}$, the system (9) is solidly controllable in time $T$ for the finite dimensional space $F$.

Then, for any $u_{0}\in\mathscr{H}$ and for any finite dimensional subspace $F\subset\mathscr{H}$ there exists a density function $\rho:F\to{\mathbb{R}}^{+}_{0}$ such that

[TABLE]

Proof.

Let us fix a finite dimensional subspace $F$ of $\mathscr{H}$ and consider the operator

[TABLE]

where $X=L^{r}(0,T;U)$, $u$ solves equation (8) and ${\mathcal{R}}_{T}$ is defined in (10).

The proof of our theorem will follow from the applicability of [1, Theorem 2.2]. Thus we just need to check that all the assumptions of [1, Theorem 2.2] are all satisfied. For this aim it is sufficient to prove the two claims below.

1. Claim1

. There exists a finite dimensional subspace $G_{m}$ of $X$ such that for any $u_{0}\in\mathscr{H}$, there exists a ball $B_{0}\subset G_{m}$ and a ball $B_{F}\subset F$ such that

[TABLE]

To prove this claim we fix a large number $R>0$ such that $u_{0}\in B_{\mathscr{H}}(R)$. By the definition of solidly controllability, we know that there exists an $\epsilon>0$ and a compact set $K_{\epsilon}\subset\mathscr{H}$ such that, any function $\Phi:K_{\epsilon}\to\mathscr{H}$ satisfying

[TABLE]

satisfies

[TABLE]

Fix $u_{0}\in B_{\mathscr{H}}(R)$, $\epsilon>0$ and the corresponding compact set $K_{\epsilon}$. Since the operator

[TABLE]

is continuous, it is uniformly continuous on $K_{\epsilon}$, and, hence, there exists a $\delta_{0}>0$ such that

[TABLE]

Since the function system $\{e_{n}:n\in{\rm I\kern-1.99997ptN}\}$ is a Schauder basis of $X$, it follows that $\cup_{m\in{\rm I\kern-1.39998ptN}}F_{m}$ is a dense subset in $X$. In particular, since $K_{\epsilon}$ is compact, for any $\delta>0$, there exists a number $m$ such that

[TABLE]

Let $m\in{\rm I\kern-1.99997ptN}$ be sufficiently large such that

[TABLE]

Let us define

[TABLE]

by

[TABLE]

From the consideration above, it follows that

[TABLE]

Hence, by the solid controllability

[TABLE]

In particular, since $\pi_{G_{m}}K_{\epsilon}$ is a bounded set of $G_{m}$, there exists a number $R_{1}>0$ such that $\{y\in G_{m}:|y|\leq R_{1}\}\supset\pi_{G_{m}}K_{\epsilon}$. Setting $B_{F}:=\{y\in F:|y|_{F}\leq R\}$ and $B_{1}:=\{y\in G_{m}:|y|\leq R_{1}\}$ we have

[TABLE]

which proves Claim1. 2. Claim2

.  The measure $\mu$ on $E$ satisfies Assumption 2.1.

Claim 2 is easy to prove. Thanks to Lemma C.1 the measure satisfies Assumption 2.3, which is equivalent to Claim 2.

4. Application to the 2D stochastic Navier-Stokes

Throughout this section $\mathbb{T}$ denotes the 2D torus, $L^{p}(\mathbb{T})$ and $W^{m,p}$ will respectively denote the usual Lebesgue space of $p$-integrable functions and Sobolev spaces. The symbol $B^{s}_{p,p}([0,1]):=B^{s}_{p,p}([0,1];\mathbb{R})$ is the Besov spaces of all $\mathbb{R}$-valued functions defined on the interval $[0,1]$.

Let $\mathcal{V}$ be the set of periodic, divergence free and infinitely differentiable function with zero mean. In what follows, we denote by $\mathscr{H}$ and $\mathbf{V}$ the closures of $\mathcal{V}$ in $L^{2}(\mathbb{T})$ and $W^{1,2}(\mathbb{T})$, respectively. We endow the space $\mathscr{H}$ with the $L^{2}$-scalar product denoted by $(\cdot,\cdot)$ and the usual $L^{2}$-norm denoted by $\lvert\cdot\rvert$. The space $\mathbf{V}$ is equipped with the gradient norm $\lvert\nabla\cdot\rvert$. We also set

[TABLE]

where $\Pi$ is the orthogonal projection from $L^{2}(\mathbb{T})$ onto $\mathscr{H}$. It is well-known that the Stokes operator $\mathcal{L}$ is positive self-adjoint with compact resolvent and its eigenfunctions $\{e_{1},e_{2},\ldots\}$, with eigenvalues $0<\lambda_{1}\leq\lambda_{2}\leq\ldots$, form an orthonormal basis of $\mathscr{H}$. It is also well-known that $\mathbf{V}=D(\mathcal{L}^{\frac{1}{2}})$, see [15, Appendix A.1 of Chapter II]. Furthermore, we see from [27, Chapter II, Section 1.2] and [15, Appendix A.3 of Chapter II] that one can define a continuous bilinear map $B$ from $\mathbf{V}\times\mathbf{V}$ with values in $\mathbf{V}^{\ast}$ such that

[TABLE]

With all these notations the Navier-Stokes equations (1) can be written in the abstract form

[TABLE]

where for the sake of simplicity we assume that $\Pi\dot{\Xi}=\dot{\Xi}$. The positive number $\kappa>0$ denotes the viscosity. Before characterizing the noise entering our system, we introduce the trigonometric basis in $\mathscr{H}$ by elements in $\mathbb{Z}$. Namely, we write $j-(j_{1},j_{2})\subset\mathbb{Z}^{2}$ and set

[TABLE]

where $j^{\perp}=(-j_{2},j_{1})$. The family $\mathcal{E}=\{e^{j}_{0},e_{j},i=1,2,j\in\mathbb{Z}\setminus\{0\}\}$ is a complete set of eigenfunctions for the Stokes operator which forms an orthonormal basis in $\mathscr{H}$.

For any symmetric set $\mathcal{K}\subset\mathbb{Z}^{2}$ containing $(0,0)$ we write $\mathcal{K}_{0}=\mathcal{K}$ and define $\mathcal{K}^{i}$ with $i\geq 1$ as the union for $\mathcal{K}^{i-1}$ and the family of vectors $l\in\mathbb{Z}^{2}$ for which there are $m,n\in\mathcal{K}^{i-1}$ such that $l=m+n$, $|m|\not=|n|$, and $|m\wedge n\not=0$, where $m\wedge n=m_{1}n_{2}-m_{2}n_{1}$.

Definition 4.1**.**

A symmetric subset $\mathcal{K}\subset\mathbb{Z}^{2}$ containing $(0,0)$ is saturating, if and only if $\cup_{i\in{\rm I\kern-1.39998ptN}}\mathcal{K}^{i-1}=\mathbb{Z}^{2}$.

Throughout we set $d={\dim{\mathcal{K}}}$ and denote by $\mathscr{H}_{d}$ the finite dimensional subspace of $\mathscr{H}$ spanned by the eigenvectors $\{e_{j};j\in\mathcal{K}\}$. The driving noise is either

[TABLE]

where $\{l_{j}:j\in\mathcal{K}\}$ is a family of identical distributed and mutual independent Lévy processes with Lévy measure $\nu_{j}$ over a probability space $(\Omega,{\mathcal{F}},{\mathbb{P}})$, or

[TABLE]

where $\{\beta^{H}_{j}:j\in\mathcal{K}\}$ is a family of identical distributed and mutual independent fractal Brownian motions with Hurst parameter $H\in(\tfrac{1}{2},1)$ over a probability space $(\Omega,{\mathcal{F}},{\mathbb{P}})$. The existence of a unique solution $u=\{u(t):t\geq 0\}$ to (16) follows from the results in [6] for example for the case of pure jump Lévy noise, and from [21] for case of fractional Brownian motion perturbation.

We can now state the main results of this section. We start with the following theorem which treats the case of Navier-Stokes equations driven by Lévy noise.

Theorem 4.1**.**

Let $\mathcal{K}$ be a saturating set and assume that the noise $\Xi$ entering the system (8) is defined by (17). We also assume that the Lévy measures $\nu_{j}$, $j=1,\ldots,d$, are symmetric and equivalent to the Lebesgue measure on ${\mathbb{R}}\setminus\{0\}$ and satisfies

[TABLE]

for some $p\in(1,2)$. In addition, we assume that there exists a number $\alpha\in(0,2]$ such that

[TABLE]

for some slow varying function $l$. Let $u=\{u(t,u_{0}):t\geq 0,u_{0}\in\mathscr{H}\}$ be the unique solution of system (8). Then for any finite dimension subspace $F\subset\mathscr{H}$, for all initial conditions $u_{0}\in\mathscr{H}$, there exists a density function $\rho_{u_{0}}:F\to{\mathbb{R}}^{+}_{0}$ such that

[TABLE]

In addition, for any sequence $\{u_{n}:n\in{\rm I\kern-1.99997ptN}\}$ with $u_{n}\to u_{0}\in\mathscr{H}$ as $n\to\infty$, we have

[TABLE]

Proof.

For simplicity, let us assume $T=1$. As in the previous section we consider map

[TABLE]

which is the solution operator that takes each function $g\in L^{2}(0,T;\mathscr{H}_{d})$ and initial condition $u_{0}\in\mathscr{H}$ to the solution $u(T,u_{0})$ of the control system (9) associated to the Navier-Stokes equations. It is proved in [1, Proposition A.2], see also [19], that the operator ${\mathcal{R}}_{T}$ is analytic. It is also known from [1, Proposition A.5 ], see also [2], that the system (9) for the Navier-Stokes is solidly controllable in time $T$ for any finite dimensional space $F\subset\mathscr{H}$. Hereafter we respectively identify $\mathscr{H}_{d}$ and $F$ to $\mathbb{R}^{d}$ and $\mathbb{R}^{\dim F}$. Let $p\in(1,2)$ such that (19) is satisfied. Let $p^{\prime}$ be the conjugate exponent to $p$ and $s<\tfrac{1}{p}-1$. For each $j\in\mathcal{K}$ let $\xi_{j}$ be the map defined by

[TABLE]

and $\mu_{j}$ be the cylindrical measure on $B_{p,p}^{s}([0,1],{\mathbb{R}})$ defined by

[TABLE]

where $n\in{\rm I\kern-1.99997ptN}$, $\phi_{1},\ldots,\phi_{n}\in{{\mathcal{S}}}({\mathbb{R}})$. In Proposition A.2 we show that the cylindrical measure is actually a Radon probability measure on $B_{p,p}^{s}([0,1])$.

From the results of Section A we infer that the probability measure $\mu_{j}$ on $B_{p,p}^{s}([0,1],{\mathbb{R}})$ is decomposable with decomposition $\{F_{n},G^{n},l_{n}\}_{n=0}^{\infty}$, where $F_{n}$ and $l_{n}$ are respectively defined by $F_{0}=V_{0}$, $F_{n}=W_{n}$, $n\geq 2$, where $V_{0}$ and $W_{n}$ are defined in (21) and the existence of $l_{n}$ is given by Lemma A.3. With $F_{n}$ at hand the space $G^{n}$ is defined as in Definition 2.1. Moreover, we infer from Lemma A.7 that for each $j$ the probability measure $\mu_{j}$ satisfies Assumptions 2.2 and 2.3. With these observation in mind, it is not difficult to check that the product measure $\mu=\otimes_{j\in\mathcal{K}}\mu_{j}$ satisfies Assumptions 2.2 and 2.3 on the Banach space $E:=B_{p,p}^{s}([0,1],{\mathbb{R}}^{d})$ where $d=\dim(\mathcal{K})$. Now, the proof of the theorem easily follows from an application of Theorem 3.1.

We now proceed to the statement and the proof of the above theorem when the noise entering the system is a fractional Brownian motion given by (18).

Theorem 4.2**.**

Let $\mathcal{K}$ be a saturating set and assume that the noise $\Xi$ is a fractional Brownian motion defined by (18) with Hurst parameter $H\in(\tfrac{1}{2},1)$. Let $u=\{u(t,u_{0}):t\geq 0,u_{0}\in H\}$ be the unique solution of system (8) with initial condition $u_{0}$. Then, for any finite dimensional space $F\subset\mathscr{H}$ and initial condition $u_{0}\in\mathscr{H}$, there exists a density function $\rho_{u_{0}}:F\to{\mathbb{R}}^{+}_{0}$ such that

[TABLE]

In addition, for any sequence $\{u_{n}:n\in{\rm I\kern-1.99997ptN}\}$ with $u_{n}\to u_{0}\in\mathscr{H}$ as $n\to\infty$, we have

[TABLE]

Proof.

Let ${\mathcal{R}}_{T}:\mathscr{H}\times L^{2}(0,T;\mathscr{H}_{d})\to\mathscr{H}$ be the solution operator defined by (20) in the proof of Theorem 4.1. It satisfies the properties enumerated in the proof of Theorem 4.1. Hereafter we respectively identify $\mathscr{H}_{d}$ and $F$ to $\mathbb{R}^{d}$ and $\mathbb{R}^{\dim F}$. Let $s\in(-\tfrac{1}{2},\,H-1)$. For each $j\in\mathcal{K}$ let $\xi_{j}$ be the map defined by

[TABLE]

and $\mu_{j}$ be the cylindrical measure on $B_{2,2}^{s}([0,1],{\mathbb{R}})$ defined by

[TABLE]

where $n\in{\rm I\kern-1.99997ptN}$, $\phi_{1},\ldots,\phi_{n}\in{{\mathcal{S}}}({\mathbb{R}})$. From the results of Section B we infer that the cylindrical measure $\mu_{j}$ on $B_{2,2}^{s}([0,1],{\mathbb{R}})$ is actually a probability measure and is decomposable with decomposition $\{F_{n},G^{n},l_{n}\}_{n=0}^{\infty}$, where $F_{n}$ and $l_{n}$ are respectively defined by $F_{0}=V_{0}$, $F_{n}=W_{n}$, $n\geq 2$, where $V_{0}$ and $W_{n}$ are defined in (33). With $F_{n}$ in mind we define $G^{n}$ as in Definition 2.1. We also infer from Lemma A.7 that for each $j$ the probability measure $\mu_{j}$ satisfies Assumptions 2.2 and 2.3. We now easily complete the proof by using a similar argument as in the proof of Theorem 4.1.

Appendix A The Lévy Noise and its Wavelet Expansion

In this section we assume that we are given a real-valued Lévy process $\ell$ with $\sigma$-additive Lévy measure $\nu$ on $\mathbb{R}\backslash\{0\}$ satisfying (19), i.e.

[TABLE]

for some $p\in(1,2)$. Our aim is to investigate the expansion of the process $\ell$ in terms of Debauchies wavelets of order $k$. To keep this section and the article short we refer to the reader for the technical jargon about wavelets to [7] or [28].

We start introducing the Daubechies wavelets, see for e.g. [7]. For such aim we fix $u>0$ and consider the Debauchies wavelets $\psi$ having continuous bounded derivatives up to order $k$. It is known, see for e.g [7], that to $\psi$ we can associate scaling function denoted by $\phi$. With these in mind, the system of wavelets is given by

[TABLE]

where $J^{\psi}_{j}=\{k\in{\rm I\kern-1.99997ptN}:\mbox{supp}(\psi_{j,k})\cap I\not=\emptyset\}$, $J^{\phi}_{j}=\{k\in{\rm I\kern-1.99997ptN}:\mbox{supp}(\phi_{j,k})\cap I\not=\emptyset\}$. The corresponding multiresolution analysis is defined by

[TABLE]

For detail on the properties of the wavelet basis we refer to [28, Theorem 1.58] or to [17]. Note that for $s\in{\mathbb{R}}$ the Daubechies wavelets of order $k$, with $k>\max(s,(1-\frac{1}{p})_{+}-s)$, form an unconditional basis of $B^{s}_{p,p}([0,1])$. In particular, for each element $f\in B^{s}_{p,p}([0,1])$ there exists a unique sequence

[TABLE]

such that $f$ can be written as

[TABLE]

Note that since we are considering the process on the time interval $[0,1]$, we only need to sum over $J_{j}^{\psi}$. We also note that $|J_{j}^{\psi}|\sim 2^{j}$.

In the next paragraph, we will construct the probability measure induced by a Lévy process which will be represented as an integral with respect to a Poisson random measure. This representation is motivated in one hand by the fact that the use of Poisson random measure simplifies many calculation. In other hand the Poisson random measure framework seems more general. We refer to [3], [24, Chapters 6-8] and [26, Chapter 4] for a precise connection between Poisson random measures and Lévy processes and stochastic integration with respect to them.

Over a probability space $\mathfrak{A}=(\Omega,{\mathcal{F}},{\mathbb{P}})$, we consider a time homogenous Poisson random measure $\eta$ on ${\mathbb{R}}$ with symmetric intensity measure $\nu$ as above.

Proposition A.1**.**

The Poisson random measure $\eta$ over a probability space $(\Omega,{\mathcal{F}},{\mathbb{P}})$ induces a Radon probability measure $\mu$ on $B_{p,p}^{s}([0,1])$.

Proof.

We will start the proof with removing jumps of size bigger than $\epsilon\in(0,1]$ and let $\epsilon$ converges to 0. For this purpose we take an arbitrary constant $\epsilon\in(0,1]$ and define a Poisson random measure $\eta_{\epsilon}$ by

[TABLE]

The family $\{\eta_{\epsilon}:\epsilon\in(0,1]\}$ induces a family of cylindrical measures on $C_{b}([0,1])^{\prime}$. Here, it is important that the to Poisson random measure $\eta_{\epsilon}$ corresponding the Lévy process can be written as a sum over finitely many jumps at certain, possibly random, jump times. To be more precise, let $\nu_{\epsilon}$ be defined by $\nu_{\epsilon}(A)=\nu(A\cap({\mathbb{R}}\setminus[-\epsilon,\epsilon]))$, $\rho_{\epsilon}=\nu({\mathbb{R}}\setminus[-\epsilon,\epsilon])$, let $N_{\epsilon}$ be a Poisson distributed random variable with parameter $\rho_{\epsilon}$, $\{\tau^{\epsilon}_{n}:n=1,\ldots,N\}$ be a family of independent uniform distributed random variables on $[0,1]$, and $\{Y_{n}:n=1,\ldots,N\}$ be a family of independent, $\nu_{\epsilon}/\rho_{\epsilon}$ distributed random variables. Denoting $\delta_{x}$ the Dirac distribution concentrated at $x$, the Poisson random measure $\eta_{\epsilon}$ can be written as

[TABLE]

and for any $f\in C_{b}([0,1])$ the mapping

[TABLE]

is well defined.

Let us define the random variables

[TABLE]

[TABLE]

Since the mother wavelet $\psi$ and the scaling function $\phi$ are continuous, the families $\{\zeta^{\epsilon}_{j,k}:j\in{\rm I\kern-1.99997ptN},\,k\in J_{j}\}\cup\{a_{0}^{\epsilon}\}$ of random variables over $\mathfrak{A}$ are well defined. In addition, by the definition of $\zeta^{\epsilon}$ and $a_{0}$ and the fact that the multiresolution analysis is a Schauder basis in $B^{s}_{p,p}([0,1])$, and $\delta\in B^{s}_{p,p}([0,1])$ (see [25, Remark 3, p. 34]), we infer that $\xi_{\epsilon}$ admits a wavelet series representation as in (22).

Note that for any $C\in{{\mathcal{B}}}({\mathbb{R}})$,

[TABLE]

Later on we will need the following proposition which will be proved at the end of the current proof.

Proposition A.2**.**

Let $\nu$ be a Lévy measure satisfying (19) for some $p\in[1,2)$ and $\epsilon\in(0,1]$. Let

[TABLE]

Then,

1. (1)

for any $s<\frac{1}{p}-1$, there exists a $C>0$ such that

[TABLE] 2. (2)

For any $s<\frac{1}{p}-1$ and $\epsilon_{1},\epsilon_{2}\in(0,1]$ we have

[TABLE]

By the choice of $s$ and $p$, we have $B^{-s}_{p^{\prime},p^{\prime}}([0,1])\hookrightarrow C_{b}([0,1])$ and $\eta$ is a finite measure. Secondly, the mappings $\xi_{\epsilon}$ induces a the family of cylindrical measures $\mu_{\epsilon}$ on $B^{s}_{p,p}([0,1])$ defined by

[TABLE]

$\phi_{1},\cdots,\phi_{n}\in(B_{p,p}^{s}([0,1]))^{\prime}=B_{p^{\prime},p^{\prime}}^{-s}([0,1])$, and $C\in{{\mathcal{B}}}({\mathbb{R}}^{n})$.

We will now show that the family of cylindrical measures $\{\mu_{\epsilon}:\epsilon\in(0,1]\}$ has a limit. In fact, the family of probability measures $\mu_{\epsilon}$ is tight on $B_{p,p}^{s}([0,1])$. To show this claim we fix a constant $s_{0}\in(s,\frac{1}{p}-1)$. We firstly note that the embedding $B_{p,p}^{s_{0}}([0,1])\hookrightarrow B_{p,p}^{s}([0,1])$ is compact. Secondly, the Chebyscheff inequality and Proposition A.2 give that for any $\delta>0$ we can find a compact $K_{\delta}:=\{x\in B_{p,p}^{s}([0,1]):|x|_{B_{p,p}^{s_{0}}}\leq\delta^{-1/p}\}$ such that

[TABLE]

It follows that the family of probability measures $\{\mu_{\epsilon}:\epsilon\in(0,1]\}$ is tight on $B_{p,p}^{s}([0,1])$. It even follows from Proposition A.2 that the sequence $\{\mu_{\epsilon_{n}}:\epsilon=\frac{1}{n}\}$ forms a Cauchy sequence and the limit $\mu$ is unique. Therefore, there exists a unique cylindrical measure $\mu$ on $B^{s}_{p,p}([0,1])$. Since there exists a constant $C>0$ such that for all $\epsilon>0$ $\mathbb{E}\left[|\xi_{\epsilon}|^{p}_{B_{p,p}^{s}}\right]\leq C$, it follows from the Lebesgue Dominated Convergence Theorem that $\mathbb{E}\left[|\xi|^{p}_{B_{p,p}^{s}}\right]\leq C$. Hence, $\mu$ is also a Radon probability measure on $B^{s}_{p,p}([0,1])$.

Now we shall consider the general case in which $\nu$ is assumed to satisfy (19) for some $p\in(1,2)$. For this purpose we consider the Poisson random measures $\eta_{1}$ and $\eta_{2}$ defined by

[TABLE]

and

[TABLE]

respectively. Since $A\cap[-1,1]\cap A\cap{\mathbb{R}}\setminus[-1,1]=\emptyset$, the Poisson random measures $\eta_{1}$ and $\eta_{2}$ are independent. Hence, the two families of coefficients in the wavelet expansion $\eta_{1}$ and $\eta_{2}$ are independent too. In addition from the first part of the proof $\eta_{1}$ induces a Radon probability measure on $B_{p,p}^{s}([0,1])$. Since the process

[TABLE]

can be written as a finite sum over jumps happen at certain, possibly random, times within the interval $[0,1]$, $\dot{L}_{t}^{2}$ consist of a sum over finitely many Dirac distributions. Since any Dirac distributions belong to $B_{p,p}^{s}([0,1])$, $\dot{L}_{t}^{2}$ is an element of $B_{p,p}^{s}([0,1])$ and induces a probability measure on $B_{p,p}^{s}([0,1])$. Hence, $\eta$ itself induces a Radon probability measure on $B_{p,p}^{s}([0,1])$.

Proof of Proposition A.2:.

We recall that $\int|z|^{p}\nu(dz)<\infty$ for some $p\in(1,2)$. By the definition of the norm we get

[TABLE]

Since

[TABLE]

we infer that there exists a constant $C>0$ such that

[TABLE]

which is finite for $s<\frac{1}{p}-1$.

Let us denote the Radon probability measure induced by $\eta$ on $B_{p,p}^{s}([0,1])$ by $\mu$ and let us define the mapping

[TABLE]

This mapping is well defined thanks to the above calculation.

We are now interested in the properties of the decomposition of $\mu$ by the multiresolution analysis. In particular, we will show that for any $n\in{\rm I\kern-1.99997ptN}$, the probability measure $\mu_{G_{n}}$ is equivalent to the Lebesgue measure.

We firstly note that since $V_{n}=W_{n}\otimes W_{n-1}\otimes\cdots\otimes W_{1}\otimes V_{0}$, given the coefficients $\{\zeta_{j,k}:j=1,\ldots,n,\,\,k\in J^{\psi}_{j}\}\cup\{a_{0}\}$, one knows the coefficient of $\phi_{n+1,k}$. For $k\in J_{n+1}^{\phi}$ let us denote $\gamma_{n,k}$ the coefficients of $\phi_{n+1,k}$. In particular, we have

[TABLE]

which implies that

[TABLE]

Let us now denote by $\bf z^{n}$ and $\bf g^{n}$ the random vectors $(\zeta_{n,0},\zeta_{n,1},\ldots,\zeta_{n,|J^{\phi}_{n}|})$ and $(\gamma_{n,1},\gamma_{n,2},\ldots,\gamma_{n,|J^{\psi}_{n}|})$, respectively. Finally, for a function $f:[0,1]\to{\mathbb{R}}$ we write

[TABLE]

Lemma A.1**.**

Let $f:[0,1]\to{\mathbb{R}}$ be a mapping such that there exists constants $\delta>0$ and $t_{1},t_{2}\in[0,1]$, $t_{1}<t_{2}$ such that $|f(t)|\geq\delta$ for all $t\in[t_{1},t_{2}]$. Then

1. (1)

$\operatorname{supp}(\xi(f))={\mathbb{R}}$; 2. (2)

the law of $\xi(f)$ is absolutely continuous with respect to the Lebesgue measure.

Proof.

Let us define the following Lévy measure

[TABLE]

Then $\xi(f1_{[t_{1},t_{2}]})$ is an infinite divisible random variable, and item (i) follows from [26, Corollary 24.4]. Item (ii) follows from [26, Theorem 27.7].

Lemma A.2**.**

For any $n\geq 1$, the measure

[TABLE]

is equivalent to the Lebesgue measure on ${\mathbb{R}}^{|J^{\phi}_{n+1}|}$.

Proof.

This follows from the fact that for all $k=\min(J^{\phi}_{n+1}),\ldots,\max(J^{\phi}_{n+1})-1$, the functions $\phi_{n+1,k}$ and $\phi_{n+1,k+1}$ have disjoint supports. Let us write $\phi_{n+1,k+1}=f_{1}+f_{2}$ with $\operatorname{supp}(f_{1})\cap\operatorname{supp}(\phi_{n+1,k})=\emptyset$, $\operatorname{supp}(f_{1})$ is an interval $[a,b]$, $\{s:f_{2}(s)>0\}\cap[a,b]=\emptyset$, and $f_{1}$ is bounded away from zero. Then $\xi(f_{1})$ and $\xi(f_{2})$ are independent, and so are $\xi(f_{1})$ and $\xi(\phi_{n+1,k})$ . In addition, by Lemma A.1 the law of $\xi(f_{1})$ is equivalent to the Lebesgue measure. Hence, from [26, Lemma 27.1-(iii)] it follows that the law of the sum of the random variables $\xi(f_{1})$ and $\xi(f_{2}+\phi_{n+1,k+1})$ is also equivalent to the Lebesgue measure. Now, one easily prove the assertion by an induction starting at $k=\min(J^{\phi}_{n+1})$.

Lemma A.3**.**

For each $U\in{{\mathcal{B}}}(G_{n})$ and ${{{\bf y}}}\in{\mathbb{R}}^{|J^{\psi}_{n}|}$, the conditioned measure

[TABLE]

is equivalent to the Lebesgue measure.

Proof.

Given the scaling function $\phi$ there exists coefficients $\{p_{j}:j=1,\ldots,u\}$, where $u$ is the order of the Daubechies wavelet, such that

[TABLE]

In addition, we have the following representation

[TABLE]

Because of the orthogonality of the wavelet basis we additionally have that

[TABLE]

Let us now consider the mapping

[TABLE]

where $f_{n+1,k}=\phi_{n+1,k}(f)$. It is not difficult to show that $\mathcal{I}$ is an isomorphism from $V_{n+1}$ onto ${\mathbb{R}}^{|J^{\phi}_{n+1}|}$. We note that since $V_{n+1}=V_{n}\otimes W_{n}$, it follows from (26) that there exists a linear mapping $T:V_{n+1}\to V_{n}$ which induces a mapping

[TABLE]

We can also define a mapping ${{\mathcal{S}}}:V_{n+1}\to W_{n}$ by ${{\mathcal{S}}}{{{\bf x}}}:=\pi_{W_{n}}(I-{{\mathcal{T}}}){\bf x}$. As above we can also find a linear mapping $S:V_{n+1}\to W_{n}$ inducing a mapping

[TABLE]

Since $V_{n+1}=V_{n}\otimes W_{n}$, we have ${{\mathcal{I}}}^{-1}\mbox{ker}(\mathcal{S})=V_{n}$ and ${{\mathcal{I}}}^{-1}\mbox{ker}(\mathcal{T})=W_{n}$. Hence, from the Bayes formula we infer that

[TABLE]

for any ${{{\bf x}}}\in{\mathbb{R}}^{|J^{\psi}_{n}|}$ and ${{{\bf y}}}\in{\mathbb{R}}^{|J^{\phi}_{n}|}$. By Lemma A.2 ${\mathbb{P}}\left({{\mathcal{I}}}^{-1}\mathcal{S}^{-1}{{{\bf y}}}\right)>0$ and ${\mathbb{P}}\left(\pi_{W_{n}}{{\mathcal{I}}}^{-1}\mathcal{S}^{-1}{{{\bf x}}}+\pi_{V_{n}}{{\mathcal{I}}}^{-1}\mathcal{S}^{-1}{{{\bf y}}}\right)>0$. In particular, there exists a density

[TABLE]

such that

[TABLE]

and $h_{n}({{{\bf x}}},{{{\bf y}}})>0$ for all ${{{\bf x}}}\in{\mathbb{R}}^{|J^{\psi}_{n}|}$ and ${{{\bf y}}}\in{\mathbb{R}}^{|J^{\phi}_{n}|}$.

In order to verify Assumption 2.3 for a point $Y$ we will show in the following Lemma that for all $n\in{\rm I\kern-1.99997ptN}$, $\pi_{G_{n}}0$ belongs to the support of the measure $\mu$. If this holds, we can set $Y=0$.

Lemma A.4**.**

Let $\alpha\in(0,2)$, $1\leq p<\alpha$ and $s<\frac{1}{p}-1$. Let $\nu$ be a $\sigma$–finite symmetric measure on ${\mathbb{R}}\setminus\{0\}$ such that there exists a number $\alpha\in(0,2]$ such that

[TABLE]

for some slow varying function $l$. Let $\eta$ be the to $\nu$ corresponding Poisson random measure over the probability space $(\Omega,{\mathcal{F}},{\mathbb{P}})$. Let $\mu$ be the from $\xi$ defined in (23) on $B_{p,p}^{s}({\mathbb{R}})$ induced probability measure. Then for any $\epsilon>0$,

[TABLE]

Proof.

Let $L$ be given by

[TABLE]

From [4], Example 2.2 we know that

[TABLE]

hence, for any $\tilde{\epsilon}>0$ there exists a $\delta>0$ such that

[TABLE]

Let $\tilde{\epsilon}>0$ be a constant to be chosen later and let us set

[TABLE]

Then,

[TABLE]

Note that on $\Omega_{\tilde{\epsilon}}$ the jump size of the process is less than $2\tilde{\epsilon}$. Hence

[TABLE]

and

[TABLE]

From these calculations we infer that

[TABLE]

Now, choosing $\tilde{\epsilon}$ such that

[TABLE]

we infer that

[TABLE]

from which the assertion follows.

For any $\mathcal{D}\in{{\mathcal{B}}}(B_{p,p}^{s}([0,1])$ we define the conditional probability $\mu(\,\cdot\,\mid\mathcal{D})$ by

[TABLE]

Lemma A.5**.**

Let $\alpha\in(0,2)$, $1\leq p<\alpha$ and $s<\frac{1}{p}-1$. Let $\nu$ be a $\sigma$–finite symmetric measure on ${\mathbb{R}}\setminus\{0\}$ such that there exists a number $\alpha\in(0,2]$ such that

[TABLE]

for some slow varying function $l$.

Let $\eta$ also be the Poisson random measure, over the probability space $(\Omega,{\mathcal{F}},{\mathbb{P}})$, associated to the Lévy measure $\nu$. Let $\mu$ be the probability measure on $B_{p,p}^{s}([0,1])$ induced by the mapping $\xi$ defined in (23). Then, for any $R>0$, $x\in B_{E}(R,0)$ and $\epsilon>0$ there exist $n\in{\rm I\kern-1.99997ptN}$ and some $\delta>0$ such that

[TABLE]

Proof.

From Lemma A.4 we infer that there exists a constant $\delta>0$ such that for

[TABLE]

where $\mathcal{D}_{\Omega}:=\left\{\sup_{0\leq t\leq 1}|L(t)|\leq 1\right\}$ and $L=\{L(t):t\in[0,1]\}$ is defined in (29), Observe that the set $\mathcal{D}:=\xi({\mathcal{D}}_{\Omega})$ satisfies $\mu(\mathcal{D})\geq\delta$. Thus,

[TABLE]

Now, from (30) we infer that

[TABLE]

Therefore,

[TABLE]

For any $\kappa<1$ there exists a number $n\in{\rm I\kern-1.99997ptN}$ sufficiently large, such that

[TABLE]

which gives the assertion.

Lemma A.6**.**

Let $\alpha\in(0,2)$, $1\leq p<\alpha$ and $s<\frac{1}{p}-1$. Let $\nu$ be a $\sigma$–finite symmetric measure on ${\mathbb{R}}\setminus\{0\}$ such that there exists a number $\alpha\in(0,2]$ such that

[TABLE]

for some slow varying function $l$.

Then, for all $N\in{\rm I\kern-1.99997ptN}$, $x_{0}\in G_{N}$, and all $\epsilon>0$, there exists a $\delta>0$ such that

[TABLE]

Proof.

Let $\epsilon>0$ be a fixed constant and $s<s_{0}<\frac{1}{p}-1$. From Lemma A.5 we deduce that there exist $n_{0}\in{\rm I\kern-1.99997ptN}$ and $\delta_{2}>0$ such that

[TABLE]

Then

[TABLE]

We now set $A_{n_{0}}=\left\{x\in B^{s}_{p,p}([0,1])\mid|x_{0}-\pi_{G_{n_{0}}}x|_{B^{s}_{p,p}}\leq\frac{\epsilon}{4}\right\}$ and observe that for $\gamma=\delta_{2}/2>0$ there exists a closed set $C_{\gamma}\subset G^{n_{0}}$ such that $\mu^{n_{0}}(G^{n_{0}}\setminus C_{\gamma})\leq\gamma$ and the function

[TABLE]

is continuous. Furthermore, since for all ${{\bf y}}\in G^{n_{0}}$ $\mu$ a.s. $l_{n}({{\bf y}},\cdot)$ is equivalent to the Lebesgue measure and $\mbox{Leb}_{G_{n_{0}}}({A_{n_{0}}})>0$, we have $l_{n_{0}}({{\bf y}},A_{n_{0}})>0$. Since the embedding $B^{s_{0}}_{p,p}([0,1])\hookrightarrow B^{s}_{p,p}([0,1])$ is compact,

[TABLE]

is a compact subset of $G^{n_{0}}$ and there exists a $\delta_{3}>0$ such that for all ${{\bf y}}\in C_{n_{0}}\cap C_{\gamma}$, $l_{n_{0}}({{\bf y}},A_{n_{0}})\geq\delta_{3}$. From the above consideration we now infer that

[TABLE]

The above discussion is summarized in the following lemma.

Lemma A.7**.**

Let $\eta$ be a time homogeneous Poisson random measure on ${\mathbb{R}}$ over a probability space $(\Omega,{\mathcal{F}},{\mathbb{P}})$. We assume that the Lévy measure $\nu$ associated to $\eta$ is symmetric, $\sigma$-additive, absolutely continuous with respect to the Lebesgue measure on ${\mathbb{R}}\setminus\{0\}$. In addition, we assume, that there exists some $p\in(1,2)$ with

[TABLE]

and there exists a number $\alpha\in(0,2]$ such that

[TABLE]

for some slow varying function $l$.

*Let $\{\phi_{j,k}:j\in{\rm I\kern-1.99997ptN}:k=1,\ldots,2^{j}\}$ be the wavelet basis in $B_{p,p}^{s}([0,1])$ described in Section A. Then, the measure $\mu$ induced by the map $\xi$ defined by (23) on $B_{p,p}^{s}([0,1])$ is decomposable with decomposition $\{F_{n},G^{n},l_{n}\}_{n=0}^{\infty}$ satisfying Assumption 2.2 and Assumption 2.3. Here the spaces $F_{n}$ are defined by $F_{0}=V_{0}$, $F_{n}=W_{n}$, $n\geq 2$, $V_{0}$ and $W_{n}$ are defined in (21), and $l_{n}$ is defined in (25). *

Proof.

The decomposability follows from the fact the wavelet basis described in section A is a Schauder basis of $B_{p,p}^{s}([0,1])$. Assumption 2.3-(1) follows from choosing $Y=(0,0,\ldots)$ and from the fact that ${\mathbb{P}}\left(\pi_{G_{j+1}}x\in\cdot\mid\pi_{G_{j}}x=y\right)$ is equivalent to the Lebesgue measure and that for any $y\in{\mathbb{R}}$ we have (see Lemma A.3)

[TABLE]

Using an induction argument one can easily show that for any open set in $G_{n}$ $\mu_{G_{n}}({\mathcal{O}})>0$ from which it follows that $\mu_{G_{n}}$ is absolutely continuous with respect to $\mbox{Leb}_{G_{n}}$. Finally, Assumption 2.3-(2) follows from Lemma A.6.

Appendix B The Fractional Brownian Noise and its Wavelet Expansion

Let $B^{H}=\{B^{H}(t):t\geq 0\}$ a the fractional Brownian motion with Hurst parameter $H\in(\frac{1}{2},1)$. Let us fix $s\in(-\frac{1}{2},H-1)$ and consider the mapping

[TABLE]

For all $n\in{\rm I\kern-1.99997ptN}$, $\phi_{1},\ldots,\phi_{n}\in{{\mathcal{S}}}({\mathbb{R}})$, $C\in{{\mathcal{B}}}({\mathbb{R}}^{n})$ we set

[TABLE]

We will firstly show that this measure $\mu$ is a Radon measure on $B_{2,2}^{s}([0,1])$. For this aim, let us consider the Haar wavelet $\psi$ defined by

[TABLE]

and the scaling function $\phi$ defined by

[TABLE]

Also, we set

[TABLE]

to which we associate the multiresolution analysis

[TABLE]

The Haar wavelet is an unconditional basis in $L^{p}([0,1])$ with $1<p<\infty$, a basis in $B_{p,q}^{s}([0,1])$ for $1<p<\infty$ and $\frac{1}{p}-1<s<\frac{1}{p}$, and a basis for $B_{p,p}^{s}([0,1])$, $\frac{1}{2}<p\leq 1$ and $\frac{1}{p}-1<s<1$ (see Triebel [28, Theorem 1.58]).

Now, let $F_{0}:=V_{0}$, $F_{n}=W_{n}$, $n\in{\rm I\kern-1.99997ptN}$, and $G_{n}:=F_{0}\oplus F_{1}\oplus\cdots\oplus F_{n}$, and ${\mathcal{F}}_{n}:=\sigma(F_{0}\oplus F_{1}\oplus\ldots\oplus F_{n})$. Let us denote the projection of $\xi$ onto $G_{n}$ by $P_{n}$ and onto $W_{n}$ by $Q_{n}$. For the time being let us assume that the Radon-Nikodym derivative of the fractional Brownian motion belongs to $B^{s}_{2,2}([0,1])$. Since the Haar wavelets are a basis of $E:=B^{s}_{2,2}([0,1])$, then for each element $x\in E$ there exists a unique sequence $\{\lambda_{j,k}:j\in{\rm I\kern-1.99997ptN},k=1,\ldots,2^{j}\}$ such that

[TABLE]

Observe also that

[TABLE]

where $\{\zeta_{j,i}:j\in{\rm I\kern-1.99997ptN},i=1,\ldots,2^{j}\}$ is a family of random variables defined by

[TABLE]

and

[TABLE]

In fact, given the coefficients $\{\zeta_{j,k}:j=1,\ldots,n,k=1,\ldots,2^{j}\}\cup\{a_{0}\}$, one know the coefficient of $\phi_{n,k}$, $k=1,\ldots,2^{n}$. Since $G_{n}$ consists of all functions $f:[0,1)\to{\mathbb{R}}$ that are constant on the intervals $[2^{-n}k,2^{-n}(k+1))$, $k=1,\ldots,2^{n}-1$, there exists random coefficients $\gamma_{n,k}$, $k=1,\ldots,2^{n}-1$ such that

[TABLE]

It is now easy to see show that

[TABLE]

Since for two functions $\phi,\psi:[0,1]\to{\mathbb{R}}$, the random variables $\xi^{H}(\phi)$ and $\xi^{H}(\psi)$ are Gaussian distributed with covariance

[TABLE]

straightforward calculations gives for $l\not=k$

[TABLE]

Hence

[TABLE]

One can also easily prove that for $l=k$

[TABLE]

Using these estimates we can prove the following proposition.

Proposition B.1**.**

For $H\in(\frac{1}{2},1)$ and $-\frac{1}{2}<s<H-1$ we have $\xi^{H}\in L^{2}(\Omega;B_{2,2}^{s}([0,1]))$.

Proof.

The proof is the result of the following straightforward calculation

[TABLE]

Now, the sum is finite if $s+1-H<0$.

Remark B.1**.**

If $H\in(\frac{1}{2},1)$ one can find a number $s\in(-\frac{1}{2},H-1)$ such that $\xi^{H}\in L^{2}(\Omega;B_{2,2}^{s}([0,1]))$. Since all coefficients of $\phi_{n,k}$ and $\psi_{n,k}$ are Gaussian distributed, their law are equivalent with respect to the Lebesgue measure. Now, since the Haar basis is a Schauder basis in $B_{2,2}^{s}([0,1])$, Assumption 2.2 is satisfied. By the same arguments as used in the proof of Lemma A.6, one can show that Assumption 2.3 is also satisfied.

Lemma B.1**.**

Let $B^{H}$ be a fractional Bownian motion with Hurst parameter $H>\frac{1}{2}$ and $\mu$ the probability measure on $B_{2,2}^{s}([0,1])$ defined by (31). Let $\{\phi_{j,k}:j\in{\rm I\kern-1.99997ptN}:k=1,\ldots,2^{j}\}$ be the wavelet basis in $B_{2,2}^{s}([0,1])$ described in (32). Then, the measure $\mu$ is decomposable with decomposition satisfying Assumption 2.2 and Assumption 2.3.

Appendix C Zero One Laws for decomposable measures with density

In this Section we generalize the Theorem 4 of [11] to decomposable measures with decomposition as defined in Definition 2.1. We will also identify the conditions under which a measure satisfies Assumption 2.1 and Assumption 2.2.

Throughout this section $E$ denotes and arbitrary a separable Banach space and ${{\mathcal{B}}}(E)$ the $\sigma$–algebra generated by its open sets. Let $\mu$ be a measure on $(E,{{\mathcal{B}}}(E))$ and $F$ and $G$ be two subsets of $E$ such that $E=F\oplus G$. Then, there is a probability measure

[TABLE]

For $A\subset E$ and $y\in G$ let $A_{(F,G)}(y)=\{x\in F:x+y\in A\}$.

As mentioned in the introduction the concept of decomposability can be extended to the notion of decomposability we introduced in Definition 2.1.

Example C.1**.**

Let $E$ be a separable Banach space and $\{e_{n}:n\in{\rm I\kern-1.99997ptN}\}$ be a Schauder basis and $F_{n}:=\{\lambda e_{n}:\lambda\in{\mathbb{R}}\}$. For each element $x\in E$ there exists a unique sequence $\{a_{n}:{n\in{\rm I\kern-1.99997ptN}}\}$ in ${\mathbb{R}}$ such that $x=\sum_{n\in{\rm I\kern-1.39998ptN}}a_{n}e_{n}$. Let $G_{n}:=F_{1}\oplus\cdots\oplus F_{n}$, $G^{n}=G_{n}^{\perp}$,

[TABLE]

be a projection from $E$ onto $F_{1}\oplus\cdots\oplus F_{n}$ and

[TABLE]

Then, the probability measure of each $E$–valued random variable is decomposable in the sense of Definition 2.1. This can be shown by the following consideration. From the Radon-Nikodym Theorem (see [18, Theorem 6.3]) for any $E$–valued random variable $X$ there exists a probability kernel

[TABLE]

such that

1. (1)

${\mathbb{P}}\left(\pi_{G_{n}}X\in U\mid\pi_{G^{n}}X=y\right)=l_{n}(y,U)$* for all $U\in{{\mathcal{B}}}(F_{1}\oplus\cdots\oplus F_{n})$;* 2. (2)

for each $U\in{{\mathcal{B}}}(F_{1}\oplus\cdots\oplus F_{n})$ the mapping

[TABLE]

is ${{\mathcal{B}}}(G^{n})$–measurable

To simplify the notation let us denote $\mu_{(G_{n},G^{n})}$ by $\mu_{n}$ and $\mu_{(G^{n},G_{n})}$ by $\mu^{n}$. Note that given a decomposition $(F_{n},G_{n},l_{n})$ of $\mu$ it is essential that the kernel $l_{n}$ has a density with respect to the Lebesgue measure on $G_{n}$ which, as we will show in the next Lemma, follows from the absolute continuity of $\mu_{n}$ with respect to $\mbox{Leb}_{G_{n}}$ for any $n\in{\rm I\kern-1.99997ptN}$.

Lemma C.1**.**

Let $E$ be a separable Banach space and $\{e_{n}:n\in{\rm I\kern-1.99997ptN}\}$ be a Schauder basis and $F_{n}:=\{\lambda e_{n}:\lambda\in{\mathbb{R}}\}$, $G_{n}:=F_{1}\oplus\cdots\oplus F_{n}$. Let us assume that for all $n\in{\rm I\kern-1.99997ptN}$ $\mu_{G_{n}}$ is absolutely continuous with respect to $\mbox{Leb}_{G_{n}}$. Then for any $n\in{\rm I\kern-1.99997ptN}$,

[TABLE]

In particular, for any $U\in{{\mathcal{B}}}(G_{n})$ with $\mu_{n}(U)=0$, we have

[TABLE]

Proof.

Fix $U\in{{\mathcal{B}}}(G_{n})$ with $\mu_{G_{n}}(U)=0$. We will show that $\mu^{n}(\{{{\bf y}}\in G^{n}:l_{n}(y,U)>0\})=0$. From [23, Theorem 4.1 ] we infer that for all $\epsilon>0$ there exists a subset $C_{n,U}^{\epsilon}\subset G^{n}$ such that $\mu^{n}(G^{n}\setminus C_{n,U}^{\epsilon})\leq\epsilon$ and

[TABLE]

is continuous. Now let us set

[TABLE]

Since l_{n}(\cdot,U)\big{|}_{C^{\epsilon}_{n,A}} is continuous and the sets $[\epsilon,1]$ and $C_{n,U}^{\epsilon}$ are closed, the set $G_{\epsilon}^{\ast}$ is closed. Hence,

[TABLE]

Since $G^{n}\supset G_{\epsilon}^{\ast}$, we additionally have

[TABLE]

By the definition of the set $G_{\epsilon}^{\ast}$ we have

[TABLE]

Since $\epsilon>0$, we have $\mu^{n}({G_{\epsilon}^{\ast}})=0$. Now, from the closedness of $G_{\epsilon}^{\ast}$ and the regularity of the measure$\mu_{G^{n}}$ we infer that

[TABLE]

Lemma C.2**.**

Let $\mu$ be a decomposable finite measure on $E$ with decomposition $\{F_{n},G^{n},l_{n}\}_{n=1}^{\infty}.$ Let us assume that $\mu_{G_{n}}$ is absolutely continuous with respect to $\mbox{Leb}_{G_{n}}$. Then, for any $U\in{\mathcal{F}}_{n}$ satisfying $\mu_{G_{n}}(U)=0$ we have

[TABLE]

Proof.

Let $n\in{\rm I\kern-1.99997ptN}$ and $U\in{\mathcal{F}}_{n}$ such that $\mu_{G_{n}}(U)=0$. We will show that $\mu_{G^{n}}(\{{{\bf y}}\in G^{n}:l_{n}({{\bf y}},U)>0\})=0$.

Firstly, note that by the Radon-Nikodym Theorem the mapping

[TABLE]

is measurable. Hence, from [23, Theorem 4.1] we infer that for all $\epsilon>0$ there exists a closed subset $C_{n,U}^{\epsilon}$ of $G^{n}$ such that $\mu^{n}(G^{n}\setminus C_{n,U}^{\epsilon})\leq\epsilon$ and the function

[TABLE]

is continuous.

Secondly, let us set

[TABLE]

From the continuity of l_{n}(\cdot,U)\big{|}_{C^{\epsilon}_{n,U}} and the fact that the sets $[\epsilon,1]$ and $C_{n,U}^{\epsilon}$ are closed we conclude the set $G_{\epsilon}^{\ast}$ is also closed. Next, thanks to the definition of $\mu_{G_{n}}$ we obtain that

[TABLE]

Furthermore, because $G_{\epsilon}^{\ast}\subset C_{n,U}^{\epsilon}$ we also have

[TABLE]

Invoking now the definition of the set $G_{\epsilon}^{\ast}$ we obtain

[TABLE]

Since $\epsilon>0$, we have $\mu_{G^{n}}({G_{\epsilon}^{\ast}})=0$. From the closedness of $G_{\epsilon}^{\ast}$ and the regularity of the measure $\mu^{n}$ we infer that

[TABLE]

Therefore,

[TABLE]

Corollary C.1**.**

Let $E$ be a separable Banach space and $\{e_{n}:n\in{\rm I\kern-1.99997ptN}\}$ be a Schauder basis. Put $F_{n}:=\{\lambda e_{n}:\lambda\in{\mathbb{R}}\}$ and $G_{n}:=F_{1}\oplus\cdots\oplus F_{n}$. Let us assume that for all $n\in{\rm I\kern-1.99997ptN}$ $\mu_{G_{n}}$ is absolutely continuous with respect to the $\mbox{Leb}_{G_{n}}$. Then for any $n\in{\rm I\kern-1.99997ptN}$ there exists a function $h_{n}:G^{n}\times F_{1}\oplus\cdots\oplus F_{n}\to{\mathbb{R}}^{+}_{0}$ such that $\mu_{G^{n}}$–a.s.

[TABLE]

Proof.

From

[TABLE]

follows the corollary’s assertion. Indeed the above identity implies the existence of a Radon-Nikodyn derivative. In particular, it holds that

[TABLE]

Definition C.1**.**

We call a set $U\in{{\mathcal{B}}}^{\mu}(E)$ a finite zero one $\mu$–set if and only if for all $n\in{\rm I\kern-1.99997ptN}$

[TABLE]

where $U_{n}({{\bf y}})=U_{(F_{1}\oplus\cdots\oplus F_{n},G^{n})}({{\bf y}})$.

Let us now present the generalization of Theorem 4 in [11].

Theorem C.1**.**

Let $\{F_{n},G^{n},l_{n}\}_{n=1}^{\infty}$ be a decomposition for $\mu$ such that for any $n\in{\rm I\kern-1.99997ptN}$ $\mu_{G_{n}}$ is absolutely continuous with respect to $\mbox{Leb}_{F_{1}\oplus\cdots\oplus F_{n}}$. Let $F_{\infty}=\cup_{n\in{\rm I\kern-1.39998ptN}}\{F_{1}+F_{2}+\cdots+F_{n}\}$. If $U$ is a finite zero one $\mu$ measurable subset of $E$, then there exists $B\in{{\mathcal{B}}}(E)$ such that $B+F_{\infty}=B$ and $\mu(B)=\mu(U)$.

Proof.

The proof is very similar to the proof of [11, Theorem 4]. Let us assume $U\in{{\mathcal{B}}}(E)$. For fix $n\in{\rm I\kern-1.99997ptN}$ we set $U^{n}=\left\{y\in G^{n}:\mu_{G_{n}}(U_{n}(y))=1\right\}$,

[TABLE]

$B_{n}=G_{n}+U^{n}$ and $B=\liminf_{n\to\infty}B_{n}=\cup_{n=1}^{\infty}\left\{\cap_{m\geq n}B_{m}\right\}$. For the time being let us assume that

[TABLE]

Then,

• •

$\mu(U\bigtriangleup B_{n})=0$ for all $n\in{\rm I\kern-1.99997ptN}$,

• •

and $\mu(U)=\mu(B_{n})\geq\mu(\cap_{m\geq n}B_{m})\geq\mu(U)$,

• •

$\mu(B)=\lim_{n\to\infty}\mu(\cap_{m\geq n}B_{m})$.

Since $\mu$ is regular we additionally have that

[TABLE]

from which the assertion of Theorem C.1 follows.

Now it remains to prove (34). To this end, observe first that because of Lemma C.1 the kernel $l_{n}$ is $\mu^{n}$–a.s.  absolutely continuous on $G_{n}$. Hence, by the Radon-Nikodym Theorem for $\mu^{n}$–a.s.  there exists a probability kernel

[TABLE]

such that

[TABLE]

Then, by using $B_{n}=G_{n}\oplus U_{n}$ we obtain that

[TABLE]