Existence of a density of the 2-dimensional Stochastic Navier Stokes Equation driven by Lévy processes or fractional Brownian motion

Journal Pre-proof Existence of a density of the 2-dimensional Stochastic Navier Stokes Equation driven by L´evy processes or fractional Brownian motion E. Hausenblas, Paul A. Razafimandimby

PII: DOI: Reference:

S0304-4149(18)30083-8 https://doi.org/10.1016/j.spa.2019.12.001 SPA 3608

To appear in:

Stochastic Processes and their Applications

Received date : 7 April 2018 Revised date : 3 November 2019 Accepted date : 7 December 2019 Please cite this article as: E. Hausenblas and P.A. Razafimandimby, Existence of a density of the 2-dimensional Stochastic Navier Stokes Equation driven by L´evy processes or fractional Brownian motion, Stochastic Processes and their Applications (2019), doi: https://doi.org/10.1016/j.spa.2019.12.001. This is a PDF file of an article that has undergone enhancements after acceptance, such as the addition of a cover page and metadata, and formatting for readability, but it is not yet the definitive version of record. This version will undergo additional copyediting, typesetting and review before it is published in its final form, but we are providing this version to give early visibility of the article. Please note that, during the production process, errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain. c 2019 Elsevier B.V. All rights reserved. ⃝

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EXISTENCE OF A DENSITY OF THE 2-DIMENSIONAL STOCHASTIC NAVIER ´ STOKES EQUATION DRIVEN BY LEVY PROCESSES OR FRACTIONAL BROWNIAN MOTION E. HAUSENBLAS AND PAUL A. RAZAFIMANDIMBY Abstract. In this article we are interested in the regularity properties of the probability measure induced by the solution process by a L´evy process or a fractional Brownian motion driven Navier– Stokes equations on the two-dimensional torus 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 ⊂ L2 (T) is absolutely continuous with respect to the Lebesgue measure on F .

1. Introduction

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We consider the Navier-Stokes equations (NSEs) subjected to the periodic boundary condition on the torus  ∂t u(t) − ν∆u(t) + u(t) · ∇u(t) + ∇p(t) = f (t),    ∇ · u(t) = 0, R (1)  T u(t, x) dx = 0    u(0) = u0 ,

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where u and 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 a divergence free R initial velocity u0 satisfying T u0 (x) dx = 0. The perturbation f can be replaced by a stochastic process. In the case when f is replaced by a Gaussian 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 significant results, see for instance [7, 11, 18, 20, 25], to cite a few results. Particularly, when f is replaced by the time derivative of ∞ X Ξ(t) = bj ej βj (t), t ≥ 0, (2) j=1

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where (bj )j∈N is a sequence of non-negative numbers, (βj )j∈N is a sequence of independent, identically distributed real-valued Brownian motions and (ej )j∈N is an orthonormal basis of the space of square integrable, periodic and divergence free functions with mean zero, the authors in [12], [2] and [28] 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 [12] a method based on Girsanov theorem is used, and the Malliavin calculus is used in [28]. In [2] a method based on controllability of (1) in finite-dimensional projections and an abstract result on an image of a decomposable measure under analytic mappings is used. This method does not use the Gaussian structure of the noise as the methods in [12] and [28]. In this paper, we are mainly interested in proving

Date: December 18, 2019. We would like the anonymous referee for their helpful remarks and Armen Shirikyan for his fruitful discussion. This work was supported by the Austrian Science Foundation (FWF), project number P 28010, Razafimandimby’s research was also partly supported by the National Research Foundation South Africa (Grant Numbers 109355 and 112084). 1

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E. HAUSENBLAS AND PAUL A. RAZAFIMANDIMBY

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the existence of densities for the laws of finite-dimensional analytic functionals of the solution of (1) when the driving process Ξ is a L´evy process or a fractional Brownian motion. For this purpose, we extend the results in [2] to our framework. Although we closely follow the approach in [2] the extension of the result therein to our setting is not trivial. The proof in [2] relies on the natural decomposability of the law of the driving process in a Hilbert space H which is not naturally satisfied by a L´evy process or a fractional Brownian motion. In fact, even if the L´evy process (or fractional Brownian motion) Ξ has a decomposition as in (2), which is one of the main assumptions in [2], it is not known whether there exists a Hilbert space H on which the law of Ξ on H is decomposable. In order to overcome this difficulty, we first weaken the assumption on the by the process induced probability measure by modifying the result of [14]. Secondly, we use wavelet analysis to verify that the measure satisfies the modified assumptions. 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). Further works related to our result are, for instance, the work of Dong et.a. [15], who studied under which conditions on the L´evy measure the Markovian semigroup generated by the process is smoothing in space by using the Kolmogorov equation. In [31], the authors prove exponential convergence to the invariant measure where the underlying process is again a solution of a stochastic (partial) differential equation on a Hilbert space H driven by an α–stable process. Here, the Harris Theorem is used to get the result. In [1] the authors prove the strong Feller property and exponential mixing for the 3 dimensional Navier-Stokes equations under different degeneracy assumptions on the driving process. Here, the main tool is the Malliavin calculus and H¨ormander’s Theorem. In the next section, i.e. Section 2, 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 Section 4. In Section 5 and Section 6, we present and prove several results related to the wavelet expansion of L´evy process and fractional Brownian motion, respectively. In Appendix A 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

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For a separable Banach space E we denote by B(E) its Borel σ–algebra. For a finite dimensional subspace E0 of E we denote by E1 the complemented subspace of E, i.e. the subspace of E such that E = E0 ⊕ E1 , i.e., E1 = E0⊥ . For two subspaces E1 and E2 of E, E1 ⊕ E2 is defined by E1 ⊕ E2 := {z = e1 + e2 : e1 ∈ E1 , e2 ∈ E2 }. Furthermore, for A ⊂ E and y ∈ E1 we put A(E0 ,E1 ) (y) = {x ∈ E0 : x + y ∈ A}.

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Let µ be a probability measure on (E, B(E)) and E0 and E1 as above. We define a probability measure µE0 on (E0 , B(E0 )) by µE0 : B(E0 ) 3 A 7→ µ(A + E1 ) ∈ [0, 1]. ˜0 ⊂ E1 (again, E ˜0 is supposed to be finite dimensional) we set For a subspace E ˜0 ) 3 A 7→ µ(A + E1 ) ∈ [0, 1]. µ ˜ : B(E (E0 ,E1 )

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If E0 is finite dimensional, then we denote by LebE0 the measure defined by LebE0 : B(E0 ) 3 U 7→ µE0 (U ) := LebRn (ι−1 (U )),

where ι is the isomorphism ι : E0 → Rn , n = dim(E0 ). In addition, if E1 is a finite dimensional subspace of E, then πE1 denotes the projection onto E1 . In case E2 is the complement subspace of E1 , the projection πE2 of E2 is also uniquely defined and continuous on E. We can now introduce the following definition. Definition 2.1. Let (Ω1 , A1 ) and (Ω2 , A2 ) be measurable spaces. A map κ : Ω1 × A2 → [0, 1] is called a probability kernel if

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(1) Ω1 3 ω 7→ κ(ω1 , A2 ) is measurable for any A2 ∈ A2 ; (2) A2 3 A2 7→ κ(ω1 , A2 ) is a probability measure for any ω1 ∈ Ω1 ; In case of a finite dimensional subspace and its complement, desintegration gives the existence of a probability kernel. Let {Fn : n ∈ IN} be a family of mutually disjoint, linearly independent, and finite dimensional subspaces of E. We would like to define a sequence of spaces Gn such that Gn is a complement to Gn := F1 ⊗ F2 ⊗ · · · ⊗ Fn , i.e. Gn := (F1 ⊕ · · · ⊕ Fn )⊥ . However, since for any finite dimensional subspace the complement need not to be unique, we suppose that the sequence {Gn : n ∈ IN} is nested, i.e. Gn+1 ⊂ Gn ⊃ E. In fact, such a sequence always exists. This can be shown by induction. Let G1 a complement of G1 = F1 . Then, the quotient space E/G1 is again a Banach space. Let F˜2 = [{f ∈ F2 }], where for an element e ∈ E the bracket [e] denotes equivalence ˜ 2 . Let class of e in E/G1 . Again, there exists a complement of F˜2 in E/G1 , which we denote by G 2 2 ˜ F := q(F ), where q : E → E/F1 denotes the quotient map. Remark 2.1. Let {Fn : n ∈ IN} be a family of mutually disjoint, linearly independent, and finite dimensional subspaces of E. We set Gn := F1 ⊕ · · · ⊕ Fn . Let {Gn : n ∈ IN} be a nested sequence of Banach spaces such that E ⊃ Gn ⊃ Gn+1 , n ∈ IN and Gn is a complement of Gn . Let µGn the probability measure induced on Gn by the projection PGn (y) := {z ∈ Gn : ∃x ∈ Gn : x + z = y}. Then, by desintegration (see e.g. Theorem 6.4 [23, p.108]) we know that for any n ∈ N there exists a kernel

µ(A) =

Z

Z

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such that

f

ln : Gn × B(F1 ⊕ · · · ⊕ Fn ) → R+ 0, Gn

ln (y, dx)µGn (dy),

An (y)

where An (y) = A(F1 ⊕···⊕Fn ,Gn ) (y). In particular, we can decompose the measure µ into the triplet {Fn , Gn , ln }∞ n=1 .

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Hereafter we fix a separable Banach space E with Schauder basis {en : n ∈ IN} and we set Fn = {λen : λ ∈ R}.

We also set

Gn = F0 ⊕ F1 ⊕ . . . ⊕ Fn .

{Gn

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Let : n ∈ IN} be a nested sequence of subspaces of E such that for any n ∈ IN the space complement to Gn . Let us consider the probability kernel

(3) (4) Gn

is a

ln : Gn × B(Gn ) → [0, 1].

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The projection onto any nontrivial subspace F ⊂ E is denoted by πF . Here, we assume that either F is finite dimensional, or F has a complement. Since in our setting we only consider pairs (E1 , E2 ) where E1 is finite dimensional, and E2 its complement, the projections onto E1 and onto E2 are uniquely defined and continuous on E. Having fixed these notations we now proceed to the statement of our assumptions.

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Analysing Theorem 2.2 of [2], one can easily verify that the following assumption is essential. Assumption 2.1. Let µ ∈ P(E) be a decomposable measure with decomposition {Fn , Gn , ln }∞ n=0 . We + n n assume that for any n ∈ IN there exists a positive function ρn : G × Gn → R0 such that µG –a.s. we have for all U ∈ B(Gn ) Z ln (y, U ) =

ρn (y, x) dx.

U

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

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Assumption 2.2. Let µ ∈ P(E) be a decomposable measure with decomposition {Fn , Gn , ln }∞ n=0

constructed by a Schauder basis {en : n ∈ IN} with Fn := {λen : λ ∈ R}, Gn := span{e1 , . . . , en } such that µGn is absolutely continuous with respect to the Lebesgue measure LebGn . Our third condition is given in the following next lines. Assumption 2.3. Let µ ∈ P(E) and let {Fn , Gn , ln }∞ n=0 be a decomposition of µ. There exists a point Y =

∞ X j=1

such that (1) for any n ∈ IN and δ > 01

yj ej ∈ E,

(5)

µGn (BGn (πGn Y, δ)) > 0. (2) for all numbers N ∈ IN there exists a RN > 0 such that for all x0 ∈ BGN (πGN Y, RN ), and all  > 0, there exists a δ > 0 such that µ ({x ∈ E | |x − x0 |E ≤ }) ≥ δ.

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

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Definition 2.2. We call a set A ∈ B µ (E) a finite zero–one µ–set if and only if for all n ∈ IN µGn ({y ∈ Gn : µn (An (y)) = 0 or 1}) = 1,

where An (y) = A(F1 ⊕···⊕Fn ,Gn ) (y).

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Let F∞ = ∪n∈IN {F0 + F1 + · · · + Fn }. Now, let us present the generalization of Theorem 4 in [14] or [2, Theorem 1.6]], whose proof requires that the measure µ is decomposable and has a finite second moment (see [2, Property (P), page 402] for the precise statement).

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Theorem 2.1. Let f : X → R be an analytic function and let µ ∈ P(X) be a decomposable measure with decomposition {Fn , Gn , ln } satisfying Assumption 2.1 and Assumption 2.3. Let Nf ⊂ E be defined by Nf := {x ∈ E : f (x) = 0}. Then, we have µ(Nf ) = 0 or 1. Furthermore, if f is not identical zero, then µ(Nf ) = 0.

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Proof of Theorem 2.1: Let Nf := {x ∈ G : f (x) = 0}. Since f is analytic, for all n ∈ IN for any y ∈ Gn the function fy (x) := f (y + x) is also analytic. Therefore, either LebGn (Nfn (y)) = 0 or LebGn (E \ Nfn (y)) = 0, where Nfn (y) = {x ∈ Gn : x + y ∈ Nf }. Thus, Nf is a finite zero–one µ set, ˜f ∈ B(E) such that N ˜f + F(∞) = N ˜f and µ(N ˜f ) = µ(Nf ). and there exists a set N To prove the second part we assume that f 6≡ 0. We will show that µ(Nfc ) > 0. For this purpose let n ∈ N be fixed and set fn := f|Gn and Yn := πGn Y where Y is the point from Assumption 2.3. Observe that fn is analytic and therefore   LebGn Gn \ {x ∈ Gn : f (x) 6= 0} = 0. We now distinguish two cases: fn (Yn ) 6= 0 and fn (Yn ) = 0. In the first case, i.e., fn (Yn ) 6= 0 we observe that by the continuity of fn there exists a number δ > 0 such that f (x) 6= 0 for all x ∈ BE (Yn , δ). 1For a Banach space E we denote by B (y, δ) the ball centered at y with radius δ, i.e. B (y, δ) = {x ∈ E : |x−y| ≤ δ}. E E E

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Hence, and taking into account item (2) of Assumption 2.3 we conclude that µ(Nfc ) > 0. To treat the second case, i.e, fn (Yn ) = 0, we first notice that, since fn is analytic, we have   LebGn {x ∈ Gn : f (x) = 0} = 0.

This implies that for any  > 0 one can find an element x0 ∈ Gn such that |x0 −Yn |E ≤  and f (x0 ) 6= 0. Since f is continuous we can find a number δ > 0 such that f (x) 6= 0 for all x ∈ BE (x0 , δ). Item (2) of Assumption 2.3 with  = R2 yields that µ(BE (x0 , δ) > 0) from which it follows that µ(Nfc ) > 0. The above theorem will, as in [2, Theorem 2.2], be used to prove the existence of the density of the law of the finite projection on a finite dimensional space of the solution of a stochastic evolution equation driven by L´evy process and fractional Brownian motion. 3. The main result In this section we consider an abstract stochastic evolution equation in a separable Banach space E 
du(t) + Lu(t) dt + B(u(t), u(t)) dt = d Ξ(t), t > 0, (6) u(0) = u0 ∈ E .

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where the driving process Ξ is either a L´evy process or a fractional Brownian motion, L : D(L) → E is a possibly unbounded operator, and B : E × E → E is a densely defined bilinear operator taking values in E . We assume that the equation above is uniquely solvable in E and we denote the solution starting from u0 ∈ E at time t = 0 by {u(t, u0 ) : t ≥ 0}. In order to formulate the main result of this section we need to introduce few concepts from control theory. For this aim, let U ⊂ E be a separable Banach space r ≥ 1 be a fixed number and let us consider the following control problem  ∂t u(t) + Lu(t) dt + B(u(t), u(t)) = v(t), t > 0, (7) u(0) = u0 ∈ E , where v ∈ Lr ([0, T ]; U ) is the control and U is the control space. For a fixed time T > 0 we denote by RT : E × Lr ([0, T ]; U ) → E

(8)

and initial condition u0 ∈ E

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the so called solution operator that maps each function v ∈ to the solution u(T, u0 ) of the system (7).

Lr ([0, T ]; U )

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Definition 3.1. A system is controllable in time T > 0 for a finite dimensional subspace F ⊂ E if and only if πF RT (u0 , Lr ([0, T ]; U )) ⊃ F for any u0 ∈ E .

we have

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Definition 3.2. A system is solid controllable in time T > 0 for a finite dimensional subspace F ⊂ E , if and only if for any R > 0 and any u0 ∈ E , there exists an  > 0 and a compact set K ⊂ Lr ([0, T ]; U ) such that for any function Φ : K → F satisfying sup |Φ(x) − πF RT (u0 , x)|F ≤ ,

x∈K

Φ(K ) ⊃ BF (R).

In our equation, the control is replaced by a stochastic perturbation Ξ, which induces a probability measure on a specific Banach space E. Let us suppose E ⊃ Lr ([0, T ]; U ). Since each finite subset F of Lr ([0, T ]; U ) is also a finite dimensional subset of E, the fact that E is larger than Lr ([0, T ]; U ) does not cause any problem. In addition, we need a Schauder basis {en : n ∈ IN} such that en ∈ Lr ([0, T ]; U ) for all n ∈ IN. To explain the by Ξ induced probability measure on E, we illustrate here the idea using

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a Wiener process, i.e. the setting of [2]. The analysis for the L´evy process and the fractal Brownian motion is more sophisticated and treated in Section 5 and Section 6. Let Ξ be a d–dimensional Wiener P process with covariance I defined over a probability space (Ω, F, P). Let U = Rd , W (t) = dj=1 ej β j (t), t ≥ 0, where e1 , . . . , ed ∈ U and {β j : j = 1, . . . , d} be a family of mutually independent Brownian motions. Let γ > 12 and let us define for a function φ ∈ H2γ (0, T ; Rd ) the random variable W (φ) by Z T φ(s)dW (s). 0

H2γ ([0, T ]; Rd )

C([0, T ]; Rd )

Since ,→ continuously, the mapping is well defined. In this way, the Wiener process W induces a cylindrical measure µ on H2−γ ([0, T ]; Rd ), which is the dual space of H2γ ([0, T ]; Rd ). To be more precise, for φ1 , . . . , φn ∈ H2γ ([0, T ]; Rd ) and C ∈ B(Rd ), the cylindrical measure µ is defined by µ({ξ ∈ H2−γ ([0, T ]; Rd ) : (ξ(φ1 ), ξ(φ2 ), . . . , ξ(φ2 ) ∈ C}) = P((W (φ1 ), . . . , W (φn )) ∈ C)

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Since EW (φ1 )W (φ2 ) = hφ1 , φ2 iL2 for φ1 , φ2 ∈ H2γ ([0, T ]; Rd ), it is straightforward to show that the cylindrical measure µ is a probability measure on H2−γ ([0, T ]; Rd ). Besides, each finite dimensional subspace K of Lr ([0, T ]; Rd ) is a finite dimensional subspace of H2−γ ([0, T ]; Rd ). On the other side, the orthonormal basis {en : n ∈ IN} of H2−γ ([0, T ]; Rd ) constructed by sinus and cosines functions is a subset of Lr ([0, T ]; Rd ). In addition, it is straightforward to verify that the measure µ on H2−γ ([0, T ]; Rd ) and the decomposition {Fn , Gn , ln }∞ n=0 with Fn := {λen : λ ∈ R} satisfy Assumption 2.2 and Assumption 2.3. With this preliminary work the following general result can be shown.

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Theorem 3.1. Let r ≥ 1 and let E be a Banach space with Schauder basis {en : n ∈ IN} such that {en : n ∈ IN} ⊂ Lr (0, T ; U ), Lr (0, T ; U ) ,→ E continuously, and the probability measure induced by the the process Ξ on E admits a decomposition {Fn , Gn , ln }∞ n=0 with Fn := {λen : λ ∈ R} (the notation used in (3) and (4) is enforced). Suppose {Fn , Gn , ln }∞ n=0 satisfies Assumption 2.1 and Assumption 2.3. For a fixed number T > 0 we also suppose that (A1) the solution operator RT defined in (8) which is generated by the system (7) is analytic, (A2) and for any finite dimensional space F ⊂ E , the system (7) is solid controllable in time T for the finite dimensional space F . Then, for any u0 ∈ E and for any finite dimensional subspace F ⊂ E there exists a density ρ : F → R+ 0 such that Z E1O (πF u(T, u0 )) = ρ(x) LebF (dx). O

Proof. Let us fix a finite dimensional subspace F of E and consider the operator f : E × X 3 (u0 , ξ) 7→ πF RT (u0 , ξ) ∈ F,

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Lr ([0, T ]; U ),

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where X = u solves equation (6), and RT is defined in (8). The proof of our theorem will follow from the applicability of [2, Theorem 2.2], respectively of [2, Theorem 2.1]. However, one has to analyse the situation carefully, since the measure µ is defined on a Banach space E being usually larger than X. Here, the fact is essential that the decomposition is constructed by a Schauder basis {en : n ∈ IN}, where all elements en , n ∈ IN, belongs to X. Due to this fact, all finite-dimensional subspaces Gn , n ∈ IN, of E are also finite-dimensional subspaces of X. Therefore, although the measure is decomposable on E, we can show that the requested properties are satisfied on X. Firstly, we will check if all assumptions of [2, Theorem 2.2] are all satisfied. Secondly, we give a sketch of the proof of Theorem 2.2 [2], to verify if the Theorem also first to our assumptions. Claim 1: For any u0 ∈ E there exists a finite dimensional subspace G = G(u0 ) of X given by some Gm , m ∈ IN, an open set (or ball) BG ⊂ G and an open set (or ball) BF ⊂ F such that πF RT (u0 , BG ) ⊃ BF .

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To prove this claim we fix a large number R > 0 such that u0 ∈ BE (R). By the definition of solid controllability, we know that there exists an  > 0 and a compact set K ⊂ X such that any function Φ : K → E satisfying sup |Φ(y) − πF RT (u0 , y)|F ≤ , y∈K

satisfies Φ(K ) ⊃ {y ∈ F : |y|F ≤ R}. Fix  > 0 and the corresponding compact set K . Since the operator RT (u0 , ·) : X → E

is continuous, the operator is uniformly continuous on K , and, hence, there exists a δ0 > 0 such that |RT (u0 , y1 ) − RT (u0 , y2 )|E ≤ ,

∀ y1 , y2 ∈ K with |y1 − y2 |X ≤ δ0 .

Since the system {en : n ∈ IN} is a Schauder basis of E, {en : n ∈ IN} ⊂ X, and X ,→ E continuously, it follows that ∪n∈IN Fn is a dense subset in X. In particular, since K is compact, for any δ > 0, there exists a number N ∈ IN such that sup |y − πGN y|X ≤ δ, y∈K

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where GN := ⊕l≤N Fl and πGN denotes the projection onto GN . The projection may not be unique, but we define the projection by the decomposition of the complement GN , which gives the uniqueness of the projection πGN . Let N ∈ IN be sufficiently large such that sup |y − πGN y|X ≤ δ0 .

y∈K

Let us put G = GN and define the map

Φ : K → E

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by

Φ(y) = πF (RT (u0 , πG y)), From the consideration above, it follows that sup |Φ(y) − πF RT (u0 , y)|F ≤ .

y∈K

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Hence, by the solid controllability

Φ(K ) ⊃ {y ∈ F : |y|F ≤ R}.

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In addition, since πG K is a bounded set of G, there exists a number R1 > 0 such that {y ∈ G : |y| ≤ R1 } ⊃ πG K . Setting BF := {y ∈ F : |y|F ≤ R} and BG := {y ∈ G : |y|G ≤ R1 } we have RT (u0 , BG ) ⊃ BF ,

which proves Claim 1. Since in finite dimensions all norms are equivalent, it does not matter if one consider the norm in G induced by X or E.

Claim 2: The measure µ on E satisfies Assumption 2.1.

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Claim 2 is easy to prove. Thanks to Lemma A.1 the measure satisfies Assumption 2.3, which is equivalent to Claim 2. Note that the first Claim is operating on X, the second Claim is operating on E. The proof of Theorem 2.2 in [2] is done by reducing the problem to a finite dimensional problem. In the first step one fix an element u0 ∈ U . By Claim 1, there exists a finite dimensional subspace G = G(u0 ) and open sets BG ⊂ G and BF ⊂ F such that πF RT (u0 , BG ) ⊃ BF . In particular, the interior of πF RT (u0 , BG ) is non-empty. Let us put dim(G) = N and dim(F ) = K. Let g0 : E × G −→ F defined by g(u, x) = πF RT (u, x), u ∈ E , x ∈ G. Then, by the Sard’s Theorem Dx g(u0 , x) has almost everywhere rank K, or, for almost all f ∈ F , there exists a N − K

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E. HAUSENBLAS AND PAUL A. RAZAFIMANDIMBY

dimensional submanifold S of G such that that S = {x ∈ G : g0 (u0 , x) = f }. In combination with the implicit function Theorem, we can find a sample of charts {(Vj1 , Vj2 ) : j = 1, . . . n}, open sets {BFj : j = 1, . . . , n} a neighbourhood BE of u0 , and mappings hj : BE × Vj2 × BFj → Vj1 such that Vj1 ⊂ RK , Vj2 ⊂ S, πF RT (u, hj (u, x2 , f ) + x2 ) = f . Using these charts and the probability measure on G given by Z Z ν(A) =

GN

A

lN (y, dx)µGN (dy),

A ∈ B(G),

a density on F can be constructed.

4. Application to the 2D stochastic Navier-Stokes Equations Throughout this section T denotes the 2D torus, Lp (T) and W m,p will respectively denote the usual s ([0, 1]) := B s ([0, 1]; R) Lebesgue space of p-integrable functions and Sobolev spaces. The symbol Bp,p p,p is the Besov spaces of all R-valued functions defined on the interval [0, 1] defined in Appendix B.

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Let V be the set of periodic, divergence free and infinitely differentiable function with zero mean. In what follows, we denote by H and V the closures of V in L2 (T) and W 1,2 (T), respectively. Since the Navier Stokes equation is here treated in the Hilbert space setting, we denote the underlying function space by H instead of E . We endow the space H with the L2 -scalar product denoted by (·, ·) and the usual L2 -norm denoted by |·|. The space V is equipped with the gradient norm k · k = |∇·|. We also set D(L) = [H 2 (T)]2 ∩ V, Lv = −Π∆v, v ∈ D(L),

rep

where Π is the orthogonal projection from L2 (T) onto H . It is well-known that the Stokes operator L is positive self-adjoint with compact resolvent and its eigenfunctions {e1 , e2 , . . .}, with eigenvalues 1 0 < λ1 ≤ λ2 ≤ . . ., form an orthonormal basis of H . It is also well-known that V = D(L 2 ), see [19, Appendix A.1 of Chapter II]. Furthermore, we see from [35, Chapter II, Section 1.2] and [19, Appendix A.3 of Chapter II] that one can define a continuous bilinear map B from V × V with values in V∗ such that Z hB(u, v), wi = [u(z) · ∇v(z)] · w(z)dz for any u, v, w ∈ V, (9)

lP

T

hB(u, v), vi = 0,

for any u, v ∈ V,

(10)

4

|hB(u, v), wi| ≤ C0 kukL4 kvkL4 kwk, for u, v ∈ L (T), w ∈ V.

(11)

rna

With all these notations the Navier-Stokes equations (1) can be written in the abstract form (
du(t) + κLu(t) + B(u(t), u(t)) dt = dΞ(t), (12) u(0) = u0 ∈ H ,

Jo u

˙ The positive number κ > 0 denotes the where for the sake of simplicity we assume that Π Ξ˙ = Ξ. viscosity. Before characterizing the process entering our system, we introduce the trigonometric basis in H by elements in Z2 . Namely, we write j = (j1 , j2 ) ∈ Z2 and set

j⊥

ej (x) = sin(jx)j ⊥

for j1 > 0 or j1 = 0, j2 > 0,

ej (x) = cos(jx)j

⊥

for j1 < 0 or j1 = 0, j2 < 0,

e10 (x)

e20 (x)

= (1, 0),

= (0, 1),

where = (−j2 , j1 ). The family E = = 1, 2, j ∈ Z2 \ {0}} is a complete set of eigenfunctions for the Stokes operator which forms an orthonormal basis in H . {ei0 , ej , i

Journal Pre-proof IRREDUCIBILITY AND THE 2-D STOCHASTIC NAVIER STOKES

9

For any symmetric set K ⊂ Z2 containing (0, 0) we write K0 = K and define Ki with i ≥ 1 as the union of Ki−1 and the family of vectors l ∈ Z2 for which there are m, n ∈ Ki−1 such that l = m + n, |m| = 6 |n|, and |m ∧ n| = 6 0, where m ∧ n = m1 n2 − m2 n1 . Definition 4.1. A symmetric subset K ⊂ Z2 containing (0, 0) is saturating if and only if ∪i∈IN Ki−1 = Z2 .

Throughout we set d = dim K and denote by Hd the finite dimensional subspace of H spanned by the eigenvectors {ej ; j ∈ K}. The driving process is either X Ξ(t) = ej lj (t), t ≥ 0, (13) j∈K

where {lj : j ∈ K} is a family of mutually independent L´evy processes with L´evy measure ν over a probability space (Ω, F, P), or X Ξ(t) = ej βjH (t), t ≥ 0, (14) j∈K

{βjH

roo

f

where : j ∈ K} is a family of identical distributed and mutual independent fractal Brownian motions with Hurst parameter H ∈ ( 21 , 1) over a probability space (Ω, F, P). The existence of a unique solution u = {u(t) : t ≥ 0} to (12) follows from the results in [8] for example for the case of pure jump L´evy process, and from [26] for case of a perturbation by the fractional Brownian motion. Before stating the main result of this section, let us introduce the following definition (compare [32, p. 13]). A measurable function l : [0, ∞) → R is slowly varying at ∞, iff for any x > 0 lim

l(tx) = 1. l(t)

rep

t→∞

We start with the following theorem which treats the case of Navier-Stokes equations driven by a L´evy process.

lP

Theorem 4.1. Let K be a saturating set and assume that the process Ξ entering the system (6) is defined by (13). We also assume that the L´evy measures νj , j = 1, . . . , d, are symmetric and equivalent to the Lebesgue measure on R \ {0}, and satisfy Z |z|p νj (dz) < ∞, (15) |z|≤1

for some p ∈ (1, 2). In addition, we assume that there exists a number α ∈ (0, 2] such that

rna

νj (R \ [−, ]) ∼ −α l()

as

 → 0,

(16)

Jo u

for some slowly varying function l. Let u = {u(t, u0 ) : t ≥ 0, u0 ∈ H } be the unique solution of system (6). Then for any finite dimensional subspace F ⊂ H , for all initial conditions u0 ∈ H , there exists a density ρu0 : F → R+ 0 such that Z E1O (πF u(T, u0 )) = ρu0 (x) LebF (dx). O

In addition, for any sequence {un : n ∈ IN} with un → u0 ∈ H as n → ∞, we have Z |ρu0 (x) − ρun (x)|dx−→0 as n → ∞. F

Remark 4.1. In fact, given α ∈ (0, 2) such that (16) holds, it is straightforward to see that (15) is satisfied for any p > α. On the other hand, if p > 0 satisfies (15), then this does not imply that there exists a number α ∈ (0, 2) such that (16) holds.

Journal Pre-proof 10

E. HAUSENBLAS AND PAUL A. RAZAFIMANDIMBY

Proof. For simplicity, let us assume T = 1. In addition, since in our example the underlying function space H = E is a Hilbert space, we denote instead of E the underlying function space by H . In addition, the control space Lr ([0, T ]; U ) is given by Lr ([0, T ]; Hd ). As in the previous section (see (8)) we consider the solution operator RT : H × L2 ([0, T ]; Hd ) → H ,

(17)

which takes each function v ∈ L2 ([0, T ]; Hd ) and initial condition u0 ∈ H to the solution u(T, u0 ) of the control system (7) associated to the Navier-Stokes equations. It is proved in [2, Proposition A.2], see also [24], that the operator RT is analytic. It is also known from [2, Proposition A.5 ], see also [3], that the system (7) for the Navier-Stokes is solid controllable in time T for any finite dimensional space F ⊂ H . In particular, Assumptions (A1) and (A2) of Theorem 3.1 are satisfied. Hereafter we respectively identify Hd and F to Rd and Rdim F . Let p ∈ (1, 2) such that (15) is satisfied. Let p0 be the conjugate exponent to p and s < p1 − 1. For each j ∈ K let ξj be the map defined by Z 1 φ(τ ) dlj (τ ) ∈ L0 (Ω; R). ([0, 1], R) 3 φ → 7 ξ (φ) = ξj : Bp−s 0 ,p0 j 0

roo

f

To see that ξj is well defined, let us assume for the time being that νj has finite measure. Let σj = νj (R), let Nj be a Poisson distributed random variable with parameter σ and {τnj : n = 1, . . . , Nj } be a family of independent and on [0, 1] uniform distributed random variables. Let {Ykj : k = 1, . . . , Nj } be a family of R–valued random variable distributed according to νj /σj . Now, l˙j = {l˙j (t) : t ∈ [0, 1]} can be represented as (see [9, Section 3.2]) ( 0 for Nj = 0, l˙j = PNj (18) j for Nj > 0. k=1 Yk δτ j

rep

k

Bp−s 0 ,p0 ([0, 1]),

Due to the embedding Cb ([0, 1]) ,→ we know that any φ ∈ Bp−s 0 ,p0 ([0, 1]) is continuous. It follows by the definition of the Dirac distribution that Z 1 φ(r)δτ (r) dr = φ(τ ), τ ∈ [0, 1]. 0

( 0 ξj (φ) = PNj

lP

Therefore, we have

for Nj = 0, for Nj > 0.

j j k=1 Zk φ(τk )

(19)

rna

In addition, since R is a Banach space of type p (see also Proposition 7.1.16 [21], or [27]), we observe that Nj Z 1 Z X
j p j p p p E|ξj (φ)| = E |Zk | |φ(τk )| = |φ| (τ ) dτ |z|p νj (dz). (20) R

0

k=1

Jo u

Now, suppose that νj is only σ–finite. Let {m : m ∈ IN} be a sequence such that m > 0 and j j m ↓ 0. For any m ∈ IN, let νm be the L´evy measure defined by νm (U ) := νj (U \ [−m , m ]), j j j j U ∈ B(R), σm = νm (R), Nm be a Poisson distributed random variable with parameter σm , and j,m j {τn : n = 1, . . . , Nm } be a family of independent random variables uniformly distributed on [0, 1]. j j Let {Zkj,m : k = 1, . . . , Nm } be a family of R–valued random variable distributed along νjm /σm . Then m ξj (φ) be defined as in (19). In addition, we have by (20) Z 1 Z
j E|ξjm (φ)|p ≤ |φ(τ )|p dτ |z|p νm (dz). 0

R

Journal Pre-proof IRREDUCIBILITY AND THE 2-D STOCHASTIC NAVIER STOKES

11

By Theorem 2.5.2 [4], hence, and since the family of random variables {ξjm (φ) : m ∈ IN} is uniformly integrable in Lq (Ω; R) for q < p, and, therefore, the limit ξj (φ) := lim ξjm (φ) m→∞

exists. s ([0, 1], R) defined by Next, let µj be the cylindrical measure µj on Bp,p 
s µj x ∈ Bp,p ([0, 1]) : (x(φ1 ), . . . , x(φn )) ∈ C := P ((ξ(φ1 ), . . . , ξ(φn )) ∈ C) , C ∈ B(Rn ),

roo

f

−s ([0, 1], R). In Proposition 5.1 (see also 5.2) we show that the cylindrical where n ∈ IN, φ1 , . . . , φn ∈ Bp,p s ([0, 1]). measure µ is actually a Radon measure on Bp,p From the results of Section 5 we infer that for all s < p1 − 1 the probability measure µj on s Bp,p ([0, 1], R) is decomposable with decomposition {Fn , Gn , ln }∞ n=0 , where the decomposition comes from the multiresolution analysis generated by Debauchie wavelets. In particular, Fn and ln are respectively defined by F0 = V0 , Fn = Wn , n ≥ 2, where V0 and Wn are defined in (21), the space Gn is defined as in Definition 2.1. The existence of a density for ln is given by Lemma 5.3. In addition, with Lemma 5.7 we know that the decomposition {Fn , Gn , ln }∞ n=0 satisfies Assumption 2.1 and Assumption 2.3. In addition, we know by Triebel [36, Theorem 1.58] that the wavelets chosen to construct s {Fn , Gn , ln }∞ n=0 form an unconditionally basis of Bp,p ([0, 1], R). With these observation in mind, it is not difficult to check that the product measure µ = ⊗j∈K µj s ([0, 1], Rd ) where d = satisfies Assumption 2.2 and Assumption 2.3 on the Banach space E := Bp,p dim(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 random process entering the system is a fractional Brownian motion given by (14).

rep

Theorem 4.2. Let K be a saturating set and assume that the process Ξ is a fractional Brownian motion defined by (14) with Hurst parameter H ∈ ( 12 , 1). Let u = {u(t, u0 ) : t ≥ 0, u0 ∈ H} be the unique solution of system (6) with initial condition u0 . Then, for any finite dimensional space F ⊂ H and initial condition u0 ∈ H , there exists a density ρu0 : F → R+ 0 such that Z E1O (πF u(T, u0 )) = ρu0 (x) LebF (dx). O

lP

In addition, for any sequence {un : n ∈ IN} with un → u0 ∈ H as n → ∞, we have Z |ρu0 (x) − ρun (x)|dx−→0 as n → ∞. F

L2 ([0, T ]; Hd )

Jo u

rna

Proof. Let RT : H × → H be the solution operator defined by (17) in the proof of Theorem 4.1. It satisfies the properties enumerated in the proof of Theorem 4.1. Hereafter we respectively identify Hd and F to Rd and Rdim F . Let s ∈ (− 12 , H − 1). For each j ∈ K let ξj be the map defined by Z 1 −s ξj : B2,2 ([0, 1]) 3 φ 7→ ξj (φ) = φ(τ ) dβjH (τ ) ∈ L2 (Ω; R), 0

s ([0, 1], R) B2,2

and µj be the cylindrical measure on defined by 
s µj x ∈ B2,2 ([0, 1]) : (x(φ1 ), . . . , x(φn )) ∈ C := P ((ξ(φ1 ), . . . , ξ(φn )) ∈ C) , C ∈ B(Rn ),

where n ∈ IN, φ1 , . . . , φn ∈ S(R). From the results of Section 6 we infer that the cylindrical meas ([0, 1], R) is actually a probability measure and is decomposable with decomposition sure µj on B2,2 n ∞ {Fn , G , ln }n=0 , where the decomposition comes by the multiresolution analysis generated by Debauchie wavelets. In particular, Fn and ln are respectively defined by F0 = V0 , Fn = Wn , n ≥ 2, where V0 and Wn are defined in (36). With Fn in mind we define Gn as in Definition 2.1. We also infer from

Journal Pre-proof 12

E. HAUSENBLAS AND PAUL A. RAZAFIMANDIMBY

Lemma 5.7 that for each j ∈ IN the probability measure µj 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. ´vy Noise and its Wavelet Expansion 5. The Le In this section we assume that we are given a real-valued L´evy process ` with σ-additive L´evy measure ν on R\{0} satisfying (15), i.e. Z |z|p ν(dz) < ∞, |z|≤1

roo

f

for some p ∈ (1, 2). As space discretization we chose a multiresolution analysis generated by Daubechies wavelets. Our aim is to investigate the expansion of the process ` in terms of Debauchies wavelets of order k > max(s, (1 − p1 )+ − s), see for e.g. [10] for the exact definition. To introduce the space discretization {Fn : n ∈ IN} let us introduce the multiresolution analysis. A multiresolution analysis is a sequence of subspaces {Vn : n ∈ Z} satisfying the following properties • Vn ⊂ Vn+1 , n ∈ Z; • ∪n∈Z Vn = L2 (Rd ); • ∩n∈Z Vn = {0}; • if f ∈ Vn , then f (2·) ∈ Vn+1 ; • if f ∈ Vn , then f (· − k) ∈ Vn , k ∈ Z; In addition, there exists a, so called, scaling function φ such that Vn is generated by translations and scaling, i.e. j φj,k := 2− 2 φ(2j t + k) and (21)

Wn := span{ψn,k : k ∈ Jnψ }

(22)

rep

Vn := span{φj,k : j = 1, . . . , n, k ∈ Jjφ }.

Also, there exists a so called mother wavelet ψ, such that Vn = Vn−1 ⊕ Wn−1 with and

j

lP

ψj,k := 2− 2 φ(2j t + k), j ∈ IN, k ∈ Jj . Observe, that due to the scaling, there exists coefficients {pk : k ∈ Z} such that X φ(x) = pk φ(2x + k), x ∈ R.

(23)

k∈Z

Jo u

rna

The reason of our choice is that Daubechies wavelets have a compact support in space. In addition, the number of coefficients in (23), which are not equal to zero, i.e. ]{k : pk 6= 0}, is finite. If the Daubechies wavelet is sufficiently smooth, i.e. the order of the Daubechies wavelet k satisfies s ([0, 1]) (see k > max(s, (1 − p1 )+ − s), the multiresolution analysis forms an unconditional basis in Bp,p [36, Theorem 1.58]). To keep this section and the article short we refer the reader for an introduction to wavelets to [10, 22] or [36]. Note that for s ∈ R the Daubechies wavelets of order k, with k > max(s, (1 − p1 )+ − s), form an s ([0, 1]). In particular, for each element f ∈ B s ([0, 1]) there exists a unique unconditional basis of Bp,p p,p sequence {λj,k : j ∈ IN, k ∈ Jjψ } such that f can be written as X X f= λj,k ψj,k + λ0 φ. (24) j∈IN k∈J ψ j

Journal Pre-proof IRREDUCIBILITY AND THE 2-D STOCHASTIC NAVIER STOKES

13

Note that since we are considering the process on the time interval [0, 1], we only need to sum over We also note that |Jjψ | ∼ 2j .

Jjψ .

In the next paragraph, we will construct the probability measure induced by a L´evy process which will be represented as an integral with respect to a Poisson random measure. This representation is motivated on one hand by the fact that the use of Poisson random measure simplifies many calculation. On the other hand, the Poisson random measure framework seems more general. We refer to [5], [30, Chapters 6-8] and [34, Chapter 4] for a precise connection between Poisson random measures and L´evy processes and stochastic integration with respect to them. Over a probability space A = (Ω, F, P), we consider a time homogenous Poisson random measure η on R with symmetric intensity measure ν as above. Proposition 5.1. The Poisson random measure η over a probability space (Ω, F, P) induces a Radon s ([0, 1]) with s < 1 − 1. probability measure µ on Bp,p p Proof. We will start the proof with removing jumps of size bigger than  ∈ (0, 1] and let  converge to 0. For this purpose we take an arbitrary constant  ∈ (0, 1] and define a Poisson random measure η by ¯ η : B(R) × B([0, 1]) → IN,

f

(A × I) 7→ η(A ∩ (R \ [−, ]) × I).

rep

roo

First we will show that the family {η :  ∈ (0, 1]} induces a family of cylindrical measures on Cb ([0, 1])0 . Here, it is important that the to the Poisson random measure η corresponding L´evy process can be written as a sum over finitely many jumps at random jump times. To be more precise, let ν be defined by ν (A) = ν(A ∩ (R \ [−, ])), ρ = ν(R \ [−, ]), let N be a Poisson distributed random variable with parameter ρ , {τn : n = 1, . . . , N } be a family of independent uniform distributed random variables on [0, 1], and {Yn : n = 1, . . . , N } be a family of independent, ν /ρ distributed random variables. Denoting δx the Dirac distribution concentrated at x, the Poisson random measure η can be written as N X

δτn (I) δYn (A),

n=1

lP

η (A × I) =

A ∈ B(Rd ), I ∈ B([0, 1]).

In addition, for any f ∈ Cb ([0, 1]) the mapping Z 1Z N X ξ (f ) := f (s)zη (dz, ds) = f (τn )Yn R

is well defined.

rna

0

n=1

and

Jo u

Let us define the random variables Z 1Z  ζj,k = ψj,k (τ ) zη (dz, dτ ), 0

R

a0 =

Z

0

1Z

R

j ∈ IN, k ∈ Jj ,

φ0,0 (τ ) zη (dz, dτ ).

Since the mother wavelet ψ and the scaling function φ are supposed to be continuous, the families  : j ∈ IN, k ∈ J } ∪ {a } of random variables over A are well defined. In addition, by the definition {ζj,k j 0 s ([0, 1]), and of ζ  and a0 and the fact that the multiresolution analysis is a Schauder basis in Bp,p s δ ∈ Bp,p ([0, 1]) (see [33, Remark 3, p. 34]), we infer that ξ admits a wavelet series representation as in (24).

Journal Pre-proof 14

E. HAUSENBLAS AND PAUL A. RAZAFIMANDIMBY

Note that for any C ∈ B(R),

s  µ ({x ∈ Bp,p ([0, 1]) : x(ψj,k ) ∈ C}) = P(ζj,k ∈ C).

Later on we will need the following proposition which will be proved at the end of the current proof. Proposition 5.2. Let ν be a L´evy measure satisfying (15) for some p ∈ [1, 2) and  ∈ (0, 1]. Let ∞ X X  ξ := ζk,j ψj,k + a0 φ0,0 . j=1 k∈J ψ j

Then, (1) for any s <

1 p

(2) for any s <

1 p

− 1, there exists a constant C > 0 such that h i E |ξ |pBp,p ≤ C. s

− 1, there exists a constant C > 0 such that for any 1 , 2 ∈ (0, 1] we have h i E |ξ1 − ξ2 |pBp,p ≤ C min(1 , 2 )2−p . s

roo

f

By the choice of s and p, we have Bp−s 0 ,p0 ([0, 1]) ,→ Cb ([0, 1]) and η is a finite measure. Secondly, the s ([0, 1]) defined by mappings ξ induces a family of cylindrical measures µ on Bp,p 
s µ x ∈ Bp,p ([0, 1]) : (x(φ1 ), . . . , x(φn )) ∈ C := P ((ξ (φ1 ), . . . , ξ (φn )) ∈ C) , s ([0, 1]))0 = B −s ([0, 1]), and C ∈ B(Rn ). φ1 , · · · , φn ∈ (Bp,p p0 ,p0

rep

We will now show that the family of cylindrical measures {µ :  ∈ (0, 1]} has a limit in the set s ([0, 1]). In fact, the family of probability measures µ is tight on of probability measures on Bp,p  s ([0, 1]). To show this claim we fix a constant s ∈ (s, 1 − 1). We firstly note that the embedding Bp,p 0 p s0 ([0, 1]) ,→ B s ([0, 1]) is compact. Secondly, the Chebyscheff inequality and Proposition 5.2 give Bp,p p,p s ([0, 1]) : |x| s ≤ δ −1/p } such that that for any δ > 0 we can find a compact Kδ := {x ∈ Bp,p 0 Bp,p i h P (ξ 6∈ Kδ ) ≤ δ E |ξ |pB s0 ≤ Cδ. p,p

rna

lP

s ([0, 1]). It even It follows that the family of probability measures {µ :  ∈ (0, 1]} is tight on Bp,p follows from Proposition 5.2 that the sequence {µn :  = n1 } forms a Cauchy sequence and the limit s ([0, 1]). Since there exists µ is unique. Therefore, there exists a unique cylindrical measure µ on Bp,p p a constant C > 0 such that for all  > 0 we have E[|ξ |Bp,p s ] ≤ C, it follows from the Lebesgue p Dominated Convergence Theorem that E[|ξ|Bp,p s ] ≤ C. Hence, µ is also a Radon probability measure s on Bp,p ([0, 1]).

Now we shall consider the general case in which ν is assumed to satisfy (15) for some p ∈ (1, 2). For this purpose we consider the Poisson random measures η1 and η2 defined by

Jo u

and

B([0, ∞)) × B(R) 3 (I × A) 7→ η1 (I × A) := η(I × A ∩ [−1, 1])

B([0, ∞)) × B(R) 3 (I × A) 7→ η2 (I × A) := η(I × A ∩ R \ [−1, 1]), respectively. Since A ∩ [−1, 1] ∩ A ∩ R \ [−1, 1] = ∅, the Poisson random measures η1 and η2 are independent. Hence, the two families of coefficients in the wavelet expansion η1 and η2 are independent s ([0, 1]). too. In addition from the first part of the proof η1 induces a Radon probability measure on Bp,p Since the process Z Z L2t :=

t

0

R

zη2 (dz, ds)

Journal Pre-proof IRREDUCIBILITY AND THE 2-D STOCHASTIC NAVIER STOKES

15

can be written as a finite sum over jumps happen at random times within the interval [0, 1], L˙ 2t consist s ([0, 1]), L ˙ 2t of a sum over finitely many Dirac distributions. Since any Dirac distributions belong to Bp,p s s is an element of Bp,p ([0, 1]) and induces a probability measure on Bp,p ([0, 1]). Hence, η itself induces s ([0, 1]). a Radon probability measure on Bp,p R Proof of Proposition 5.2: We recall that |z|p ν(dz) < ∞ for some p ∈ (1, 2). By the definition of the norm we get ∞ X X p p j(s− p1 )p p ζ  2j 2 . = E E|ξ |Bp,p 2 s k,j j=1

Since

 p E|ζj,k |

≤ Cν

Z

k∈Jjψ

1

0

jp

|ψj,k (s)|p ds = 2 2 2−j ,

we infer that there exists a constant C > 0 such that ∞ ∞ X X p p j(ps−1) j j( p2 −1) j p2 2 ≤ C 2j(ps+ 2 −1+ 2 ) , E|ξ |pBp,p ≤ C 2 2 2 s j=1

j=1

f

− 1.

roo

which is finite for s <

1 p

s ([0, 1]) by µ and let us define the Let us denote the Radon probability measure induced by η on Bp,p mapping Z 1Z φ(τ ) z η(dz, dτ ). (25) ([0, 1]) 3 φ → 7 ξ(φ) = ξ : Bp−s 0 ,p0

rep

0

R

This mapping is well defined thanks to the above calculation.

lP

We are now interested in the properties of the decomposition of µ by the multiresolution analysis. In particular, we will show that for any n ∈ IN, the probability measure µGn is equivalent to the Lebesgue measure. Let us put Gn = Vn ,

n ∈ IN.

(26)

Jo u

which implies that

rna

We firstly note that since Vn+1 = Wn ⊗ Wn−1 ⊗ · · · ⊗ W1 ⊗ V0 , given the coefficients {ζj,k : j = 1, . . . , n, k ∈ Jjψ } ∪ {a0 }, one knows the coefficient of φn+1,k . For k ∈ Jnφ let us denote γn,k the coefficients of φn,k . In particular, we have Z 1Z γn,k := φn,k (t)zη(dz, dt), R

0

π Gn ξ =

X

γn,k φn,k .

k∈Jnφ

Let us now denote by zn and gn the random vectors (ζn,0 , ζn,1 , . . . , ζn,|J φ | ) and (γn,1 , γn,2 , . . . , γn,|J ψ | ), n n respectively. Finally, for a function f : [0, 1] → R we write Z 1Z ξ(f ) := f (s)z η(dz, ds). 0

R

Lemma 5.1. Let f : [0, 1] → R be a function such that there exists constants δ > 0 and t1 , t2 ∈ [0, 1], t1 < t2 such that |f (t)| ≥ δ for all t ∈ [t1 , t2 ]. Then

Journal Pre-proof 16

E. HAUSENBLAS AND PAUL A. RAZAFIMANDIMBY

(1) supp(ξ(f )) = R; (2) the law of ξ(f ) is absolutely continuous with respect to the Lebesgue measure. Proof. Let us define the following L´evy measure Z νt1 ,t2 : B(R) 3 B 7→

t2

t1

Z

R

1B (f (t)z)ν(dz) dt.

Then ξ(f 1[t1 ,t2 ] ) is an infinite divisible random variable. Item (i) follows from [34, Corollary 24.4]. Item (ii) follows from [34, Theorem 27.7]. Lemma 5.2. For any n ≥ 1, the measure φ

B(R|Jn+1 | ) 3 U 7→ P gn+1 ∈ U φ | |Jn+1

is equivalent to the Lebesgue measure on R

.




(27)

Proof. This follows from the fact that the scaling function φ has compact support and, therefore, that φ φ for all k = min(Jn+1 ), . . . , max(Jn+1 ) − 1 the set supp(φn+1,k ) ∪ supp(φn+1,k+1 ) \ (supp(φn+1,k ) ∪ supp(φn+1,k+1 ))

roo

f

has non–empty interior. Let us write φn+1,k+1 = f1 + f2 with supp(f1 ) ∩ supp(φn+1,k ) = ∅, supp(f1 ) is an interval [a, b], {s : f2 (s) > 0} ∩ [a, b] = ∅, and f1 is bounded away from zero. Then ξ(f1 ) and ξ(f2 ) are independent, and so are ξ(f1 ) and ξ(φn+1,k ). In addition, by Lemma 5.1 the law of ξ(f1 ) is equivalent to the Lebesgue measure. Hence, from [34, Lemma 27.1-(iii)] it follows that the law of the sum of the random variables ξ(f1 ) and ξ(f2 + φn+1,k+1 ) is also equivalent to the Lebesgue measure. φ ). Now, one easily prove the assertion by an induction starting at k = min(Jn+1 ψ

Lemma 5.3. For each U ∈ B(Gn ) and y ∈ R|Jn | , the conditioned measure

rep

B(|Jnψ |) 3 U 7→ ln (y, U ) = P (zn ∈ U | gn = y)

(28)

is equivalent to the Lebesgue measure.

lP

Proof. First, note that given the scaling function φ there exists coefficients {pj : j = 1, . . . , u}, where u is the order of the Daubechies wavelet, such that (see (23)) u X φ(x) = pj φ(2x + j), x ∈ R. (29) j=1

rna

In addition, we have the following representation u X φ(x) = (−1)j pj ψ(2x + j), j=1

x ∈ R.

(30)

Jo u

Because of the orthogonality of the wavelet basis we additionally have that ( k X √1 for l = j, 2 pj p¯j+2l = 0 for l 6= j. j=1

(31)

Let us now consider the mapping

φ

I : Vn+1 3 f 7→ (fn+1,1 , . . . , fn+1,2n+1 ) ∈ R|Jn+1 | ,

where fn+1,k = φn+1,k (f ). It is not difficult to show that I is an isomorphism from Vn+1 onto φ

R|Jn+1 | . We note that since Vn+1 = Vn ⊗ Wn , it follows from (29) that there exists a linear mapping T : Vn+1 → Vn which induces a mapping φ

φ

T : R|Jn+1 | → R|Jn | .

Journal Pre-proof IRREDUCIBILITY AND THE 2-D STOCHASTIC NAVIER STOKES

17

We can also define a mapping S : Vn+1 → Wn by Sx := πWn (I − T )x. As above we can also find a linear mapping S : Vn+1 → Wn inducing a mapping φ

ψ

S : R|Jn+1 | → R|Jn | .

Since Vn+1 = Vn ⊗ Wn , we have I −1 ker(S) = Vn and I −1 ker(T ) = Wn . Hence, from the Bayes formula we infer that
P πWn I −1 S −1 x + πVn I −1 S −1 y P (ζ = x | γ = y) = , P (I −1 S −1 y)

ψ φ for any x ∈ R|Jn | and y ∈ R|Jn | . By Lemma 5.2 P I −1 S −1 y > 0 and P πWn I −1 S −1 x + πVn I −1 S −1 y > 0. In particular, there exists a density hn (x, y) = P (ζ = x | γ = y) ,

such that ln (y, U ) =

Z

hn (x, y) dx,

U

ψ

φ

and hn (x, y) > 0 for all x ∈ R|Jn | and y ∈ R|Jn | .

f

In order to verify Assumption 2.3 for a point Y we will show in the following Lemma that for all n ∈ IN, πGn 0 belongs to the support of the measure µ. If this holds, we can set Y = 0.

roo

Lemma 5.4. Let α ∈ (0, 2), 1 ≤ p < α and s < p1 − 1. Let ν be a σ–finite symmetric measure on R \ {0} such that there exists a number α ∈ (0, 2] such that ν(R \ [−, ]) ∼ −α l()

as

 → 0,

rep

for some slow varying function l. Let η be the to ν corresponding Poisson random measure over the s (R) induced probability measure. probability space (Ω, F, P). Let µ be the from ξ defined in (25) on Bp,p Then for any  > 0,   s ≤ } > 0. µ {x ∈ Bp,p (0, 1) : |x|pBp,p s

Proof. Let L be given by

lP

L(t) :=

Z tZ

From [6], Example 2.2 we know that

rna

R

0

− log P



z η(dz, ds),

t ∈ [0, 1].

 sup |L(t)| ≤  ∼ Kα ,

0≤t≤1

Jo u

hence, for any ˜ > 0 there exists a δ > 0 such that   P sup |L(t)| ≤ ˜ ≥ δ. 0≤t≤1

Let ˜ > 0 be a constant to be chosen later and let us set   Ω˜ := sup |L(t)| ≤ ˜ . Then,

0≤t≤1

      ≤  | Ω \ Ω˜ P (ω \ Ω˜) P |ξ|pBp,p ≤ = P |ξ|pBp,p ≤  | Ω˜ P (Ω˜) + P |ξ|pBp,p s s s     ≥ P |ξ|pBp,p ≤  | Ω˜ P (Ω˜) ≥ δP |ξ|pBp,p ≤  | Ω˜ . s s

(32)

Journal Pre-proof 18

E. HAUSENBLAS AND PAUL A. RAZAFIMANDIMBY

Note that on Ω˜ the jump size of the process is less than 2˜ . Hence Z 1Z ψj,k (s)1|z|≤2˜ η(dz, ds) E [|ζj,k |p | Ω˜] ≤ E 0 R Z (2˜ )p−α ( p −1)j (2˜ )p−α 1 |ψj,k (s)|p ds ≤ Cp ≤ 2 2 , p−α 0 p−α and

  ∞ 1 X X jp j(s− p )p   ≤ E 2 |ζj,k |p 2 2 | Ω˜

i

h

(33)

| Ω˜ E |ξ|pBp,p s

j=0

≤ Cp

k∈Jjψ

∞ (2˜ )p−α X j(s− p1 )p X ( p −1)j jp (2˜ )p−α 2 2 2 2 2 ≤ C˜p . p−α p−α ψ j=0

k∈Jj

From these calculations we infer that     p >  | Ω P |ξ|pBp,p ≤  | Ω = 1 − P |ξ| s s ˜ ˜ Bp,p

f



(2˜ )p−α ≥ 1 − C˜p . (p − α)

roo

≥1−

E|ξ|pBp,p s

Now, choosing ˜ such that

(2˜ )p−α 1 C˜p = , (p − α) 2

 1  ≤  | Ω˜ ≥ , P |ξ|pBp,p s 2

rep

we infer that

from which the assertion follows.

lP

s ([0, 1]) we define the conditional probability µ( · | D) by For any D ∈ B(Bp,p ( µ(U ∩D) if µ(D) > 0, s µ(D) B(Bp,p ([0, 1])) 3 U 7→ µ(U | D) := 1 if µ(D) = 0.

rna

Lemma 5.5. Let α ∈ (0, 2), 1 ≤ p < α and s < p1 − 1. Let ν be a σ–finite symmetric measure on R \ {0} such that there exists a number α ∈ (0, 2] such that ν(R \ [−, ]) ∼ −α l()

as

 → 0,

Jo u

for some slow varying function l. Let η also be the Poisson random measure, over the probability space (Ω, F, P), associated to the s ([0, 1]) induced by the mapping ξ defined in L´evy measure ν. Let µ be the probability measure on Bp,p (25). Then, for any R > 0, x ∈ BE (R, 0) and  > 0 there exist n ∈ IN and some δ > 0 such that   s µ {x ∈ Bp,p ([0, 1]) : |πGn x|pBp,p ≤ } > δ. s

Proof. From Lemma 5.4 we infer that there exists a constant δ > 0 such that for P (DΩ ) ≥ δ,

Journal Pre-proof IRREDUCIBILITY AND THE 2-D STOCHASTIC NAVIER STOKES

19

 where DΩ := sup0≤t≤1 |L(t)| ≤ 1 and L = {L(t) : t ∈ [0, 1]} is defined in (32), Observe that the set D := ξ(DΩ ) satisfies µ(D) ≥ δ. Thus,   s ≤ } µ {x ∈ Bp,p ([0, 1]) : |πGn x|pBp,p s     s s ≤ } | D . ≤ } | D µ (D) ≥ δ · µ {x ∈ Bp,p : |πGn ξ|pBp,p ≥ µ {x ∈ Bp,p : |πGn ξ|pBp,p s s

Now, from (33) we infer that   ∞ i h X j(s− 1 )p X jp   p ≤ E 1Ω˜ E |πGn ξ|pBp,p 2 |ζj,k |p 2 2  s 1Ω ˜ j=n+1

≤ Cp Therefore,

k∈Jjψ

∞ ∞ X (2˜ )p−α X j(sp+p−1) 2 ≤ C˜p 2n(sp+p−1) 2j(sp+p−1) ≤ Cˆp 2n(sp+p−1) . p−α j=n+1

j=0

    P |πGn ξ|pBp,p ≤  | Ω˜ = 1 − P |πGn ξ|pBp,p >  | Ω˜ s s E|πGn ξ|pBp,p s

roo

f

≥ 1 − Cˆp 2n(sp+p−1) /.  For any κ < 1 there exists a number n ∈ IN sufficiently large, such that ≥ 1−

Cˆp 2n(sp+p−1) / ≤ 1 − κ.

rep

which gives the assertion.

Lemma 5.6. Let α ∈ (0, 2), 1 ≤ p < α and s < p1 − 1. Let ν be a σ–finite symmetric measure on R \ {0} such that there exists a number α ∈ (0, 2] such that νj (R \ [−, ]) ∼ −α l()

as

 → 0,

lP

for some slow varying function l. Then, for all N ∈ IN, x0 ∈ GN , and all  > 0, there exists a δ > 0 such that n o s s µ x ∈ Bp,p ([0, 1]) | |x − x0 |Bp,p ≤ ≥ δ.

Jo u

rna

Proof. Let  > 0 be a fixed constant and s < s0 < p1 − 1. From Lemma 5.5 we deduce that there exist n0 ∈ IN and δ2 > 0 such that n  o s s0 ≤ µ x ∈ Bp,p ([0, 1]) | |πGn0 x|Bp,p ≥ δ2 . 4 Then n o s s µ x ∈ Bp,p ([0, 1]) | |x0 − x|Bp,p ≤ n o n  o s s s0 ≤ s ≥ µ x ∈ Bp,p ([0, 1]) | |x0 − πGn0 x|Bp,p ≤ ∩ x ∈ Bp,p ([0, 1]) | |πGn0 x|Bp,p . 4 4 n o s ([0, 1]) | |x − π s We now set An0 = x ∈ Bp,p ≤ 4 and observe that for γ = δ2 /2 > 0 there 0 Gn0 x|Bp,p exists a closed set Cγ ⊂ Gn0 such that µn0 (Gn0 \ Cγ ) ≤ γ and the function Cγ 3 y 7→ ln0 (y, An0 ) ∈ [0, 1]

Journal Pre-proof 20

E. HAUSENBLAS AND PAUL A. RAZAFIMANDIMBY

is continuous. Furthermore, since for all y ∈ Gn0 µ a.s. ln (y, ·) is equivalent to the Lebesgue measure s0 ([0, 1]) ,→ B s ([0, 1]) is and LebGn0 (An0 ) > 0, we have ln0 (y, An0 ) > 0. Since the embedding Bp,p p,p compact, n o s s0 ≤ ([0, 1]) | |πGn0 x|Bp,p Cn0 = x ∈ Bp,p ∩ Cγ 4 is a compact subset of Gn0 and there exists a δ3 > 0 such that for all y ∈ Cn0 ∩ Cγ , ln0 (y, An0 ) ≥ δ3 . From the above consideration we now infer that Z 
s µ x ∈ Bp,p ([0, 1]) | |x − x0 | ≤  ≥ ln0 (y, An0 ) µn0 (dy)  {|πGn0 x|B s0 ≤ 4 }∩Cγ p,p

n o   s0 ≤ |πGn0 x|Bp,p ∩ C ≥ δ3 µn0 γ 4   n o  n0  s0 ≤ ≥ δ3 1 − µn0 Gn0 \ |πGn0 x|Bp,p ∪ (G \ C ) γ 4 ≥ δ3 (1 − (1 − δ2 + γ)) = δ3

δ2 . 2

The above discussion is summarized in the following lemma.

roo

f

Lemma 5.7. Let η be a time homogeneous Poisson random measure on R over a probability space (Ω, F, P). We assume that the L´evy measure ν associated to η is symmetric, σ-additive, absolutely continuous with respect to the Lebesgue measure on R \ {0}. In addition, we assume, that there exists some p ∈ (1, 2) with Z |z|≤1

|z|p ν(dz) < ∞

rep

and there exists a number α ∈ (0, 2] such that

ν(R \ [−, ]) ∼ −α l()

as

 → 0,

lP

for some slow varying function l. s ([0, 1]) described in Section 5. Then, Let {φj,k : j ∈ IN : k = 1, . . . , 2j } be the wavelet basis in Bp,p s the measure µ induced by the map ξ defined by (25) on Bp,p ([0, 1]) is decomposable with decomposition {Fn , Gn , ln }∞ n=0 satisfying Assumption 2.2 and Assumption 2.3. Here the spaces Fn are defined by F0 = V0 , Fn = Wn , n ≥ 2, V0 and Wn are defined in (21), and ln is defined in (28).

rna

Proof. The decomposability follows from the fact the wavelet basis described in section 5 is a Schauder s ([0, 1]). Assumption 2.3-(1) follows from choosing Y = (0, 0, . . .) and from the fact that basis of Bp,p
P πGj+1 x ∈ · | πGj x = y is equivalent to the Lebesgue measure and that for any y ∈ R we have (see Lemma 5.3)
P πGj+1 x ∈ · | πGj x = y > 0.

Jo u

Using an induction argument one can easily show that for any open set in Gn µGn (O) > 0 from which it follows that µGn is absolutely continuous with respect to LebGn . Finally, Assumption 2.3-(2) follows from Lemma 5.6. 6. The Fractional Brownian Noise and its Wavelet Expansion

Let B H = {B H (t) : t ≥ 0} a the fractional Brownian motion with Hurst parameter H ∈ ( 12 , 1). Let us fix s ∈ (− 12 , H − 1) and consider the mapping Z 1 ξ H : S([0, 1]) 3 φ 7→ ξ H (φ) = φ(t) dB H (t). 0

Journal Pre-proof IRREDUCIBILITY AND THE 2-D STOCHASTIC NAVIER STOKES

21

For all n ∈ IN, φ1 , . . . , φn ∈ S(R), C ∈ B(Rn ) we set 
µ x ∈ S 0 ([0, 1]) : (x(φ1 ), . . . , x(φn )) ∈ C := P ((ξ(φ1 ), . . . , ξ(φn )) ∈ C) . We will firstly show that this measure µ is a Radon measure on consider the Haar wavelet ψ defined by   for t ∈ [0, 21 ), 1; ψ(t) := −1; for t ∈ [ 12 , 1],   0 elsewhere,

s ([0, 1]). B2,2

(34)

For this aim, let us

and the scaling function φ defined by

( 1; φ(t) := 0;

for t ∈ [0, 1], elsewhere.

Also, we set j

j

ψj,k := 2− 2 ψ(2j t + k) and φj,k := 2− 2 φ(2j t + k), j = 1, . . . , n, k = 1, . . . , 2j ,

(35)

to which we associate the multiresolution analysis Vn := span{φj,k : j = 1, . . . , n, k = 1, . . . , 2j },

Wn := span{ψn,k : k = 1, . . . , 2n }.

Lp ([0, 1])

(36)

s ([0, 1]) Bp,q

roo

f

The Haar wavelet is an unconditional basis in with 1 < p < ∞, a basis in for 1 1 1 1 s 1 < p < ∞ and p − 1 < s < p , and a basis for Bp,p ([0, 1]), 2 < p ≤ 1 and p − 1 < s < 1 (see Triebel [36, Theorem 1.58]).

rep

Now, let F0 := V0 , Fn = Wn , n ∈ IN, and Gn := F0 ⊕ F1 ⊕ · · · ⊕ Fn , and Fn := σ(F0 ⊕ F1 ⊕ . . . ⊕ Fn ). Let us denote the projection of ξ onto Gn by Pn and onto Wn by Qn . For the time being let us assume s ([0, 1]). Since that the Radon-Nikodym derivative of the fractional Brownian motion belongs to B2,2 s ([0, 1]), then for each element x ∈ E there exists a unique the Haar wavelets are a basis of E := B2,2 j sequence {λj,k : j ∈ IN, k = 1, . . . , 2 } such that n

x=

2 XX

λj,k ψj,k + λ0 φ.

lP

j∈IN k=1

Observe also that

rna

ξ(t) :=

j

∞ X 2 X

ζj,k ψj,k (t) + a0 φ(t),

j=1 k=1

t ∈ [0, 1].

and

Jo u

where {ζj,i : j ∈ IN, i = 1, . . . , 2j } is a family of random variables defined by Z 1 d ζj,k = ψj,k (s) dB H (s), 0

d

a0 =

Z

1

φ0,0 (s) dB H (s).

0

In fact, given the coefficients {ζj,k : j = 1, . . . , n, k = 1, . . . , 2j } ∪ {a0 }, one know the coefficient of φn,k , k = 1, . . . , 2n . Since Gn consists of all functions f : [0, 1) → R that are constant on the intervals [2−n k, 2−n (k + 1)), k = 1, . . . , 2n − 1, there exists random coefficients γn,k , k = 1, . . . , 2n − 1 such that Pn ξ =

n −1 2X

k=0

γn,k φn,k .

Journal Pre-proof 22

E. HAUSENBLAS AND PAUL A. RAZAFIMANDIMBY

It is now easy to see show that γn,k :=

Z

1

φn,k (t)dB H (t).

0

Since for two functions φ, ψ : [0, 1] → R, the random variables ξ H (φ) and ξ H (ψ) are Gaussian distributed with covariance Z 1Z 1  H  H φ(s)φ(t) |t − s|2H−2 dt ds, E ξ (φ) ξ (ψ) = 0

straightforward calculations gives for l 6= k Z 2−j (k+1) Z  H  H j E ξ (ψj,k ) ξ (ψj,l ) = 2 2−j k

0

2−j (l+1)

2−j l

ψj,k (s)ψj,l (t)|t − s|2H−2 dt ds

Z 2−j (k+1)   1 = 2 (t − 2−j l)2H−1 − (t − 2−j (l + 1)) dt 2H − 1 2−j k 1 1  −j = 2j (2 k − 2−j l)2H−1 2H − 1 2H    − (2−j k − 2−j (l + 1)) − (2−j (k + 1) − 2−j l)2H−1 − (2−j (k + 1) − 2−j (l + 1)) 1 1  −j = 2j 2(2 k − 2−j l)2H−1 2H − 1 2H  − (2−j k − 2−j (l + 1)) − (2−j (k + 1) − 2−j l)2H−1 ∼ 2−j |k − j|2H−1 2−j .

Hence

roo

f

j

rep

Eζj,k ζj,l ∼ 2−j |k − j|2H−1 . One can also easily prove that for l = k   E ξ H (ψj,k ) ξ H (ψj,k ) Z 2−j (k+1) Z 2−j (k+1) ψj,k (s)ψj,l (t)|t − s|2H−2 dt ds = 2j 2−j k

2−j k

lP

1  −j 1 = 2j (2 k − 2−j l)2H−1 ∼ 21−2Hj . 2H − 1 2H Using these estimates we can prove the following proposition.

s ([0, 1])). Proposition 6.1. For H ∈ ( 12 , 1) and − 12 < s < H − 1 we have ξ H ∈ L2 (Ω; B2,2

Proof. The proof is the result of the following straightforward calculation =E

∞ X

j

2

2sj

2 X

rna

s E|ξ|2B2,2

j=0

E ζj,k .

k=0

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

∞ X j=0

22sj 2j 21−2Hj .

∞ X

2j(2s+2−2H) 2j 21−2Hj

j=0

Jo u

s ([0, 1])). Remark 6.1. If H ∈ ( 12 , 1) one can find a number s ∈ (− 12 , H − 1) such that ξ H ∈ L2 (Ω; B2,2 Since all coefficients of φn,k and ψn,k are Gaussian distributed, their law are equivalent with respect to s ([0, 1]), Assumption 2.2 the Lebesgue measure. Now, since the Haar basis is a Schauder basis in B2,2 is satisfied. By the same arguments as used in the proof of Lemma 5.6, one can show that Assumption 2.3 is also satisfied.

Lemma 6.1. Let B H be a fractional Bownian motion with Hurst parameter H > 12 and µ the probs ([0, 1]) defined by (34). Let {φ j ability measure on B2,2 j,k : j ∈ IN : k = 1, . . . , 2 } be the wavelet basis s in B2,2 ([0, 1]) described in (35). Then, the measure µ is decomposable with decomposition satisfying Assumption 2.2 and Assumption 2.3.

Journal Pre-proof IRREDUCIBILITY AND THE 2-D STOCHASTIC NAVIER STOKES

23

Appendix A. Zero One Laws for decomposable measures with density In this Section we generalize the Theorem 4 of [14] 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 B(E) the σ–algebra generated by its open sets. Let µ be a measure on (E, B(E)) and F and G be two subsets of E such that E = F ⊕ G. Then, there is a probability measure µ(F,G) : B(F ) 3 A 7→ µ(A + G) ∈ [0, 1].

For A ⊂ E and y ∈ G let A(F,G) (y) = {x ∈ F : x + y ∈ 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 A.1. Let E be a separable Banach space and {en : n ∈ IN} be a Schauder basis and Fn := {λeP n : λ ∈ R}. For each element x ∈ E there exists a unique sequence {an : n ∈ IN} in R such that x = n∈IN an en . Let Gn := F1 ⊕ · · · ⊕ Fn , Gn = G⊥ n, πGn : E 3 x 7→ a1 e2 + · · · + an en → Gn X

aj+n ej+n ∈ Gn .

j∈IN

roo

πGn : E 3 x 7→

f

be a projection from E onto F1 ⊕ · · · ⊕ Fn and

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 [23, Theorem 6.3]) for any E–valued random variable X there exists a probability kernel

rep

ln : Gn × B(F1 ⊕ · · · ⊕ Fn ) → [0, 1],

such that (1) P (πGn X ∈ U | πGn X = y) = ln (y, U ) for all U ∈ B(F1 ⊕ · · · ⊕ Fn ); (2) for each U ∈ B(F1 ⊕ · · · ⊕ Fn ) the mapping

lP

is B(Gn )–measurable

Gn 3 y 7→ ln (y, U )

rna

To simplify the notation let us denote µ(Gn ,Gn ) by µn and µ(Gn ,Gn ) by µn . Note that given a decomposition (Fn , Gn , ln ) of µ it is essential that the kernel ln has a density with respect to the Lebesgue measure on Gn which, as we will show in the next Lemma, follows from the absolute continuity of µn with respect to LebGn for any n ∈ IN.

Jo u

Lemma A.1. Let E be a separable Banach space and {en : n ∈ IN} be a Schauder basis and Fn := {λen : λ ∈ R}, Gn := F1 ⊕ · · · ⊕ Fn . Let us assume that for all n ∈ IN the measure µGn is absolutely continuous with respect to LebGn . Then for any n ∈ IN, µn ({y ∈ Gn : ln (y, ·) is abs. continuous with respect to LebGn }) = 1.

In particular, for any U ∈ B(Gn ) with µn (U ) = 0, we have

µn ({y ∈ Gn : ln (y, U ) = 0}) = 1.

Proof. Fix U ∈ B(Gn ) with µGn (U ) = 0. We will show that µn ({y ∈ Gn : ln (y, U ) > 0}) = 0.  From [29, Theorem 4.1 ] we infer that for all  > 0 there exists a subset Cn,U ⊂ Gn such that n n  µ (G \ Cn,U ) ≤  and  ln (·, A) : Cn,U 3 y 7→ ln (y, U ) ∈ [0, 1],

Journal Pre-proof 24

E. HAUSENBLAS AND PAUL A. RAZAFIMANDIMBY

is continuous. Now let us set Since ln (·, U ) C 

 G∗ = {y ∈ Gn ∩ Cn,U : ln (y, U ) ≥ }.

n,A

 is continuous and the sets [, 1] and Cn,U are closed, the set G∗ is closed. Hence, Z ln (y, U ) µn (dy). 0 = µGn (U ) = µ(U + Gn ) = Gn

Since

Gn

⊃

G∗ ,

we additionally have

0 = µn (U ) = µ(U + Gn ) = By the definition of the set G∗ we have

Z

Gn

n

ln (y, U ) µ (dy) ≥

Z

G∗

ln (y, U ) µn (dy).

0 ≥  µn (G∗ ). Since  > 0, we have µn (G∗ ) = 0. Now, from the closedness of G∗ and the regularity of the measureµGn we infer that µn ({y ∈ Gn : ln (y, U ) > 0}) = lim µGn (G∗ ) = 0. →0

roo

f

Lemma A.2. Let µ be a decomposable finite measure on E with decomposition {Fn , Gn , ln }∞ n=1 . Let us assume that µGn is absolutely continuous with respect to LebGn . Then, for any U ∈ Fn satisfying µGn (U ) = 0 we have µGn ({y ∈ Gn : ln (y, U ) = 0}) = 1.

Proof. Let n ∈ IN and U ∈ Fn such that µGn (U ) = 0. We will show that µGn ({y ∈ Gn : ln (y, U ) > 0}) = 0.

rep

Firstly, note that by the Radon-Nikodym Theorem the mapping  ln : Cn,U 3 y 7→ ln (y, U ) ∈ [0, 1],

is measurable. Hence, from [29, Theorem 4.1] we infer that for all  > 0 there exists a closed subset   ) ≤  and the function Cn,U of Gn such that µn (Gn \ Cn,U  ln : Cn,U 3 y 7→ ln (y, U ) ∈ [0, 1],

lP

is continuous. Secondly, let us set

 G∗ = {y ∈ Cn,U : ln (y, U ) ≥ }.

From the continuity of ln (·, U ) C 

n,U

 and the fact that the sets [, 1] and Cn,U are closed we conclude

rna

the set G∗ is also closed. Next, thanks to the definition of µGn we obtain that Z 0 = µGn (U ) = µ(U + Gn ) = ln (y, U ) µGn (dy).

Jo u

Furthermore, because

G∗

⊂

 Cn,U

0=

Z

Gn

we also have

Gn

ln (y, U ) µGn (dy) ≥

Invoking now the definition of the set G∗ we obtain

Z

G∗

ln (y, U ) µGn (dy).

0 ≥ µGn (G∗ ).

Since  > 0, we have µGn (G∗ ) = 0. From the closedness of G∗ and the regularity of the measure µn we infer that µGn ({y ∈ Gn : ln (y, U ) > 0}) = lim µGn (G∗ ) = 0. →0

Journal Pre-proof IRREDUCIBILITY AND THE 2-D STOCHASTIC NAVIER STOKES

25

Therefore, µGn ({y ∈ Gn : ln (y, U ) = 0}) = 1. Corollary A.1. Let E be a separable Banach space and {en : n ∈ IN} be a Schauder basis. Put Fn := {λen : λ ∈ R} and Gn := F1 ⊕ · · · ⊕ Fn . Let us assume that for all n ∈ IN µGn is absolutely continuous with respect to the LebGn . Then for any n ∈ IN there exists a function hn : Gn × F1 ⊕ · · · ⊕ Fn → R+ 0 such that µGn –a.s. Z ln (y, U ) =

U

hn (y, x)µGn (dx).

Proof. From

µGn ({y ∈ Gn : ln (y, U ) = 0}) = 1, for any U ∈ B(Gn ), follows the corollary’s assertion. Indeed the above identity implies the existence of a Radon-Nikodyn derivative. In particular, it holds that  µGn y ∈ Gn : there exists a mapping hn (y, ·) : Gn → R Z  such that ln (y, U ) = hn (y, x)µGn (dx) = 1.

f

U

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Definition A.1. We call a set U ∈ B µ (E) a finite zero one µ–set if and only if for all n ∈ IN µGn {y ∈ Gn : µGn (Un (y)) = 0 or 1} = 1,

where Un (y) = U(F1 ⊕···⊕Fn ,Gn ) (y).

rep

Let us now present the generalization of Theorem 4 in [14]. Theorem A.1. Let {Fn , Gn , ln }∞ n=1 be a decomposition for µ such that for any n ∈ IN µGn is absolutely continuous with respect to LebF1 ⊕···⊕Fn . Let F∞ = ∪n∈IN {F1 + F2 + · · · + Fn }. If U is a finite zero one µ measurable subset of E, then there exists B ∈ B(E) such that B + F∞ = B and µ(B) = µ(U ).

lP

Proof. The proof is very similar to the proof of [14, Theorem 4]. Let us assume U ∈ B(E). For fix n ∈ IN we set U n = {y ∈ Gn : µGn (Un (y)) = 1}, Gn = F1 ⊕ F2 ⊕ · · · ⊕ Fn = linear span of ∪nk=1 Fk ,

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Bn = Gn + U n and B = lim inf n→∞ Bn = ∪∞ n=1 {∩m≥n Bm }. For the time being let us assume that µ(U ) = µ(Bn ).

(37)

Jo u

Then, • µ(U 4 Bn ) = 0 for all n ∈ IN, • and µ(U ) = µ(Bn ) ≥ µ(∩m≥n Bm ) ≥ µ(U ), • µ(B) = limn→∞ µ(∩m≥n Bm ). Since µ is regular we additionally have that µ(B) = lim µ(∩m≥n Bm ) ≥ lim µ(Bn ) = µ(U ), n→∞

n→∞

from which the assertion of Theorem A.1 follows. Now it remains to prove (37). To this end, observe first that because of Lemma A.1 the kernel ln is µn –a.s. absolutely continuous on Gn . Hence, by the Radon-Nikodym Theorem for µn –a.s. there exists a probability kernel hn : Gn × Gn → R+ 0,

Journal Pre-proof 26

E. HAUSENBLAS AND PAUL A. RAZAFIMANDIMBY

such that µ(U ) =

Z

Gn

Z

Un (y)

hn (y, x) µGn (dx)µGn (dy).

Then, by using Bn = Gn ⊕ Un we obtain that Z Z 1U (x + y)hn (y, x) µGn (dx) µGn (dy) µ(U ) = Gn Gn Z Z 1Un (y)⊕U n (x + y)hn (y, x) µGn (dx) µGn (dy) = Gn Gn Z Z = 1(Un (y)⊕U n )∩Bn (x + y)hn (y, x) µGn (dx) µGn (dy) n ZG ZGn = 1(Un (y)∩Gn )⊕U n (x + y)hn (y, x) µGn (dx) µGn (dy) Gn Gn Z = ln (y, Un (y) ∩ Gn ) µGn (dx) µGn (dy) Un Z = ln (y, Gn ) dµGn (x) = µ(Bn ).

f

Un

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Appendix B. Besov spaces and their properties

  φ0 (x) = ψ(x), x ∈ Rd , φ (x) = ψ( x2 ) − ψ(x), x ∈ Rd ,  1 φj (x) = φ1 (2−j+1 x), x ∈ Rd , j = 2, 3, . . . .

lP

Then put

rep

Let us recall the definition of the Besov spaces as given in [33, Definition 2, pp. 7-8]. First we choose a function ψ ∈ S(Rd ) such that 0 ≤ ψ(x) ≤ 1, x ∈ Rd and  1, if |x| ≤ 1, ψ(x) = 0 if |x| ≥ 23 .

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We will use the definition of the Fourier transform F = F +1 and its inverse F −1 as in [33, p. 6]. In particular, with h·, ·i being the scalar product in Rd , we put Z (F ±1 f )(ξ) := (2π)−d/2 e∓ihx,ξi f (x) dx, f ∈ S(Rd ), ξ ∈ Rd . Rd

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

Jo u

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

|f |Bp,∞ s

j=0

  = sup 2sj F −1 [φj Ff ] Lp = k 2sj F −1 [φj Ff ] Lp j∈N

s (Rd ) the space of all f ∈ S 0 (Rd ) for which |f | We denote by Bp,q is finite. s Bp,q

j∈N

j∈N

kl ∞ .

Journal Pre-proof IRREDUCIBILITY AND THE 2-D STOCHASTIC NAVIER STOKES

Let us mention that straightforward computations give that for any a ∈ Rd , p ≥ 1, and p∗ = |δa |p −

d

d p∗ Bp,∞

− pd∗

= (2π)− 2 2

.

27

p p−1

(38)

(Rd )

References

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f

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roo

Department of Mathematics, Montanuniversity Leoben, Fr. Josefstr. 18, 8700 Leoben, Austria Email address: [email protected]

Jo u

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rep

Department of Mathematics and Applied Mathematics, University of Pretoria, Lynwood Road, Hatfield, Pretoria 0083, South Africa Email address: [email protected]