INVARIANT MEASURES OF STOCHASTIC 2D NAVIER-STOKES EQUA- TIONS DRIVEN BY α -STABLE PROCESSES

in PROBABILITY

INVARIANT MEASURES OF STOCHASTIC 2D NAVIER-STOKES EQUA- TIONS DRIVEN BY α -STABLE PROCESSES

ZHAO DONG¹

Institute of Applied Mathematics, Academy of Mathematics and Systems Sciences, Academia Sinica, P.R.China

email: [email protected] LIHU XU

Department of Mathematics, Brunel University, Uxbridge UB8 3PH, ENGLAND email: [email protected]

XICHENG ZHANG²

School of Mathematics and Statistics, Wuhan University, Wuhan, Hubei 430072, P.R.China email: [email protected]

SubmittedMarch 25, 2011, accepted in final formSeptember 23, 2011 AMS 2000 Subject classification: 60H15

Keywords: α-stable process, Stochastic Navier-Stokes equation, Invariant measure Abstract

In this note we prove the well-posedness for stochastic 2D Navier-Stokes equation driven by gen- eral Lévy processes (in particular,α-stable processes), and obtain the existence of invariant mea- sures.

1 Introduction and Main Result

In this article we are concerned with the following stochastic 2D Navier-Stokes equation in torus T²= (0, 1]²:

du_t= [∆u_t−(u_t· ∇)u_t+∇p_t]dt+dL_t, divu_t=0, u₀=ϕ∈H⁰, (1.1) whereu_t(x) = (u¹_t(x),u²_t(x))is the 2D-velocity field, p is the pressure, and(L_t)t¾0is an infinite dimensional cylindrical Lévy process given by

L_t =X

j∈N

βjL^(j)_t e_j,

1RESEARCH SUPPORTED BY 973 PROGRAM (2011CB808000), KEY LABORATORY OF RANDOM COMPLEX STRUC- TURES AND DATA SCIENCE, ACADEMY OF MATHEMATICS & SYSTEMS SCIENCE, CHINESE ACADEMY OF SCIENCES (NO.2008DP173182), SCIENCE FUND FOR CREATIVE RESEARCH GROUPS (10721101), NSF OF CHINA (NO. 11071008)

2CORRESPONDING AUTHOR, RESEARCH SUPPORTED BY NSF OF CHINA (NO. 10971076) AND THE PROGRAM FOR NEW CENTURY EXCELLENT TALENTS IN UNIVERSITY (NCET-10-0654).

678

where{(L⁽_t^j))t¾0,j∈N}is a sequence of independent one dimensional purely discontinuous Lévy processes defined on some filtered probability space(Ω,F,(Ft)t¾0;P)and with the same Lévy measureν, {βj,j ∈N} is a sequence of positive numbers and {e_j,j ∈ N} is a sequence of or- thonormal basis of Hilbert spaceH⁰, where forγ∈R,H^γwith the normk · k_γand inner product

〈·,·〉_γdenotes the usual Sobolev space of divergence free vector fields onT²(see Section 2 for a definition).

As a continuous model, stochastic Navier-Stokes equation driven by Brownian motion has been extensively studied in the past decades (cf. [9, 3, 5, 8], etc.). Meanwhile, stochastic partial differential equation with jump has also been studied recently (cf. [11, 6]). However, in the well-known results, the assumption that the jump process has finite second order moments was required in order to obtain the usual energy estimate. This excludes the interestα-stable process.

In this note, we establish the well posedness for stochastic 2D Navier-Stokes equation (1.1) driven by a general cylindrical Lévy process, and obtain the existence of invariant measures for this discontinuous model. More precisely, we shall prove that:

Theorem 1.1. Suppose that for someθ ∈(0, 1], (H_θ): H_θ:=

Z

|x|>1

|x|^θν(dx) + Z

|x|¶1

|x|^2θν(dx) +X

j∈N

|βj|^θ<+∞.

Then for anyϕ∈H⁰, there exists a unique solution(u_t)t¾0= (u_t(ϕ))t¾0to equation (1.1) satisfying that for P-almost allωand for any t>0,

(i) t7→u_t(ω)is right continuous and has left-hand limit inH⁰, andRt

0k∇u_s(ω)k²₀ds<+∞; (ii) it holds that for anyφ∈H¹,

〈u_t(ω),φ〉0=〈ϕ,φ〉0+ Zt

0

[〈∆u_s(ω),φ〉0+〈u_s(ω)⊗u_s(ω),∇φ〉0]ds+〈L_t(ω),φ〉0.

Moreover, there exists a constant C=C(H_θ,θ)>0such that for any t>0, E

‚ sup

s∈[0,t]ku_sk^θ₀

Π+E

‚Z t

0

k∇u_sk²₀ (ku_sk²₀+1)^1−θ/2ds

Œ

¶C(1+kϕk^θ₀+t). (1.2) In particular, there exists a probability measure µ on (H⁰,B(H⁰)) called invariant measure of (u_t(ϕ))t¾0such that for any bounded measurable functionalΦonH⁰,

Z

H⁰

EΦ(u_t(ϕ))µ(dϕ) = Z

H⁰

Φ(ϕ)µ(dϕ).

Remark 1.2. Assumption(H_θ)implies that cylindrical Lévy process(L_t)t¾0admits a càdlàg version inH⁰and for any t>0(cf.[12, p.159, Theorem 25.3]),

EkL_tk^θ₀<+∞.

In fact, forθ∈(0, 1], by the elementary inequality(a+b)^θ¶a^θ+b^θ, we have EkL_tk^θ₀¶E



 X

j∈N

|βj| · |L⁽_t^j)|





θ

¶ X

j∈N

|βj|^θ·E|L⁽_t^j)|^θ=E|L⁽¹⁾_t |^θX

j∈N

|βj|^θ<+∞. Moreover,(H_θ)admitsν(dx) =dx/|x|^1+αwithα∈(θ, 2θ).

Remark 1.3. By estimate (1.2) and Poincàre’s inequality, we have

E

‚Z t

0

k∇u_sk^θ₀ds

Œ

¶E

‚Z t

0

k∇u_sk^θ₀(ku_sk^2−θ₀ +1) (ku_sk²₀+1)^1−θ/2 ds

Œ

¶CE

‚Z t

0

k∇u_sk²₀+1 (ku_sk²₀+1)^1−θ/2ds

Œ

¶C(1+kϕk^θ₀+t).

This estimate in particular yields the existence of invariant measures by the classical Bogoliubov- Krylov’s argument (cf.[4]).

Remark 1.4. An obvious open question is about the uniqueness of invariant measures (i.e. ergodicity) for discontinuous system (1.1). The notion of asymptotic strong Feller property in [9] is perhaps helpful for solving this problem.

This paper is organized as follows: In Section 2, we give some necessary materials. In Section 3, we prove the main result.

2 Preliminaries

In this section we prepare some materials for later use. LetC₀^∞(T²)²be the space of all smooth R²-valued function onT²with vanishing mean and divergence, i.e.,

Z

T²

f(x)dx=0, divf(x) =0.

Forγ∈R, letH^γbe the completion ofC₀^∞(T²)²with respect to the norm kfk_γ=

‚Z

T²

|(−∆)^γ/2f(x)|²dx

Œ1/2

,

where (−∆)^γ/2 is defined through Fourier’s transform. Thus, (H^γ,k · kγ)is a separable Hilbert space with the obvious inner product

〈f,g〉_γ:= Z

T²

(−∆)^γ/2f(x)·(−∆)^γ/2g(x)dx.

Below, we shall fix an orthonormal basis{e_j,j∈N} ⊂C₀^∞(T²)²ofH⁰consisting of the eigenvec- tors of∆, i.e.,

∆e_j=−λje_j, 〈e_j,e_j〉0=1, j=1, 2,· · ·, (2.1) where 0< λ1¶· · ·¶λj↑ ∞.

Let {(L^(j)_t )t¾0,j∈N} be a sequence of independent one dimensional purely discontinuous Lévy processes with the same characteristic function, i.e.,

Ee^iξL(j)t =e^−tψ(ξ), ∀t¾0,j=1, 2,· · ·,

whereψ(ξ)is a complex valued function called Lévy symbol given by ψ(ξ) =

Z

R\{0}

(e^iξy−1−iξy1_|y|¶1)ν(dy), whereνis the Lévy measure and satisfies that

Z

R\{0}

1∧ |y|²ν(dy)<+∞.

Fort>0 andΓ∈ B(R\ {0}), the Poisson random measure associated with L⁽_t^j)is defined by N^(j)(t,Γ):= X

s∈(0,t]

1_Γ(L⁽_s^j)−L⁽_s−^j)). The compensated Poisson random measure is given by

N˜^(j)(t,Γ) =N^(j)(t,Γ)−tν(Γ). By Lévy-Itô’s decomposition (cf.[2, p.108, Theorem 2.4.16]), one has

L⁽_t^j)= Z

|x|¶1

xN˜^(j)(t, dx) + Z

|x|>1

x N^(j)(t, dx).

For a Polish space(G,ρ), letD(R+;G)be the space of all right continuous functions with left-hand limits fromR+toG, which is endowed with the Skorohod topology:

d_G(u,v):= inf

λ∈Λ

– sup

s6=t

logλ(t)−λ(s) t−s

∨

Z∞

0

sup

t¾0(ρ(u_t∧r,v_λ(t)∧r)∧1)e^−rdr

™

, (2.2) whereΛis the space of all continuous and strictly increasing function fromR+→R+withλ(0) = 0 andλ(∞) =∞. Thus,(D(R+;G);d_G)is again a Polish space (cf.[7, p.121, Theorem 5.6]).

We need the following tightness criterion, which is a direct combination of[10, Corollary 5.2]and Aldous’s criterion[1].

Theorem 2.1. Let{(Xⁿ_t)t¾0,n∈N} be a sequence ofH⁻¹-valued stochastic processes with càdlàg path. Assume that

(i) for eachφ∈C₀^∞(T²)²and t>0,lim_K→∞sup_n∈NPn

sup_s∈[0,t]|〈X_sⁿ,φ〉−1|¾Ko

=0;

(ii) for eachφ∈C₀^∞(T²)²and t,a>0,

"→0+lim sup

n∈N τ∈Ssupt

Pn

|〈X_τⁿ−X_τ+"ⁿ ,φ〉−1|¾ao

=0, whereSt denotes the class of all (Ft)-stopping times with bound t;

(iii) for every" >0and t>0,

mlim→∞sup

n∈N

P



sup

s∈[0,t]

X∞

j=m

〈X_sⁿ,e_j〉²₋₁¾"



=0.

Then the laws of(Xⁿ_t)t¾0inD(R+;H⁻¹)are tight.

The following result comes from[7, p.131 Theorem 7.8].

Theorem 2.2. Suppose that stochastic processes sequence {(Xⁿ_t)t¾0,n ∈ N} weakly converges to (X_t)t¾0inD(R+;H⁻¹). Then, for any t>0andφ∈H¹, there exists a sequence t_n↓t such that for any bounded continuous function f ,

nlim→∞Ef(〈X_tⁿ

n,φ〉−1) =Ef(〈X_t,φ〉−1). We also need the following technical result.

Lemma 2.3. Suppose that sequence{uⁿ,n∈N}converges to u inD(R₊;H⁻¹). Then for any T>0 and m∈N,

sup

t∈[0,T]ku_tk0¶ lim

n→∞

sup

t∈[0,T+_m¹]kuⁿ_tk0. (2.3)

If in addition, for Lebesgue almost all t, uⁿ_t converges to u_t inH⁰, then for anyβ >0, ZT

0

k∇u_tk²₀

(1+ku_tk²₀)^βdt¶ lim

n→∞

ZT

0

k∇uⁿ_tk²₀

(1+kuⁿ_tk²₀)^βdt. (2.4) Proof. Without loss of generality, we assume that the right hand side of (2.3) is finite. For any φ ∈H¹, it is clear that t 7→ 〈u_t,φ〉0is a càdlàg real valued function, and by definition (2.2) of Skorohod metric, we have

d_R(〈uⁿ,φ〉0,〈u,φ〉0)¶(2+kφk1)d_H−1(uⁿ,u),

and so 〈uⁿ,φ〉0 converges to〈u,φ〉0 inD(R₊;R)as n→ ∞. Since the discontinuous points of

〈u_·,φ〉0are at most countable, for anyT >0 andm∈N, there exists a time T_m∈(T,T+1/m) such that〈u_·,φ〉0is continuous atT_m. Thus, we have (cf.[7, p.119, Proposition 5.3])

n→∞lim sup

t∈[0,Tm]|〈uⁿ_t,φ〉0|= sup

t∈[0,Tm]|〈u_t,φ〉0|.

Hence,

sup

t∈[0,T]ku_tk0= sup

t∈[0,T]

sup

φ∈H¹;kφk0¶1

|〈u_t,φ〉0|

¶ sup

φ∈H¹;kφk0¶1

sup

t∈[0,Tm]|〈u_t,φ〉0|

= sup

φ∈H¹;kφk0¶1

nlim→∞ sup

t∈[0,Tm]|〈uⁿ_t,φ〉0|

¶ lim

n→∞

sup

φ∈H¹;kφk0¶1

sup

t∈[0,Tm]|〈uⁿ_t,φ〉0|

= lim

n→∞

sup

t∈[0,Tm]kuⁿ_tk0. Thus, (2.3) is proven.

For proving (2.4), letN be the Lebesgue null set such that for all t∈ N/ ,uⁿ_t converges tou_t in H⁰. Fixing at∈ N/ , then as above, we have

k∇u_tk²₀ (1+ku_tk²₀)^β ¶

lim_n→∞k∇uⁿ_tk²₀

(1+lim_n→∞kuⁿ_tk²₀)^β ¶ lim

n→∞

k∇uⁿ_tk²₀ (1+kuⁿ_tk²₀)^β. Estimate (2.4) now follows by Fatou’s lemma.

3 Proof of Theorem 1.1

We first give the following definition about the weak solutions to equation (1.1).

Definition 3.1. A probability measure P onD(R+;H⁻¹)is called a weak solution of equation (1.1) if

(i) for any t>0, P

u∈D(R₊;H⁻¹): sup_s∈[0,t]ku_sk0+Rt

0k∇u_sk²₀ds<+∞

=1;

(ii) for any j∈N,

M_t^(j)(u):=〈u_t,e_j〉0− 〈u₀,e_j〉0− Z t

0

[〈u_s,∆e_j〉0+〈u_s⊗u_s,∇e_j〉0]ds (3.1) is a Lévy process with the characteristic function

Ee^iξMt(j)=exp (

t Z

R\{0}

(e^iξyβj−1−iξyβj1_|y|¶1)ν(dy) )

, and{(M_t^(j))t¾0,j∈N}is a sequence of independent Lévy processes.

Proof of Existence of Weak Solutions: We use Galerkin’s approximation to prove the existence of weak solutions and divide the proof into three steps.

(Step 1): Forn∈N, set

H⁰n:=span{e₁,e₂,· · ·,e_n}, and letΠnbe the projection fromH⁰toH⁰nand define

Lⁿ_t :=

n

X

j=1

βjL⁽_t^j)e_j=

n

X

j=1

Z

|y|¶1

yβje_jN˜^(j)(t, dy) +

n

X

j=1

Z

|y|>1

yβje_jN^(j)(t, dy). Consider the following finite dimensional SDE driven by finite dimensional Lévy processLⁿ_t:

duⁿ_t = [∆uⁿ_t −Πn((uⁿ_t · ∇)uⁿ_t)]dt+dLⁿ_t, uⁿ₀= Πnϕ. (3.2) Since for anyR>0 andu,v∈H⁰nwithkuk0,kvk0¶R,

kΠn((u· ∇)u−(v· ∇)v)k0¶C_R,nku−vk0

and

〈u,∆u−Πn((u· ∇)u)〉0=−k∇uk0, ∀u∈H⁰n, (3.3) finite dimensional SDE (3.2) is thus well-posed.

Define a smooth function f_nonH⁰nby

f_n(u):= (kuk²₀+1)^θ/2, u∈H⁰n. By simple calculations, we have

∇f_n(u) = θu

(kuk²₀+1)^1−θ/2, ∇²f_n(u) = θPn

i=1e_i⊗e_i

(kuk²₀+1)^1−θ/2− θ(2−θ)u⊗u

(kuk²₀+1)^2−θ/2, (3.4)

and for allu,v∈H⁰n,

|f_n(u)−f_n(v)|¶|(kuk²₀+1)^1/2−(kvk²₀+1)^1/2|^θ¶ku−vk^θ₀. (3.5) By (3.2), (3.3), (3.4) and Itô’s formula (cf.[2, p.226, Theorem 4.4.7]), we have

f_n(uⁿ_t) =f_n(uⁿ₀)− Z t

0

θk∇u_sⁿk²₀

(kuⁿ_sk²₀+1)^1−θ/2ds+

n

X

j=1

Zt

0

Z

|y|¶1

[f_n(uⁿ_s+yβje_j)−f_n(uⁿ_s)]N˜^(j)(ds, dy)

+

n

X

j=1

Zt

0

Z

|y|¶1

–

f_n(uⁿ_s+yβje_j)−f_n(uⁿ_s)− θ〈uⁿ_s,yβje_j〉0

(|uⁿ_s|²+1)^1−θ/2

™

ν(dy)ds

+

n

X

j=1

Zt

0

Z

|y|>1

”f_n(uⁿ_s+yβje_j)−f_n(uⁿ_s)—

N^(j)(ds, dy)

=:f_n(uⁿ₀)−I₁ⁿ(t) +I₂ⁿ(t) +I₃ⁿ(t) +I₄ⁿ(t). ForI₂ⁿ(t), by Burkholder’s inequality and (3.5), we have

E

‚ sup

t∈[0,T]

I₂ⁿ(t)

Œ

¶C

n

X

j=1

E Z T

0

Z

|y|¶1

|f_n(u_sⁿ+yβje_j)−f_n(uⁿ_s)|²N^(j)(ds, dy)

!1/2

¶C

n

X

j=1

E Z T

0

Z

|y|¶1

|f_n(u_sⁿ+yβje_j)−f_n(uⁿ_s)|²ν(dy)ds

!1/2

¶C T^1/2

n

X

j=1

|βj|^θ Z

|y|¶1

|y|^2θν(dy)

!1/2

¶C T^1/2 X∞

j=1

|βj|^θ. where we have used condition(H_θ). Here and after, the constantCis independent ofn,T. ForI₃ⁿ(t), by Taylor’s expansion and (3.4), we have

E

‚ sup

t∈[0,T]

I₃ⁿ(t)

Œ

¶C

n

X

j=1

β²_j ZT

0

Z

|y|¶1

|y|²ν(dy)ds¶C T X∞

j=1

|βj|^θ Z

|y|¶1

|y|²ν(dy). ForI₄ⁿ(t), by (3.5), we have

E

‚ sup

t∈[0,T]

I₄ⁿ(t)

Œ

¶

n

X

j=1

E ZT

0

Z

|y|>1

|f_n(uⁿ_s+yβje_j)−f_n(u_sⁿ)|N^(j)(ds, dy)

!

=

n

X

j=1

E ZT

0

Z

|y|>1

|f_n(uⁿ_s+yβje_j)−f_n(u_sⁿ)|ν(dy)ds

!

¶C T X∞

j=1

|βj|^θ Z

|y|>1

|y|^θν(dy). Combining the above calculations, we obtain that

E

‚ sup

t∈[0,T](kuⁿ_tk²₀+1)^θ/2

Π+E

ZT

0

θk∇uⁿ_sk²₀

(kuⁿ_sk²₀+1)^1−θ/2ds¶(kϕk²0+1)^θ/2+C T+C T^1/2. (3.6)

(Step 2): In this step, we use Theorem 2.1 to show that{(uⁿ_t)t¾0,n∈N}is tight inD(R+;H⁻¹). For anyφ∈C₀^∞(T²)², by equation (3.2), we have

〈uⁿ_t,φ〉−1=〈uⁿ₀,φ〉−1+ Zt

0

[〈∆uⁿ_s,φ〉−1− 〈(uⁿ_s· ∇)uⁿ_s,φ〉−1]ds+〈Lⁿ_t,φ〉−1

=〈uⁿ₀,φ〉₋₁+ Zt

0

[〈uⁿ_s,∆φ〉₋₁+〈u_sⁿ⊗u_sⁿ,∇φ〉₋₁]ds+〈Lⁿ_t,φ〉₋₁. Thus, for" >0 and any stopping timeτbounded byt, we have

〈uⁿ_τ+"−uⁿ_τ,φ〉−1= Zτ+"

τ

[〈u_sⁿ,∆φ〉−1+〈uⁿ_s⊗uⁿ_s,∇φ〉−1]ds+〈Lⁿ_τ+"−L_τⁿ,φ〉−1

¶" sup

s∈[0,t]

€kuⁿ_sk0· kφk0+kuⁿ_sk²₀· k∇(−∆)⁻¹φk_∞Š

+

n

X

j=1

|βj| · |L_τ+"^(j) −L⁽_τ^j)| · k(−∆)⁻¹φk0.

Using(a+b)^θ¶a^θ+b^θ provided thatθ∈(0, 1], we get

E|〈uⁿ_τ+"−uⁿ_τ,φ〉−1|^θ/2¶C_φE

‚ sup

s∈[0,t]kuⁿ_sk^θ₀+1

Œ

"^θ/2+C_φ



E

n

X

j=1

|βj|^θ· |L⁽_τ+"^j) −L^(j)_τ |^θ





1 2

. By the strong Markov property of Lévy process (cf.[12, p.278, Theorem 40.10]), we have

E|L⁽_τ+"^j) −L⁽_τ^j)|^θ=E|L⁽_"^j)|^θ=E|L⁽¹⁾_" |^θ, ∀j∈N. Thus, by (3.6) and(H_θ),

E|〈uⁿ_τ+"−u_τⁿ,φ〉₋₁|^θ/2¶Ch

"^θ/2+ (E|L⁽¹⁾_" |^θ)^1/2i

, (3.7)

where the constantCis independent ofn,τand". On the other hand, by (2.1), we have E



sup

s∈[0,t]

X∞

j=m

〈uⁿ_s,e_j〉²₋₁





θ/2

=E



sup

s∈[0,t]

X∞

j=m

〈uⁿ_s,e_j〉²₀ λ²j





θ/2

¶ 1 λ^θm

E

‚ sup

s∈[0,t]kuⁿ_sk^θ₀

Œ

. (3.8) By Theorem 2.1 and (3.6)-(3.8), one knows that the law of(uⁿ_t)t¾0inD(R+;H⁻¹)denoted byP_n is tight.

(Step 3): LetPbe any accumulation point of{P_n,n∈N}. In this step, we show thatPis a weak solution of equation (1.1) in the sense of Definition 3.1. First of all, by Skorohod’s embedding theorem, there exists a probability space(Ω˜, ˜F, ˜P)andD(R₊;H⁻¹)-valued random variablesXⁿ andX such that

(i) Law ofXⁿunder ˜PisP_nand law ofX under ˜PisP.

(ii)Xⁿconverges toX inD(R+;H⁻¹)a.s. asn→ ∞. Thus, by (3.6), we have

E˜

‚ sup

t∈[0,T]kX_tⁿk^θ₀

Œ +E˜

ZT

0

θk∇X_sⁿk²₀ (kX_sⁿk²₀+1)^1−θ/2ds

!

¶C(1+kϕk^θ₀+T). (3.9)

By Lemma 2.3 and Fatou’s lemma, for anym∈N, we have E^P

‚ sup

t∈[0,T]ku_tk^θ₀

Œ

=E˜

‚ sup

t∈[0,T]kX_tk^θ₀

Œ

¶ lim

n→∞

E˜

‚ sup

t∈[0,T+1/m]kX_tⁿk^θ₀

Œ

¶(kϕk²₀+1)^θ/2+C(T+1/m) +C(T+1/m)^1/2. (3.10) On the other hand, for anyδ∈(0,θ/4), by Hölder’s inequality and (3.9), we have

E˜ Z T

0

kX_sⁿ−X_sk^δ₀ds

!

¶E˜ ZT

0

kX_sⁿ−X_sk^δ/2₋₁kX_sⁿ−X_sk^δ/2₁ ds

!

¶ E˜ ZT

0

kX_sⁿ−X_sk^δ₋₁ds

!1/2

E˜ Z T

0

kX_sⁿ−X_sk^δ₁ds

!1/2

→0.

So, there exists a subsequence still denoted by n such that for ˜P×dt-almost all(ω,s), X_sⁿ(ω) converges toX_s(ω)inH⁰. By Lemma 2.3 and (3.9), we then obtain

E^P ZT

0

θk∇u_sk²₀ (ku_sk²₀+1)^1−θ/2ds

!

=E˜ Z T

0

θk∇X_sk²₀ (kX_sk²₀+1)^1−θ/2ds

!

¶ lim

n→∞

E˜ Z T

0

θk∇X_sⁿk²₀ (kX_sⁿk²₀+1)^1−θ/2ds

!

¶C(1+kϕk^θ₀+T). (3.11)

Combining (3.10) and (3.11) gives (1.2). In particular, sup_t∈[0,T]ku_tk0andRT 0

θk∇u_sk²0

(ku_sk²₀+1)^1−θ/2dsare finiteP-almost surely, which produces (i) of Definition 3.1.

Fixing j∈N, in order to show thatM_t^(j)defined by (3.1) is a Lévy process, we only need to prove that for any 0¶s<t,

E^Pe^{iξ(Mt(j)−Ms(j))}=E˜e^{iξ(M˜(j)t −M˜s(j))}=exp (

(t−s) Z

R\{0}

(e^iξyβj−1−1_|y|¶1iξyβj)ν(dy) )

, (3.12) where

M˜_t^(j):=〈X_t,e_j〉0− 〈X₀,e_j〉0− Z t

0

[〈X_r,∆e_j〉0+〈X_r⊗X_r,∇e_j〉0]dr.

Fix 0¶s<tbelow. By Theorem 2.2, there exists(s_n,t_n)↓(s,t)such that

n→∞lim

E˜e^{iξ〈Xntn,ej〉0}=E˜e^{iξ〈Xt,ej〉0}, lim

n→∞

E˜e^{iξ〈Xsnn,ej〉0}=E˜e^{iξ〈Xs,ej〉0}. By equation (3.2), it is well-known that for anyn¾j,

E˜exp (

iξ



〈X_tⁿ

n−X_sⁿ

n,e_j〉0− Zt_n

s_n

[〈X_rⁿ,∆e_j〉0+〈X_rⁿ⊗X_rⁿ,∇e_j〉0]dr



 )

=E^Pnexp (

iξ



〈uⁿ_t

n−u_sⁿ

n,e_j〉0− Z ^tn

s_n

[〈uⁿ_r,∆e_j〉0+〈uⁿ_r⊗uⁿ_r,∇e_j〉0]dr



 )

=exp (

(t_n−s_n) Z

R\{0}

(e^iξyβj−1−1_|y|¶1iξyβj)ν(dy) )

. Thus, for proving (3.12), it suffices to prove the following limits:

n→∞lim E˜

exp

¨ iξ

Z t

s

〈Xⁿ_r⊗X_rⁿ,∇e_j〉0dr

«

−exp

¨ iξ

Zt

s

〈X_r⊗X_r,∇e_j〉0dr

«

=0,

nlim→∞

E˜

exp

¨ iξ

Z t

s

〈Xⁿ_r,∆e_j〉0dr

«

−exp

¨ iξ

Zt

s

〈X_r,∆e_j〉0dr

«

=0,

nlim→∞

E˜

exp (

iξ Z t_n

s_n

〈Xⁿ_r⊗X_rⁿ,∇e_j〉0dr )

−exp

¨ iξ

Z t

s

〈Xⁿ_r⊗X_rⁿ,∇e_j〉0dr

«

=0,

nlim→∞

E˜

exp (

iξ Z t_n

s_n

〈Xⁿ_r,∆e_j〉0dr )

−exp

¨ iξ

Zt

s

〈X_rⁿ,∆e_j〉0dr

«

=0.

Let us only prove the first limit, the others are similar. Noticing that for anyδ∈(0, 1)anda,b∈R,

|e^ia−e^ib|¶2(|a−b| ∧1)¶2|a−b|^δ, by Hölder’s inequality andkuk0¶kuk^1/2₋₁kuk^1/2₁ , we have forδ < θ/4,

E˜

exp

¨ iξ

Z t

s

〈X_rⁿ⊗X_rⁿ,∇e_j〉0dr

«

−exp

¨ iξ

Z t

s

〈X_r⊗X_r,∇e_j〉0dr

«

¶2|ξ|^δE˜

Z t

s

〈X_rⁿ⊗X_rⁿ−X_r⊗X_r,∇e_j〉0dr

δ

¶CE˜

‚Z t

s

kX_rⁿ−X_rk0(kX_rⁿk0+kX_rk0)dr

Œδ

¶CE˜

‚ sup

r∈[s,t](kXⁿ_rk0+kX_rk0) Z t

s

kXⁿ_r−X_rk^1/2₋₁kXⁿ_r−X_rk^1/2₁ dr

Œδ

¶CE˜

‚ sup

r∈[s,t](kXⁿ_rk0+kX_rk0+1)^{2δ−(θδ/2)}

‚Z t

s

kX_rⁿ−X_rk₋₁dr

Œδ/2

×

‚Z t

s

(kX_rⁿk1+kX_rk1) (kX_rⁿk²₀+kX_rk²₀+1)^1−θ/2dr

Œδ/2Œ

¶C



E˜

‚Z t

s

kXⁿ_r−X_rk−1dr

Œ2δ



1/4

→0,

as n→ ∞, where in the last inequality, we have used (3.9) and Hölder’s inequality. As for the independence ofM^(j)for differentj∈N, it can be proved in a similar way.

Proof of Theorem 1.1: The pathwise uniqueness follows by the classical result for 2D deterministic Navier-Stokes equation. As for the existence of invariant measures, basing on (1.2) (see Remark 1.3), it follows by the classical Bogoliubov-Krylov’s argument.

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