Invariant sample measures and random Liouville type theorem for the two-dimensional stochastic Navier-Stokes equations

Available online at www.sciencedirect.com ScienceDirect Journal of Differential Equations 317 (2022) 474–494 www.elsevier.com/locate/jde Invariant sample measures and random Liouville type theorem for the two-dimensional stochastic Navier-Stokes equations ✩ Caidi Zhao a,∗, Jintao Wang a, Tomás Caraballo b aDepartment of Mathematics, Wenzhou University, Wenzhou, Zhejiang Province, 325035, People’s Republic of China bDepartmento de Ecuaciones Diferenciales y Análisis Numérico, Facultad de Matemáticas, Universidad de Sevilla, c/Tarfia s/n, 41012-Sevilla, Spain Received 30 August 2021; revised 18 January 2022; accepted 4 February 2022 Available online 17 February 2022 Abstract In this article, we first prove some sufficient conditions guaranteeing the existence of invariant sample measures for random dynamical systems via the approach of global random attractors. Then we consider the two-dimensional incompressible Navier-Stokes equations with additive white noise as an example to show how to check the sufficient conditions for concrete stochastic partial differential equations. Our results generalize the Liouville type theorem to the random case and reveal that the invariance of the sample measures is a particular situation of the random Liouville type theorem. ©2022 Elsevier Inc. All rights reserved. MSC: 35B41; 34D35; 76F20 Keywords: Invariant sample measures; Random Liouville type theorem; Random dynamical system; Global random attractor; Stochastic Navier-Stokes equations ✩Supported by NSF of China with No. 11971356, 11271290 and by NSF of Zhejiang Province with No. LY17A010011. Also supported by FEDER and the Spanish Ministerio de Ciencia, Innovación y Universidades project PGC2018-096540-B-I00, and Junta de Andalucía (Spain) under the projects US-1254251 and P18-FR-4509. *Corresponding author. E-mail addresses: [email protected], [email protected] (C. Zhao), [email protected] (J. Wang), [email protected] (T. Caraballo). https://doi.org/10.1016/j.jde.2022.02.007 0022-0396/©2022 Elsevier Inc. All rights reserved. C. Zhao, J. Wang and T. Caraballo Journal of Differential Equations 317 (2022) 474–494 1. Introduction Invariant measures are one of the fundamental objective in the theory of turbulence. This is due to the fact that the measurements of some aspects, say the velocity, kinetic energy and turbulent boundary layer of the turbulent flows are indeed measurements of time-averaged quantities (see e.g. [13,20]). The invariant measures for deterministic evolution equations have been extensively studied, one can refer to [4,10,13,14,16,18,23,26]for well-posed systems and to [3,15,21,24,25] for ill-posed ones. Especially, Łukaszewicz, Real and Robinson [17]used the notion of Generalized Banach limit to construct the invariant measures for general continuous dynamical system on metric spaces. Later, Chekroun and Glatt-Holtz [10] improved the results of [17]to construct invariant measures for dissipative autonomous dynamical systems, and Łukaszewicz and Robinson [18] extended the result of [10]to construct invariant measures for dissipative non-autonomous dynamical systems. Recently, Zhao, Li and Caraballo [22] established some sufficient conditions ensuring the existence of trajectory statistical solutions for general evolution equations, including those systems which possess global weak solutions but without a known result of global uniqueness, say, the three-dimensional (3D) incompressible Navier-Stokes equations. The original motivation of the current article is to investigate the invariant sample measures for dissipative random dynamical systems. Here we will adopt the definition and theory of random dynamical system from [11,27]. In this article, we let (X, d) be a separable and complete metric space and use B(•)to denote the Borel σ-algebra over the space •. Also let (, F, P)be a complete probability space and {θt: −→ , t∈R}be a family of measure preserving transformations on (, F, P). If the mapping (t, ω) → θtωis B(R ×F, F)measurable and {θt}t∈R satisfies the group property, then we call (, F, P, {θt}t∈R)a measurable dynamical system and {θt}t∈Rthe metric dynamical system over the complete probability space (, F, P). Definition 1.1. ([11,27]) A family of mappings ψ(t, τ; ω) :X−→ X, −∞ <τ<t<+∞, parameterized by ω∈, is called a random dynamical system over the measurable dynamical system (, F, P, {θt}t∈R)with state space X, if it satisfies for Palmost surely (Pa.s. for short) ω∈, (a) ψ(t, τ; ω)ψ(τ, s; ω)u =ψ(t, s; ω)u for all s⩽τ⩽tand u ∈X; (b) ψ(t, τ; ω)·is continuous on Xfor all τ⩽t; (c) for all t∈R, u ∈Xthe mapping (s, ω) → ψ(t, s; ω)u is measurable from ((−∞, t] × , B((−∞, t] ×F)to (X, B(X)); (d) for all s<t and u ∈X, the mapping ω→ ψ(t, s; ω)u is measurable from (, F)to (X, B(X)). For a given random dynamical system {ψ(t, τ; ω)}t⩾τ,ω∈over the measurable dynamical system (, F, P, {θt}t∈R)with state space X, we set φ(t −τ,θτω) =ψ(t,τ;ω) and (t) :(ω, u) −→ (θtω,φ(t,ω)u). If φ(t, ω) satisfies the so-called cocycle property (see e.g. [1]), then {(t)}t∈Rmeets the semigroup property (t +τ) =(t)(τ) (see [9]). {(t)}t∈Ris called the skew product on the extended phase space  ×X. The random invariant measures, which plays the essential role in the theory of random dynamical systems, are intimately related to random attractors. If {(t)}t∈R possesses a global random attractor A ⊂ ×X, then Asupports all the invariant measures μof 475 C. Zhao, J. Wang and T. Caraballo Journal of Differential Equations 317 (2022) 474–494 {(t)}t∈Ron the product space  ×X, and μis invariant under the action of the skew product {(t)}t∈R, that is, (t)μ =μfor all t∈R. This result is indeed the same as the deterministic situation. At a glimpse, it seems that one recovers the approach (see e.g. [10,18,22]) of constructing the invariant measure for the deterministic dynamical system. In fact, it is not the case. There produces an additional difficulty, because in the space  ×Xone can only utilize measurability on without any topological tools available. Notice that the invariant measure μlift the probability measure P, which is defined on the sample space , into the extended phase space  ×X, and the projection of μonto equals P. It may be more convenient to work on the phase space X, rather than on the extended phase space  ×X. The invariant property of μon  ×Xcorresponds to the use of random measures ω−→ μωon Xcalled sample measures (cf. [9]). In fact, we can establish that there exists a one-to-one correspondence between any μωon Xand any μon  ×X. Particularly, μ(A) =1if and only if μω(A(ω)) =1, in other words, each sample A(ω) of Asupports the sample measure μω. Definition 1.2. Let {ψ(t, τ; ω)}t⩾τ,ω∈be a random dynamical system over the measurable dynamical system (, F, P, {θt}t∈R)with state space X. A family of Borel probability measures {μθtω}t∈Ron Xis called the invariant sample measures for {ψ(t, τ; ω)}t⩾τ,ω∈, if for Pa.s. ω∈and for all E∈B(X), μθtω(E) =μθτω(ψ−1(t, τ ;ω)E), t,τ ∈R,t⩾τ. The main results of the current article are to present a general approach to construct the invariant sample measures for random dynamical systems, with application to stochastic partial differential equations (PDEs for short). Firstly, we establish some sufficient conditions guaranteeing the existence of invariant sample measures for random dynamical systems via global random attractors. Then we investigate the two-dimensional incompressible stochastic NaiverStokes equations, showing how to check the sufficient conditions for concrete stochastic PDEs. We prove the existence of invariant sample measures for the two-dimensional incompressible Navier-Stokes equations with additive white noise. Our results generalize the Liouville type theorem to the random case and reveal that the invariance of the sample measures is a particular situation of the random Liouville type theorem. We want to point out that there exists an essential difference between the invariant sample measures and the invariant measures for stochastic PDEs. The invariant measures for stochastic PDEs have been extensively studied, see e.g. [6,19] and the references therein. To investigate the invariant measures for stochastic PDEs on its phase space X, loosely speaking, one generally considers the associated Markov transition semigroup {P(t)}t⩾0defined on the set Bb(X)of bounded Borel functions. Then the invariant measures for this stochastic PDEs refer to a probability measure ρon Xsuch that P∗ tρ=ρ, t⩾0, where {P∗ t}t⩾0is the dual semigroup of {Pt}t⩾0. Here, our investigations rely heavily on the theory of infinite dimensional systems and functional analysis. We construct the invariant sample measures via the global random attractor of the random dynamical systems generated by the stochastic PDEs. Our proofs rely on the novel use of a general but elementary functional analysis, valid in any metric space, which concerns the growth of continuous functions in the neighborhood of random compact sets. We want to remark that our idea is inspired by that of [10,18,22], and our abstract result can also be applied to other dissipative stochastic PDEs including those on unbounded domains (see e.g. [2]). 476 C. Zhao, J. Wang and T. Caraballo Journal of Differential Equations 317 (2022) 474–494 The rest of the article is organized as follows. Section 2is devoted to the proof of the sufficient conditions guaranteeing the existence of invariant sample measures for general random dynamical systems via the approach of global random attractors. In Section 3, we first recall some known results concerning the 2D stochastic Navier-Stokes equations, including the wellposedness and the existence of the global random attractor. Then we establish that the generated random dynamical system is continuous with respect to the initial time. Finally, we apply the abstract result obtained in Section 2to the 2D stochastic Navier-Stokes equations. We prove the existence of the invariant sample measures and establish that the 2D stochastic Navier-Stokes equations satisfy the random Liouville type theorem. Moreover, we reveal that the invariance of the sample measures is exactly a particular situation of the random Liouville type theorem. 2. Sufficient condition guaranteeing the existence of invariant sample measures In this section, we first recall some definitions relative to the random dynamical system. Then we prove the sufficient condition guaranteeing the existence of invariant sample measures for random dynamical system via the approach of random attractor. We have introduced the separable and complete metric space (X, d) and its Borel σ-algebra B(X) over X, the measurable dynamical system (, F, P, {θt}t∈R)with the complete probability space (, F, P)and the metric dynamical system {θt}t∈Ron (, F, P). Besides these, we denote by distX(A, B) =sup a∈A inf b∈Bd(a, b) the Hausdorff semidistance between A⊂Xand B⊂X. Particularly, distX(a, B) =inf b∈Bd(a, b) and distX(a, b) =d(a, b). Also, we will use some other definitions relative to the random dynamical system. A random set can be regarded as a family of sets parameterized by the random parameter ωand satisfies some measurability property. Precisely, a random set Bcan be identified by the family of its ω-fibers B(ω), defined by B(ω) ={u∈X:(x, ω) ∈B},ω∈. As a random set B⊂X×possesses closed fibers, it is said to be a closed random set if and only if for every u ∈Xthe mapping ω∈ −→ distX(u, B(ω)) is measurable (cf. [7,8]). When the fibers of Bare compact, Bis called to be a random compact set. Definition 2.1. Let {ψ(t, τ; ω)}t⩾τ,ω∈be a random dynamical system over the measurable dynamical system (, F, P, {θt}t∈R)with the state space (X, d). A random subset {A(ω)}ω∈of X is called a global random attractor for {ψ(t, τ; ω)}t⩾τ,ω∈on (X, d), if the following conditions hold (1) (Random compactness) For Pa.s. ω∈, A(ω) is compact in X; (2) (Invariance) For Pa.s. ω∈, {A(ω)}ω∈is invariant in the sense that ψ(t,τ;ω)A(θτω) =A(θtω), ∀τ⩽t; (3) (Attracting property) For Pa.s. ω∈, for every t∈Rand B⊂Xbounded, there holds lim τ→−∞ distX(ψ(t, τ;ω)B,A(θtω)) =0. 477 C. Zhao, J. Wang and T. Caraballo Journal of Differential Equations 317 (2022) 474–494 We now begin to prove two auxiliary lemmas which will play the key role when we construct the invariant sample measures. In the sequel, we use C(•)to denote the set of continuous functions defined on the space •. Lemma 2.1. Let (, F, P)be a complete probability space and {K(ω)}ω∈be a random compact subset of the separable and complete metric space (X, d). Then for every g∈C(X), for P a.s. ω∈there corresponds an ω>0such that sup v∈O(K(ω);ω) |g(v)|<+∞, where O(K(ω); ω) ={v∈X:distX(v, K(ω)) < ω}. Proof. Let {K(ω)}ω∈be a random compact subset of the separable and complete metric space (X, d). Without loss of generality, we consider a fixed g∈C(X) and a fixed ω∈. Then for every u ∈K(ω) one can pick δ(u, ω) such that for every v∈O(u;δ(u,ω)) ={v∈X:d(v,u) <δ(u,ω)} there holds |g(u) −g(v)| <1. Choosing numbers δ(u, ω) in this way, we obtain an open covering ω=O(u;δ(u,ω)/3):u∈K(ω) for K(ω). Note that K(ω) is compact in X. We can extract from the open covering ωa finite one (m) ω=O(u1;δ(u1,ω)/3), O(u2;δ(u2,ω)/3), ··· ,O(um;δ(um,ω)/3). Choose ω=min δ(u1,ω)/3,δ(u 2,ω)/3,··· ,δ(u m,ω)/3,c=1+max 1⩽j⩽m |g(uj)|. Now for any given v∈O(K(ω); ω), we can pick u ∈K(ω) so that d(v, u) <2ω. Also we can choose ujmeeting d(u, uj) <δ(u j, ω)/3 because (m) ωcovers K(ω). Therefore, we have d(v,uj)⩽d(v,u)+d(u,uj)<2ω+δ(uj,ω)/3⩽δ(uj,ω) and |g(v)|⩽1+|g(uj)|⩽1+max 1⩽j⩽m |g(uj)|=c. This ends the proof.  478 C. Zhao, J. Wang and T. Caraballo Journal of Differential Equations 317 (2022) 474–494 Lemma 2.2. Let (, F, P)be a complete probability space and {K(ω)}ω∈a random compact subset of the separable and complete metric space (X, d), and let g, h ∈C(X) satisfying for P a.s. ω∈g(ξ) =h(ξ) for every ξ∈K(ω). Then for every >0there corresponds for Pa.s. ω∈a γ(, ω) >0such that sup v∈O(K(ω);γ(,ω)) |g(v) −h(v)|<. Proof. Let (, F, P)be a complete probability space and {K(ω)}ω∈a random compact subset of the separable and complete metric space (X, d). Consider given g, h ∈C(X). Fix >0 and ω∈. For every ξ∈K(ω) we can pick γ(ξ, , ω) yielding |g(ξ) −g(v)|+|h(ξ) −h(v)|< whenever v∈O(ξ;γ (ξ, , ω)). Obviously, {O(ξ; γ(ξ, , ω)) :ξ∈K(ω)}is an open covering of K(ω). Due to the compactness of K(ω) in X, one can cover K(ω) with finite collection O(ξ1;γ(ξ 1,,ω)/3), O(ξ2;γ(ξ 2,,ω)/3), ··· ,O(ξk;γ(ξ k,,ω)/3). Put γ(, ω) =min 1⩽j⩽k γ(ξ j,,ω) 3and we have O(K(ω);γ(,ω))⊂O k  j=1 O(ξj;γ(ξ j,,ω) 3;γ(,ω) ⊂ k  j=1 O(ξj;γ(ξ j,,ω)). Now for any v∈O(K(ω); γ(, ω)), we may pick jsuch that v∈O(ξj; γ(ξ j, , ω)). Note that g(ξj) =h(ξj). Hence |g(v) −h(v)|⩽|g(v) −g(ξj)|+|h(ξj)−h(v)|<. The proof is complete.  To state and prove the main result of this section, we need to recall the definition of generalized Banach limit. Definition 2.2. ([13,18]) A generalized Banach limit is any linear functional, denoted by LIMt→+∞, defined on the space of all bounded real-valued functions on [0, +∞)and satisfying (1) LIMt→+∞ζ(t) ⩾0for nonnegative functions ζ(·)on [0, +∞); (2) LIMt→+∞ζ(t) =lim t→+∞ ζ(t) if the usual limit lim t→+∞ ζ(t) exists. Let B+be the collection of all bounded real-valued functions on [0, +∞). For any generalized Banach limit LIMt→+∞, the following useful property |LIMt→+∞ζ(t)|⩽lim sup t→+∞ |ζ(t)|,∀ζ(·)∈B+,(2.1) 479 C. Zhao, J. Wang and T. Caraballo Journal of Differential Equations 317 (2022) 474–494 is presented in [13, (1.38)] and in [10, (2.3)]. Notice that we will consider the asymptotic behavior τ→−∞of ψ(t, τ; ω)•. Therefore, we require generalized limits as τ→−∞. For a given real-valued function ζdefined on (−∞, 0] and a given Banach limit LIMT→+∞, we define LIMt→−∞ζ(t)=LIMt→+∞ζ(−t). In the sequel, for a given Borel probability measure μon Xand a function ∈C(X), we use  X (u)dμ(u) to denote the Bochner integral. The main result of this section reads as follows. Theorem 2.1. Let (X, d) be a complete metric space and {ψ(t, τ; ω)}t⩾τ,ω∈be a random dynamical system over the measurable dynamical system (, F, P, {θt}t∈R)with state space (X, d). Suppose that (i) {ψ(t, τ; ω)}t⩾τ,ω∈possesses a global random attractor {A(ω)}ω∈on X; (ii) for each given t∈R, u ∈Xand for Pa.s. ω∈, the X-valued mapping τ−→ ψ(t, τ; ω)u is continuous and bounded on (−∞, t]. Then for a given continuous mapping v(·) :R → Xand a generalized Banach limit LIMt→+∞, there exists for Pa.s. ω∈a family of Borel probability measures {μθtω}t∈Ron Xsuch that the support of μθtωis contained in A(θtω) and  X (u)dμθtω(u) = A(θtω) (u)dμθtω(u) =LIMτ→−∞ 1 t−τ t  τ (ψ(t,s;ω)v(s))ds(2.2) =LIMτ→−∞ 1 t−τ t  τ X (ψ(t,s;ω)u)dμθsω(u)ds(2.3) for any nonnegative, real-valued continuous functional on X. Moreover, for Pa.s. ω∈, μθtω is invariant under the action of the random dynamical system {ψ(t, τ; ω)}t⩾τ,ω∈in the sense that  A(θtω) (u)dμθtω(u) = A(θτω) (ψ(t,τ;ω)u)dμθτω(u), ∀t⩾τ. (2.4) Proof. Let LIMt→+∞ be a given generalized Banach limit, v(·) :R → Xa continuous map and (·)a nonnegative, real-valued continuous functional on X. We first prove that for each given t∈Rand for Pa.s. ω∈, the function s−→ (ψ(t, s; ω)v(s)) is bounded on (−∞, t]. Indeed, we claim that there exists some negative t0 480 C. Zhao, J. Wang and T. Caraballo Journal of Differential Equations 317 (2022) 474–494 sufficiently large such that, for Pa.s. ω∈, the function s−→ (ψ(t, s; ω)v(s)) is bounded on (−∞, t0]. Assume that this is not the case. Then, there is a sequence {sn}∞ n=1with sn→−∞as n →∞such that |(ψ(t,sn;ω)v(sn))|→+∞,n→∞.(2.5) Now by condition (i), the random dynamical system {ψ(t, τ; ω)}t⩾τ,ω∈possesses a global random attractor {A(ω)}ω∈on X. From the attracting property of the global random attractor, we see that, for Pa.s. ω∈and every >0, there exists a time s(, ω, t) such that ψ(t,τ;ω)v(τ) ∈O(A(θtω);), ∀τ⩽s(,ω,t). (2.6) By Lemma 2.1, we can choose ω>0 such that Cω=sup O(A(θtω);ω) |(u)|<+∞.(2.7) Then (2.6) and (2.7) contradict with (2.5). At the same time, from condition (ii) we see that, for Pa.s. ω∈, the X-valued mapping s−→ (ψ(t, s; ω)v(s)) is continuous on (−∞, t]. Thus it is bounded on each compact interval [t0, t]. Secondly, for each given t∈Rand for Pa.s. ω∈, we define Lω,v() =LIMs→−∞ 1 t−s t  s (ψ(t,η;ω)v(η))dη(2.8) for nonnegative function ∈C(X). Then, by the above analysis and the property of the generalized Banach limit, we see that the function s−→ 1 t−s t  s (ψ(t,η;ω)v(η))dη is bounded on (−∞, t]for Pa.s. ω∈. Hence Lω,v() defined by (2.8)is well defined as a positive linear functional on C(X) for Pa.s. ω∈. Thirdly, we prove that, for Pa.s. ω∈, Lω,v() depends only on the values of on A(θtω). Factually, take nonnegative 1and 2in C(X) with 1(·) =2(·)on A(θtω) for Pa.s. ω∈. Then for any >0, by Lemma 2.2 we can find for Pa.s. ω∈a γ(, ω) >0 such that |1(u) −2(u)|</2,whenever u∈O(A(θtω);γ (, ω)). (2.9) By the attracting property of the global random attractor, we can pick s0such that distX(ψ(t, s;ω)v(s),A(θtω)) ⩽γ(,ω) for all s⩽s0. Then, for every s⩽s0there corresponds a vs∈A(θtω) such that 481 C. Zhao, J. Wang and T. Caraballo Journal of Differential Equations 317 (2022) 474–494 d(ψ(t,s;ω)v(s),vs)⩽γ(,ω). Thus, by Lemma 2.2 we have for s⩽s0that |1(ψ(t, s;ω)v(s)) −2(ψ(t, s;ω)v(s))| ⩽|1(ψ(t, s;ω)v(s)) −1(vs)|+|1(vs)−2(vs)|+|2(ψ(t, s;ω)v(s)) −2(vs)|<. By condition (ii) and noticing that v(·) :R → Xis a continuous map, we find that sup η∈[s0,t] {|1(ψ(t, η;ω)v(η))|+|2(ψ(t, η;ω)v(η))|} is bounded by a constant independent of s. Therefore, by the property of the generalized Banach limit, we have |Lω,v(1−2)|= LIMs→−∞ 1 t−s t  s1(ψ(t, η;ω)v(η)) −2(ψ(t, η;ω)v(η))dη =LIMs→−∞ 1 t−s s0  s1(ψ(t, η;ω)v(η)) −2(ψ(t, η;ω)v(η))dη +LIMs→−∞ 1 t−s t  s01(ψ(t, η;ω)v(η)) −2(ψ(t, η;ω)v(η))dη ⩽lim sup s→−∞ s0−s t−s +lim sup s→−∞ (t −s0) t−ssup η∈[s0,t] {|1(ψ(t, η;ω)v(η))|+|2(ψ(t, η;ω)v(η))|} =. By the arbitrariness of , we obtain Lω,v(1−2) =0. Fourthly, for Pa.s. ω∈, we define Gω,v() =Lω,v(() for ∈C(A(θtω)), where () is the extension of from C(A(θtω)) to C(X) given by the Tietze theorem (see [13, Theorem A.7]). Then Gω,v(·)is a positive linear functional on C(A(θtω)). Notice that A(θtω) is compact in X. A(θtω) is obviously a locally compact topological space. By the Kakutani-Riesz Representation Theorem (see [13, Theorem A.1]), we obtain that there exists a unique positive, finite, Borel measure μθtωon A(θtω) such that Gω,v() = A(θωt) (u)dμθtω(u). (2.10) We extend μθtωby zero to Borel measure on X, which is still denoted by μθtω: μθtω(E) =μθtω(E ∩A(θtω)), E ∈B(X). 482 C. Zhao, J. Wang and T. Caraballo Journal of Differential Equations 317 (2022) 474–494 We next estimate the term (v(τ, s; ω, v∗) −v∗, v∗)in (3.9). From (3.15) and (3.16), we see that there exists a positive M(ω, s∗, v∗)independent of ssuch that max s∗−1⩽η⩽s∗+1 v(η,s;ω,v∗)⩽M(ω,s∗,v ∗), ∀s∈[s∗−1,η]. Then by the density of Vin H, we find that for above there exists an element ˜v∈Vsuch that ˜v−v∗ ⩽2 8(M(ω,s∗,v∗)+v∗). Thus we have for s∈(s∗, s∗+δ1)and τ∈(s, s∗+δ1)that |(v(τ, s;ω,v∗)−v∗,v ∗)|⩽|(v(τ, s;ω,v∗)−v∗,˜v)|+|(v(τ, s;ω,v∗)−v∗,˜v−v∗)| ⩽|v(τ,s;ω,v∗)−v∗,˜v| + 2 8. (3.18) We shall estimate the term |v(τ, s; ω, v∗) −v∗, ˜v| in (3.18). Observe that |v(τ,s;ω,v∗)−v∗,v ∗| =  τ  s d dηv(η,s;ω,v∗)dη, ˜v ˜vVτ  s d dηv(η,s;ω,v∗)2 V∗dη1/2 (τ −s)1/2. (3.19) By (3.5) and the embedding V→V∗, we have dv(η,s;ω,v∗) dη2 V∗ Av(η, s;ω,v∗)2 V∗+B(v(η,s;ω,v∗)+z(η,ω))2 V∗+f2+z(η, ω)2+Az(η, ω)2 V∗. (3.20) Now using the property of the operators A, Band the embedding V→H→V∗, we obtain ⎧ ⎨ ⎩ Av(η, s;ω,v∗)2 V∗=v(η,s;ω,v∗)2 V,z(η, ω)2 V∗eα 2|η|r(ω), Az(η, s;ω,v∗)2 V∗=z(η, ω)2eα 2|η|r(ω), B(v(η,s;ω,v∗)+z(η, ω))2 V∗v(η,s;ω,v∗)2 V+eα 2|η|r(ω). (3.21) Inserting (3.20) and (3.21)into (3.19)gives |v(τ,s;ω,v∗)−v∗,v ∗| ˜vVτ  s (v(η,s,ω,v∗)2 V+eα 2|η|r(ω)+f2)dη1/2 (τ −s)1/2.(3.22) It then follows form (3.15), (3.16) and (3.22) that for above >0 there exists a positive constant δ2=δ2(, s∗, v∗, ω) such that for Pa.s. ω∈, 489 C. Zhao, J. Wang and T. Caraballo Journal of Differential Equations 317 (2022) 474–494 |v(τ,s;ω,v∗)−v∗,˜v| ⩽2 8,whenever s∈(s∗,s ∗+δ2), τ ∈(s, s∗+δ2). (3.23) Picking δ=min{δ1, δ2}, we obtain (3.8) from (3.9), (3.17), (3.18) and (3.23). This ends the proof of Lemma 3.1. Similarly to Lemma 3.1, we can also prove that for given u∗∈H, t∈R, and for Pa.s. ω∈, the H-valued mapping s→ ψ(t, s; ω)u∗is left continuous on (−∞, t]. Therefore, we conclude that for given u∗∈H, t∈R, and for Pa.s. ω∈, the H-valued mapping s→ ψ(t, s; ω)u∗is continuous on (−∞, t]. From this continuity and the attracting property of the global random attractor, we see that H-valued mapping s→ ψ(t, s; ω)u∗is bounded on (−∞, t]. Now, thanks to the abstract theory of Theorem 2.1, we can claim that the 2D incompressible Navier-Stokes equations with additive white noise possesses a family of invariant sample measures on the phase space H. This result reads as follows. Theorem 3.1. Suppose that condition (H) is satisfied and f∈H. Let {ψ(t, τ; ω)}t⩾τ,ω∈be the random dynamical system generated by problem (3.1)-(3.3)over the metric dynamical system (, F, P, {θt}t∈R)with the state space H. Let {A(ω)}ω∈be the global random attractor guaranteed by Proposition 3.1(3). Then for a given generalized Banach limit LIMt→+∞ and given continuous function v(·) :R → H, there exists for Pa.s. ω∈a family of Borel probability measures {μθtω}t∈Ron Hsuch that the support of μθtωis contained in A(θtω) and  H (u)dμθtω(u) = A(θtω) (u)dμθtω(u) =LIMτ→−∞ 1 t−τ t  τ (ψ(t,s;ω)v(s))ds =LIMτ→−∞ 1 t−τ t  τ H (ψ(t,s;ω)u)dμθsω(u)ds (3.24) for any nonnegative, real-valued continuous functional on H. Moreover, for Pa.s. ω∈, μθtω is invariant under the action of the random dynamical system {ψ(t, τ; ω)}t⩾τ,ω∈in the sense that  A(θtω) (u)dμθtω(u) = A(θτω) (ψ(t,τ;ω)u)dμθτω(u), ∀t⩾τ. (3.25) We next investigate the random Liouville type theorem for the 2D incompressible NavierStokes equations with additive white noise. To this end, we need the definition of the class of test functions. Write equation (3.2)as du=F(u,t,ω)=−(νAu +B(u))dt+fdt+ m  j=1 jdwj.(3.26) 490 C. Zhao, J. Wang and T. Caraballo Journal of Differential Equations 317 (2022) 474–494 Definition 3.1. We define the class Tof test functions as the set of nonnegative, real-valued continuous functionals ϒ=ϒ(w) on Hthat are bounded on bounded subset of Hand satisfy (1) for any w∈V, the Fréchet derivative ϒ(w) exists: for each w∈Vthere exists an element ϒ(w) such that |ϒ(w +v) −ϒ(w) −(ϒ(w), v)| vV −→ 0as vV→0,v∈V; (2) ϒ(w) ∈Vfor all w∈V, and the mapping w−→ ϒ(w) is continuous and bounded as a functional from Vto V; (3) for every global solution u(t, ω) of equation (3.2), there holds for Pa.a. ω∈ d dtϒ(u(t, ω)) =F(u,t,ω),ϒ(u).(3.27) For the existence of functions satisfying Definition 3.1, one can refer to [22, Definition 2.5]. Here we omit the details. Theorem 3.2. Let the conditions of Theorem 3.1 hold. Then, for Pa.s. ω∈, the following random Liouville type equation  A(θtω) ϒ(u)dμθtω(u) − A(θτω) ϒ(u)dμθτω(u) = t  τ H F(u,η,ω),ϒ(u)dμθηω(u)dη, ∀t⩾τ, (3.28) holds for all test functions ϒ∈T. Proof. Let ϒ∈Tbe given. By (3.27), we have for Pa.s. ω∈that ϒ(ψ(t, s;ω)u) −ϒ(ψ(τ,s;ω)u) = t  τ F(u,η,ω),ϒ(u)dη, ∀t⩾τ. (3.29) Now for any s<τ, let u∗∈Hand u(η, ω) =ψ(η, s; ω)u∗for η⩾s. By (3.29), ϒ(ψ(t, s;ω)u∗)−ϒ(ψ(τ,s;ω)u∗)= t  τ F(ψ(η,s;ω)u∗,η,ω),ϒ(ψ(η, s;ω)u∗)dη. (3.30) Using (3.24), (3.30) and Fubini’s theorem, we arrive at 491 C. Zhao, J. Wang and T. Caraballo Journal of Differential Equations 317 (2022) 474–494  H ϒ(u)dμθtω(u) − H ϒ(u)dμθτω(u) = A(θtω) ϒ(u)dμθtω(u) − A(θτω) ϒ(u)dμθτω(u) =LIMγ→−∞ 1 τ−γ τ  γ H (ϒ(ψ(t, s;ω)u∗)) −ϒ(ψ(τ,s;ω)u∗))dμθsω(u∗)ds =LIMγ→−∞ 1 τ−γ τ  γ H t  τ F(ψ(η,s;ω)u∗,η,ω),ϒ(ψ(η, s;ω)u∗)dηdμθsω(u∗)ds =LIMγ→−∞ 1 τ−γ τ  γ t  τ H F(ψ(η,s;ω)u∗,η,ω),ϒ(ψ(η, s;ω)u∗)dμθsω(u∗)dηds. (3.31) Now using the invariance of the random dynamical system ψ(η, s; ω) =ψ(η, τ; ω)ψ(τ, s; ω) and (3.25), we obtain  H F(ψ(η,s;ω)u∗,η,ω),ϒ(ψ(η, s;ω)u∗)dμθsω(u∗) = H F(ψ(η,τ;ω)ψ(τ,s;ω)u∗,η,ω),ϒ(ψ(η, τ ;ω)ψ(τ,s;ω)u∗)dμθsω(u∗) = H F(ψ(η,τ;ω)u∗,η,ω),ϒ(ψ(η, τ ;ω)u∗)dμθτω(u∗), which is independent of s. It then follows from (3.31) that  A(θtω) ϒ(u)dμθtω(u) − A(θτω) ϒ(u)dμθτω(u) = t  τ H F(ψ(η,τ;ω)u∗,η,ω),ϒ(ψ(η, τ ;ω)u∗)dμθτω(u∗)dη = t  τ H F(u,η,ω),ϒ(u)dμθηω(u)dη. The proof is complete.  The result of Theorem 3.2 can be regarded as the random Liouville type theorem. If the random statistical equilibrium has been reached by the addressed stochastic Navier-Stokes system, 492 C. Zhao, J. Wang and T. Caraballo Journal of Differential Equations 317 (2022) 474–494 then the statistical informations do not change with time, that is (u(·, ω)) =0. In this situation, we follow from (3.25) and (3.28) that for Pa.s. ω∈  A(θtω) ϒ(u)dμθtω(u) = A(θτω) ϒ(ψ(t, τ;ω)u)dμθτω(u) = A(θτω) ϒ(u)dμθτω(u), τ ∈R.(3.32) (3.32) describes exactly the invariant property of the sample measures {μθtω}t∈Runder the action of the random dynamical system {ψ(t, τ; ω)}t⩾τ,ω∈. It reveals that for Pa.s. ω∈, the shape of the global random attractor A(θtω) could change randomly with the evolution of time from τ to t, along with the sample point ω∈, but the measures of A(θτω) and A(θtω) coincide with each other. This is the random version of the Liouville Theorem in Statistical Mechanics. Thus we say that the invariant sample measures {μθtω}t∈Rof the stochastic Navier-Stokes equations satisfies a random Liouville type theorem. Acknowledgments The authors warmly thank the anonymous referee for his/her careful reading of the article and many pertinent remarks that lead to various improvements to this article. References [1] A. Arnold, Random Dynamical Systems, Springer, Berlin, 1998. [2] P. Bates, K. Lu, B. Wang, Random attractors for stochastic reaction-diffusion equations on unbounded domains, J. Differ. Equ. 246 (2009) 845–869. [3] A. Bronzi, C.F. Mondaini, R. Rosa, Trajectory statistical solutions for three-dimensional Navier-Stokes-like systems, SIAM J. Math. Anal. 46 (2014) 1893–1921. [4] A. Bronzi, C.F. Mondaini, R. Rosa, Abstract framework for the theory of statistical solutions, J. Differ. Equ. 260 (2016) 8428–8484. [5] Z. Brze´zniak, Y. Li, Asymptotic compactness and absorbing sets of 2D stochastic Navier-Stokes equations on some unbounded domains, Trans. Am. Math. Soc. 358 (2006) 5587–5629. [6] Z. Brze´zniak, E. Motyl, M. Ondreját, Invariant measures for the stochastic Navier-Stokes equations in unbounded 2D domains, Ann. Probab. 45 (2017) 3145–3201. [7] T. Caraballo, X. Han, Applied Nonautonomous and Random Dynamical Systems, Springer, BCAM SpringerBriefs, 2016. [8] C. Castaing, M. Valadier, Convex Analysis and Measurable Multifunctions, Lecture Notes in Mathematics, vol. 58, Springer-Verlag, Berlin-New York, 1977. [9] M. Chekroun, E. Simonner, M. Ghil, Stochastic climate dynamics: random attractors and time-dependent invariant measures, Physica D 240 (2011) 1685–1700. [10] M. Chekroun, N.E. Glatt-Holtz, Invariant measures for dissipative dynamical systems: abstract results and applications, Commun. Math. Phys. 316 (2012) 723–761. [11] H. Crauel, A. Debussche, F. Flandoli, Random attractors, J. Dyn. Differ. Equ. 9(2) (1997) 307–341. [12] F. Flandoli, B. Maslowski, Ergodicity of the 2D Navier-Stokes equation under random perturbation, Commun. Math. Phys. 172 (1995) 119–141. [13] C. Foias, O. Manley, R. Rosa, R. Temam, Navier-Stokes Equations and Turbulence, Cambridge University Press, Cambridge, 2001. [14] C. Foias, R. Rosa, R. Temam, Properties of stationary statistical solutions of the three-dimensional Navier-Stokes equations, J. Dyn. Differ. Equ. 31 (2019) 1689–1741. 493 C. Zhao, J. Wang and T. Caraballo Journal of Differential Equations 317 (2022) 474–494 [15] H. Jiang, C. Zhao, Trajectory statistical solutions and Liouville type theorem for nonlinear wave equations with polynomial growth, Adv. Differ. Equ. 3(4) (2021) 107–132. [16] G. Łukaszewicz, Pullback attractors and statistical solutions for 2-D Navier-Stokes equations, Discrete Contin. Dyn. Syst., Ser. B 9 (2008) 643–659. [17] G. Łukaszewicz, J. Real, J.C. Robinson, Invariant measures for dissipative dynamical systems and generalised Banach limits, J. Dyn. Differ. Equ. 23 (2011) 225–250. [18] G. Łukaszewicz, J.C. Robinson, Invariant measures for non-autonomous dissipative dynamical systems, Discrete Contin. Dyn. Syst. 34 (2014) 4211–4222. [19] U. Manna, D. Mukherjee, Weak solutions and invariant measures of stochastic Oldroyd-B type model driven by jump noise, J. Differ. Equ. 272 (2021) 760–818. [20] X.M. Wang, Upper-semicontinuity of stationary statistical properties of dissipative systems, Discrete Contin. Dyn. Syst. 23 (2009) 521–540. [21] C. Zhao, T. Caraballo, Asymptotic regularity of trajectory attractor and trajectory statistical solution for the 3D globally modified Navier-Stokes equations, J. Differ. Equ. 266 (2019) 7205–7229. [22] C. Zhao, Y. Li, T. Caraballo, Trajectory statistical solutions and Liouville type equations for evolution equations: abstract results and applications, J. Differ. Equ. 269 (2020) 467–494. [23] C. Zhao, Y. Li, G. Łukaszewicz, Statistical solution and partial degenerate regularity for the 2D non-autonomous magneto-micropolar fluids, Z. Angew. Math. Phys. 71 (2020) 141. [24] C. Zhao, Z. Song, T. Caraballo, Strong trajectory statistical solutions and Liouville type equations for dissipative Euler equations, Appl. Math. Lett. 99 (2020) 105981. [25] C. Zhao, Y. Li, Z. Song, Trajectory statistical solutions for the 3D Navier-Stokes equations: the trajectory attractor approach, Nonlinear Anal., Real World Appl. 53 (2020) 103077. [26] C. Zhao, T. Caraballo, G. Łukaszewicz, Statistical solution and Liouville type theorem for the Klein-GordonSchrödinger equations, J. Differ. Equ. 281 (2021) 1–32. [27] S. Zhou, F. Yin, Z. Ouyang, Random attractor for damped nonlinear wave equations with white noise, SIAM J. Appl. Dyn. Syst. 4 (2005) 883–903. 494