Random Dynamics of the Stochastic Boussinesq Equations Driven by Lévy Noises

Abstract

This paper is devoted to the investigation of random dynamics of the stochastic Boussinesq equations driven by Lévy noise. Some fundamental properties of a subordinator Lévy process and the stochastic integral with respect to a Lévy process are discussed, and then the existence, uniqueness, regularity, and the random dynamical system generated by the stochastic Boussinesq equations are established. Finally, some discussions on the global weak solution of the stochastic Boussinesq equations driven by general Lévy noise are also presented.

1. Introduction

Dynamical systems driven by non-Gaussian processes, such as Lévy processes, have attracted a lot of attention recently. Ordinary differential equations driven by Lévy processes have been summarized in [1]. Peszat and Zabczyk [2] have presented a basic framework for partial differential equations driven by Lévy processes.

The Navier–Stokes fluid equations are often coupled with other equations, especially, with the scalar transport equations for fluid density, salinity, or temperature. These coupled equations model a variety of phenomena arising in environmental, geophysical, and climate systems. The related Boussinesq fluid equations [3–5] under Gaussian fluctuations have been recently studied, for example, existence and uniqueness of solutions [6], stochastic flow, dynamical impact under random dynamical boundary conditions [7, 8], and large deviation principles [9, 10], among others.

Motivated by a recent work on a simple stochastic partial differential equation with Lévy noise [11], we study the stochastic Boussinesq equations driven by some special Lévy noises, and we consider the random dynamics of this stochastic system. Specifically, for a given bounded 𝐶¹-smooth domain 𝐷⊂ℝ² with sufficient smooth boundary, we consider the following stochastic Boussinesq equations driven by subordinator Lévy noise: 𝑑𝑢=1𝑑𝑡1ReΔ𝑢−∇𝑝−𝑢⋅∇𝑢−Fr²𝜃𝑒₂+𝑑𝑌₁(𝑡),on𝐷×ℝ₊,𝑑𝜃=1𝑑𝑡RePrΔ𝜃−𝑢⋅∇𝜃+𝑑𝑌₂(𝑡),on𝐷×ℝ₊,div𝑢=0,on𝐷×ℝ₊,𝑢(0)=𝑢₀,𝜃(0)=𝜃₀, (1) where 𝑢=𝑢(𝑥,𝑡)=(𝑢¹,𝑢²)∈ℝ² is the velocity vector, 𝜃=𝜃(𝑡,𝑥)∈ℝ is salinity, 𝑝(𝑡,𝑥)∈ℝ is the pressure term, 𝑥=(𝜉,𝜂)∈𝐷⊂ℝ²,  Δ denotes the Laplacian operator, and ∇ denotes the gradient operator. Moreover, Fr is the Froude number, Re is the Reynolds number, Pr is the Prandtl number, and 𝑒₂∈ℝ² is a unit vector in the upward vertical direction. The initial data 𝑢₀, 𝜃₀ are given. Both 𝑌₁(𝑡) and 𝑌₂(𝑡) are subordinator Lévy processes on Hilbert spaces 𝐻₁ and 𝐻₂, which will be specified later. The present paper is devoted to the existence, uniqueness, regularity, and the cocycle property of solution for stochastic Boussinesq equations (1).

This paper is organized as follows. In Section 2, we first present some properties of the subordinator Lévy process 𝑌(𝑡), then review some fundamental properties of the stochastic integral with respect to Lévy process 𝑌(𝑡). Section 3 is devoted to the existence, uniqueness, regularity, and the cocycle property of the stochastic Boussinesq equations. Finally, some discussions on the global weak solution of stochastic Boussinesq equations driven by general Lévy noise are also presented in Section 4.

2. Preliminaries

In this section, we introduce some operators and fraction spaces and then present some properties of the subordinator Lévy process 𝑌(𝑡) and the stochastic integral with respect to Lévy process 𝑌(𝑡).

In order to reformulate the stochastic Boussinesq equations (1) as an abstract stochastic evolution, we introduce the following function spaces.

Denote 𝐿²(𝐷) to be the space of functions defined on 𝐷, which are 𝐿²-integrable with respect to the Lebesgue measure 𝑑𝑥=𝑑𝑥₁𝑑𝑥₂, endowed with the usual scalar product and norm, that is, for 𝑢,𝑣∈𝐿²(𝐷), (𝑢,𝑣)=_𝐷𝑢(𝑥)𝑣(𝑥)𝑑𝑥,|𝑢|={(𝑢,𝑢)}^1/2. (2)

For 𝑚∈ℤ⁺∪{0} and 𝑞∈(1,∞), define 𝐻^𝑚,𝑞(𝐷)=𝑢∈𝐿^𝑞(𝐷)∶𝐷^𝛼𝑢∈𝐿^𝑞(𝐷),𝛼∈ℕ²,0≤|𝛼|≤𝑚 (3) as the usual Soblev space with scalar product (𝑢,𝑣)_𝑚=_{|𝛼|0≤≤𝑚}𝐷^𝛼𝑢,𝐷^𝛼𝑣_{𝐿𝑞(𝐷)} (4) and the induced norm |𝑢|_𝑚=‖𝑢‖_𝐻^𝑚_(𝐷)=_{|𝛼|0≤≤𝑚}𝐷^𝛼𝑢^𝑞_{𝐿𝑞(𝐷)}^1/2, (5) where 𝐷^𝛼𝑢 is the 𝛼th order weak derivative of 𝑢.

For 𝑠∈ℝ, let 𝐻^𝑠,𝑞(𝐷) be defined by the complex interpolation method [12] as follows. 𝐻^𝛽,𝑞(𝐷)=𝐻^𝑘,𝑞(𝐷),𝐻^𝑚,𝑞(𝐷)_𝜃, (6) where 𝑘,𝑚∈ℕ, 𝜃∈(0,1), and 𝑘<𝑚 are chosen to satisfy 𝛽=(1−𝜃)𝑘+𝜃𝑚. (7) The closure of 𝐶^∞₀(𝐷) in the Banach space 𝐻^𝑠,𝑞(𝐷), 𝑠≥0, 𝑞∈(1,∞), will be denoted by 𝐻₀^𝑠,𝑞(𝐷).

The following product spaces are needed: 𝒱=𝑢=𝑢¹,𝑢²∈(𝐶^∞(𝐷))²×𝐶^∞(𝐷),𝕃,∇⋅𝑢=0_𝑞(𝐷)=𝐿^𝑞(𝐷)²×𝐿^𝑞(𝐷),ℍ^𝑠,𝑞(𝐷)=𝐻^𝑠,𝑞(𝐷)²×𝐻^𝑠,𝑞(𝐷)=𝑢=𝑢¹,𝑢²∈𝐻^𝑠,𝑞(𝐷)²×𝐻^𝑠,𝑞(𝐷),,𝕍∇⋅𝑢=0_𝑠,𝑞(𝐷)=𝑢=𝑢¹,𝑢²∈ℍ₀^𝑠,𝑞(𝐷),∇⋅𝑢=0. (8)

Let 𝐇^𝑠,𝑞(𝐷) denote the closure of 𝒱 with respect to the ℍ^𝑠,𝑞-norm, 𝑉^𝑠,𝑞(𝐷) denote the closure of 𝒱 with respect to the 𝕍^𝑠,𝑞-norm, and 𝑉^′ be the dual space of 𝑉^𝑠,𝑞(𝐷). In particular, we denote by 𝐇^1,2 and 𝑉^1,2  𝐻 and 𝑉, respectively.

Denote 𝐴₁𝑢=Δ𝑢(𝑡),𝐴₂(𝑡),𝐵𝜃=Δ𝜃₁𝑢₁,𝑢₂=𝑢₁⋅∇𝑢₂,𝐵₂𝑢₁,𝜃₂=𝑢₁⋅∇𝜃₂,𝑈₀=𝑢₀𝜃₀(𝑡)=𝑢(𝑡)𝜃(𝑡)𝑅(𝑈)=−1∈𝐻,𝑈∈𝑉,Fr²𝜃𝑒₂,0,𝑌(𝑡)=𝑌₁(𝑡)𝑌₂(𝑡)∈𝐻=𝐻₁×𝐻₂, (9) where 𝜈=1/Re and 𝑘=1/RePr.

Now, we define the following two operators: 𝐴∶𝑉⟶𝑉^′∶𝑉∋𝑈=(𝑢,𝜃)⟼𝐴𝑈=𝜈𝐴₁𝑢𝑘𝐴₂𝜃,𝐵∶𝑉×𝑉⟶𝑉^′∶𝑉×𝑉∋𝑈₁,𝑈₂⟼𝐵𝑈₁,𝑈₂=𝐵₁𝑢₁,𝑢₂𝐵₂𝑢₁,𝜃₂. (10)

Then, the stochastic Boussinesq system (1) can be rewritten as the following abstract stochastic evolution equation: (𝑡)+[(𝑡)(𝑈(𝑡)(𝑡(𝑈(𝑡(𝑡),𝑈(0)𝑑𝑈𝐴𝑈+𝐵,𝑈))+𝑅))]𝑑𝑡=𝑑𝑌=𝑈₀. (11)

In order to apply the technique of the reproducing Kernel Hilbert space, it is better to introduce the definition 𝛾-radonifying.

Definition 1 (see [13]). Let 𝐾 and 𝑋 be Banach spaces, a bounded linear operator 𝐿∶𝐾→𝑋 is called 𝛾-radonifying if and only if 𝐿(𝛾_𝐾) is 𝜎-additive, where 𝛾_𝐾 is the canonical cylindrical finitely additive set-valued function (also called a Gaussian distribution) on 𝐾.

The following is our standing assumption:

Assumption 1. Space 𝐸⊂𝐻∩𝕃⁴ is a Hilbert space such that for some 𝛿∈(0,1/2), 𝐴^−𝛿∶𝐸⟶𝐻∩𝕃⁴is𝛾-radonifying. (12)

Remark 2. Under the above assumption, we have the facts that 𝐸⊂𝐻 and the Banach space 𝑈 is taken as 𝐻∩𝕃⁴ (see [11, 14, 15] for more details and related results). In fact, space 𝐸 is the reproducing kernel Hilbert space of noise 𝑊(𝑡) on 𝐻∩𝕃⁴.

It is well known that subordinators form the subclass of increasing Lévy processes, which can be thought of as a random model of time evolution (see [16]). We will present some properties of the subordinator Lévy process 𝑌(𝑡), 𝑡≥0, then review briefly the stochastic integral with respect to Lévy process 𝑌(𝑡).

Definition 3 (see [1, 2, 11]). Let 𝐸 be a Banach space, and let 𝑌=(𝑌(𝑡),𝑡≥0) be an 𝐸-valued stochastic process defined on a probability space (Ω,ℱ,ℙ). Stochastic process 𝑌 is called a Lévy process if (L1)𝑌(0)=0, a.s.;(L2) process 𝑌 has independent and stationary increments; and(L3) process 𝑌 is stochastically continuous, that is, for all 𝛿>0 and for all 𝑠≥0, lim_𝑡→𝑠ℙ(|𝑌(𝑡)−𝑌(𝑠)|>𝛿)=0. (13)
A subordinator Lévy process is an increasing one-dimensional Lévy process.
For 𝑝>0, Sub(𝑝) denotes the set of all subordinator Lévy processes 𝑍, whose intensity measure 𝜌 satisfies the condition ∫¹₀𝜂^𝑝/2𝜌(𝑑𝜂)<∞.

In the most interesting cases, the space 𝐸 is a subspace of 𝐻, that is, 𝐸⊂𝐻, and 𝑌(𝑡)=𝑊(𝑍(𝑡)),𝑡≥0, (14) where 𝑍=(𝑍(𝑡))_𝑡≥0 is an independent subordinator process belonging to class Sub(𝑝), 𝑝∈(1,2], 𝑊=(𝑊(𝑡))_𝑡≥0 is an 𝐻-valued cylindrical Wiener process defined on some Banach space 𝑈.

We decompose the 𝐻-valued Lévy process 𝑌(𝑡) into two parts 𝑁₁(𝑡) and 𝑁₂(𝑡), the first one with small jumps and the second one with (relatively) large jumps, that is, 𝑌(𝑡)=𝑁₁(𝑡)+𝑁₂(𝑡),𝑡≥0, (15) with 𝜈 being the intensity measure of Lévy process 𝑌, 𝑁₁ being the Lévy process with the intensity measure: 𝜈₁(Γ)=𝜈Γ∩𝐵_𝑈()(𝑈),𝐵0,1,Γ∈ℬ_𝑈(0,1)denotestheunitballin𝑈, (16) and 𝑁₂ be the Lévy process with the intensity measure 𝜈₂=𝜈−𝜈₁. Then 𝑁₂ can be defined as a compound Poisson process with the intensity measure 𝜈₂, and 𝑁₁, 𝑁₂ can be defined by the Poisson random measure 𝜋 which is defined as follows: 𝜋([0,1]×Γ)=Σ_𝑠≤𝑡1_ΓΔ𝑌(𝑠),Γ∈ℬ(𝑈), (17) where Δ𝑌(𝑠)=𝑌(𝑠⁺)−𝑌(𝑠⁻), 𝑠≥0. Here, the symbol Δ denotes the increment of 𝑌.

We assume that the process 𝑌 is right-continuous with left-hand side limits. Thus Δ𝑌(𝑠)=𝑌(𝑠)−𝑌(𝑠⁻),𝑠≥0. (18)

Notice that as 𝜋 is a time homogenous Poisson random measure, 𝑌 can be expressed as 𝑌(𝑡)=Σ_𝑠≤𝑡Δ𝑌(𝑠)=^𝑡_0𝑈𝑢𝜋𝑑𝑦,𝑑𝑠,𝑡≥0. (19) Hence, 𝑁₁(𝑡)=Σ_𝑠≤𝑡1_{|Δ𝑌(𝑠)|<1}Δ𝑌(𝑠)=^𝑡_0|𝑢|<1,𝑁𝑢𝜋𝑑𝑦,𝑑𝑠₂(𝑡)=Σ_𝑠≤𝑡1_{|Δ𝑌(𝑠)|≥1}Δ𝑌(𝑠)=^𝑡_0|𝑢|≥1𝑢𝜋𝑑𝑦,𝑑𝑠. (20)

Assume that the operator Ψ(𝑡), 𝑡∈[0,𝑇], is a strongly measurable function taking values in the space of all bounded linear operator from 𝑈 to 𝐸. Let 0<𝜏₁<𝜏₂<𝜏₃<⋯→∞ be the jump times for 𝑁₂ and Δ𝑁₂(𝜏_𝑘)=Δ𝑌(𝜏_𝑘)=𝑌(𝜏_𝑘)−𝑌(𝜏_𝑘−), 𝑘=1,2,…. Then, the stochastic integral of Ψ(𝑡) with respect to jump process 𝑁₂(𝑡), 𝑡≥0, can be defined as ^𝑡₀Ψ(𝑠)𝑑𝑁₂(𝑠)=Σ_{𝜏𝑘≤𝑡}Ψ𝜏_𝑘Δ𝑁₂𝜏_𝑘. (21)

Since the operator Ψ is taking values in 𝐸, it follows from the decomposition of 𝑌 that the sum of sequences is finite. Hence the stochastic integral of the operator Ψ with respect to 𝑁₂ is taking values in 𝐸. Moreover, the stochastic integral of the operator Ψ(𝑡), 𝑡∈[0,𝑇], with respect to Lévy process 𝑌 can be defined by ^𝑡₀Ψ(𝑠)𝑑𝑌(𝑠)=^𝑡₀Ψ(𝑠)𝑑𝑁₁(𝑠)+^𝑡₀Ψ(𝑠)𝑑𝑁₂(𝑠) (22) and takes values in 𝐸 as well (see [11] for more details).

Next, we recall some basic definitions and properties for general random dynamical systems, which are taken from [7]. Let (𝐻,𝑑) be a complete separable metric space and (Ω,ℱ,ℙ) be a probability space.

Definition 4. (Ω,ℱ,ℙ,(𝜃)_𝑡∈ℝ) is called a metric dynamical system if the mapping 𝜃∶ℝ×Ω→Ω is (ℬ(ℝ)×ℱ,ℱ) measurable, 𝜃₀=𝐼, 𝜃_𝑠+𝑡=𝜃_𝑠∘𝜃_𝑡 for all 𝑡,𝑠∈𝑅, and 𝜃_𝑡ℙ=ℙ for all 𝑡∈ℝ.

Definition 5. A random dynamical system (RDS) with time 𝑇 on (𝐻,𝑑) over {𝜃_𝑡} on (Ω,ℱ,ℙ,(𝜃_𝑡)_𝑡∈𝑅) is a (ℬ(𝑅⁺)×𝔉×ℬ(𝐻),𝐵(𝐻))-measurable map: Π∶𝑇×𝐻×Ω⟶𝐻×Ω,Π(𝑡,𝑠,𝜔)=𝑆(𝑡,𝜔)𝑥,𝜃_𝑡𝜔 (23) such that (i)𝑆(0,𝜔)=𝐼𝑑 (identity on 𝐻) for any 𝜔∈Ω,(ii) (Cocycle property) 𝑆(𝑡+𝑠,𝜔)=𝑆(𝑡,𝜃_𝑠𝜔)∘𝑆(𝑠,𝜔) for all 𝑠,𝑡∈𝑇 and 𝜔∈Ω.

An RDS is said to be continuous or differentiable if for all 𝑡∈𝑇, and an arbitrary outside outside ℙ-nullset 𝐵⊂Ω, 𝑤∈𝐵 the map 𝑆(𝑡,𝜔)∶𝐻→𝐻 is continuous or differentiable, respectively.

Assume that the bounded linear operator 𝐴 generates a 𝐶₀-semigroup 𝑆=(𝑒^𝑡𝐴)_𝑡≥0 on a Hilbert space 𝐸 and 𝑌 defined on a filtered probability space (Ω,ℱ,(ℱ)_𝑡≥0,ℙ) is a subordinator Lévy process taking values in a Hilbert space 𝑈.

Consider the following stochastic Langevin equation: 𝑑𝑋(𝑡)=𝐴𝑋(𝑡)𝑑𝑡+𝑑𝑌(𝑡),𝑡≥𝑡₀,𝑋𝑡₀=𝑥∈𝐸. (24)

Definition 6. Let 𝑥∈𝐸 be a square integrable ℱ_𝑡0-measurable random variable in 𝐸. A predicable process 𝑋∶[𝑡₀,∞)×Ω→𝐸 is called a mild solution of the Langevin equation (24) with initial data (𝑡₀,𝑥) if it is an adapted 𝐸-valued stochastic process and satisfies 𝑋(𝑡)=𝑆𝑡−𝑡₀𝑥+^𝑡_𝑡0𝑆(𝑡−𝑠)𝑑𝑌(𝑠),𝑡≥𝑡₀. (25)

It is well known that the Ornstein-Uhlenbeck process 𝑋(𝑡), 𝑡≥0, has some important integrability. Here we need the Banach space to be of type 𝑝, for 𝑝∈(1,2]. First we recall the definition briefly (see [14] for more details).

Definition 7 (see [14]). For 𝑝∈(1,2], the Banach space 𝐸 is called as type 𝑝, if and only if there exists a constant 𝐾_𝑝(𝐸)>0 such that for any finite sequence of symmetric independent identically distribution random variables 𝜀₁,…,𝜀_𝑛∶Ω→[−1,1], 𝑛∈ℕ, and any finite sequence 𝑥₁,…,𝑥_𝑛 from 𝐸, satisfying 𝔼^𝑛_𝑖=1𝜀_𝑖𝑥_𝑖^𝑝≤𝐾_𝑝(𝐸)^𝑛_𝑖=1𝑥_𝑖^𝑝. (26)
Moreover, if there exists a constant 𝐿_𝑝(𝐸)>0 such that for every 𝐸-valued martingale {𝑀_𝑛}^𝑁_𝑛=0, 𝑁∈ℕ, satisfying sup_𝑛𝔼𝑀_𝑛^𝑝≤𝐿_𝑝(𝐸)^𝑁_𝑛=0𝔼𝑀_𝑛−𝑀_𝑛−1^𝑝,𝑀₋₁=0, (27) the separable Banach space 𝐸 is called a separable martingale type 𝑝-Banach space.

Lemma 8 (see [11, Corollary 8.1, Proposition 8.4]). Assume that 𝑝∈(1,2], 𝑍 is a subordinator Lévy process from the class Sub(𝑝), 𝐸 is a separable type 𝑝-Banach, 𝑈 is a separable Hilbert space 𝑈, 𝐸⊂𝑈, and 𝑊=(𝑊(𝑡))_𝑡≥0 is a 𝑈-valued Wiener process.
Define the 𝑈-valued Lévy process as 𝑌(𝑡)=𝑊(𝑍(𝑡)),𝑡≥0, (28) and define the process as 𝑋(𝑡)=^𝑡₀𝑒^{(𝑡−𝑠)𝐴}𝑑𝑌(𝑠). (29) Then, with probability 1, for all 𝑇>0, ^𝑇₀|𝑋(𝑡)|^𝑝_𝐸𝑑𝑡<∞,_𝑇0|𝑋(𝑡)|⁴_𝐿4𝑑𝑡<∞. (30)

We have the following existence and regularity results, which have been studied in [2, 11].

Theorem 9. Assume that 𝐸=𝑈, 𝑆=𝑆(𝑡), 𝑡≥0 is the 𝐶₀ semigroup generated by the bounded linear operator 𝐴 in the space 𝐸. Then, if one of the following conditions is satisfied: (i)𝑝∈(0,1] or (ii)𝑝∈(1,2] and the Banach space 𝐸 is of separable martingale type 𝑝-Banach space, the Langevin equation (24) admits one mild solution 𝑋(𝑡)∈𝐸, 𝑡>0. Moreover, if 𝑝∈(1,2], 𝑆=𝑆(𝑡), 𝑡≥0, is a 𝐶₀-group in the separable martingale type 𝑝-Banach space 𝐸, then the mild solution 𝑋 of the Langevin equation is a cádlág (right-continuous with left-hand side limits) process.

Proof. As 𝑆=𝑆(𝑡), 𝑡≥0, is a 𝐶₀-group in the separable martingale type 𝑝-Banach space 𝐸, the Hilbert space 𝐻 is the reproducing kernel Hilbert space of 𝑊(1), and the embedding operator 𝑖∶𝐻↪𝐸 satisfies the 𝛾-radonifying property. The proof of Theorem 9 is just a simple application of Theorems  4.1 and 4.4 in [11].

3. Cocycle Property of the Stochastic Boussinesq Equations

In this section, we will show the existence, uniqueness, regularity, and the cocycle property of the stochastic Boussinesq equations (11).

It is well known that both 𝐴₁ and 𝐴₂ are positive definite, self-adjoint operators, and denote 𝐷(𝐴₁) and 𝐷(𝐴₂) to be the domains of 𝐴₁ and 𝐴₂, respectively. Hence, the domain of the operator 𝐴 can be represented as 𝐷(𝐴)=𝐷(𝐴₁)×𝐷(𝐴₂). It follows from Lemma  2.2 in [7] that there exists positive numbers 𝜇₁,𝜇₂, such that 𝐴₁𝑢,𝑢≥𝜇₁‖𝑢‖²_𝐿22,𝐴₂(𝑢,𝜃),(𝑢,𝜃)≥𝜇₂‖(𝑢,𝜃)‖². (31) Let 𝜆=min(𝜇₁,𝜇₂). Then (𝐴𝑈,𝑈)≥𝜆‖𝑈‖². (32)

For any arbitrary 𝑈,𝑉,𝑊∈𝕍, we can define the following trilinear form 𝑏∶𝑈×𝑉×𝑊→ℝ by 𝑏()=⟨𝐵()⟩,𝑏()𝑢,𝑣,𝑤𝑢,𝑣,𝑤𝑈,𝑉,𝑊=𝑏₁(𝑢,𝑣,𝑤)+𝑏₂(𝑤),𝑏𝑢,𝑣,₁(𝑢,𝑣,𝑤)=_𝐷𝜎²_𝑖,𝑗𝑢_𝑖𝜕𝑣_𝑗𝜕𝑥_𝑖𝑤_𝑗𝑏𝑑𝑥,₂(𝑢,𝑣,𝑤)=_𝐷𝜎²_𝑖𝑢_𝑖𝜕𝑣_𝑗𝜕𝑥_𝑖𝑤_𝑗𝑑𝑥. (33) We have the following results.

Lemma 10 (see [7, Lemma  2.3]). If  𝑈,𝑉,𝑊∈𝕍, then𝑏()(),(𝐵())()𝑈,𝑉,𝑊=−𝑏𝑈,𝑊,𝑉𝑉,𝑈,𝑈=𝑏𝑉,𝑈,𝑈=0. (34)

Lemma 11 (see [7, Lemma  2.4]). There exists a constant 𝑐_𝐵>0 such that if 𝑢∈𝑉₁, 𝜃,𝜂∈𝑉₂, 𝜙=(𝑢,𝜃), then(1)|𝑏₁(𝑢,𝑣,𝑤)|≤𝑐_𝐵‖𝑢‖_𝐻¹‖𝑣‖_𝐻²‖𝑤‖, 𝑢∈𝑉, 𝑣∈𝐷(𝐴), 𝑤∈𝐻,(2)|𝑏₁(𝑢,𝑣,𝑤)|≤𝑐_𝐵‖𝑢‖_𝐿^1/2₂‖𝑢‖_𝐻^1/2₁‖𝑣‖_𝐻²‖𝑤‖^1/2‖𝑤‖_𝐻^1/2₁, 𝑢∈𝑉, 𝑣∈𝐷(𝐴), 𝑤∈𝑉,(3)|𝑏₁(𝑢,𝑣,𝑢)|≤𝑐_𝐵‖𝑢‖_𝐻¹‖𝑣‖_𝐻¹‖𝑣‖, 𝑢∈𝑉, 𝑣∈𝑉,(4)|𝑏₂(𝑢,𝜃,𝑤)|≤𝑐_𝐵‖𝑢‖^1/2‖𝑢‖_𝐻^1/2₁‖𝜃‖_𝐻¹‖𝑤‖^1/2‖𝑤‖_𝐻^1/2₁, 𝑢∈𝑉, 𝜃∈𝑉, 𝑤∈𝑉,(5)|𝑏₂(𝑢,𝜃,𝑤)|≤𝑐_𝐵‖𝑢‖‖𝜃‖_𝐻²‖𝑤‖_𝐻¹, 𝑢∈𝐻, 𝜃∈𝐷(𝐴), 𝑤∈𝑉.

Definition 12. An 𝐻-valued (ℱ_𝑡)_𝑡≥0 adapted and 𝐇^4,2(𝐷)-valued cádlág process 𝑢(𝑡)(𝑡≥0) is considered as a solution to (11), if for each 𝑇>0, sup_{0≤𝑡≤𝑇}|𝑈(𝑡)|²_𝐻+^𝑇₀|𝑈(𝑡)|⁴_𝕃4(𝐷)𝑑𝑡<∞,a.s., (35) and for any 𝜓∈𝑉∩𝐇^2,2(𝐷), and for any 𝑡>0, ℙ-a.s., 𝑈(𝑡),𝜓−𝑈₀,𝜓−^𝑡₀𝑈(𝑠)+,Δ𝜓𝑑𝑠_𝑡0𝐵(𝑈,𝑈),𝜓(𝑠)𝑑𝑠+^𝑡₀𝑅(𝑈),𝜓𝑑𝑠=𝜓,𝑌(𝑡). (36)

Denote ℋ^1,2(0,𝑇)=thespaceofallfunctions𝑣∈𝐿²(0,𝑇;𝑉)∩ℍ^2,2(𝐷)satisfying𝑣^′∈𝐿²0,𝑇;𝑉^′. (37)

Lemma 13. Assume that 𝑧∈𝐿⁴(0,𝑇;𝕃⁴(𝐷)), 𝑔∈𝐿²(0,𝑇,𝑉^′), and 𝑣₀∈𝐻. Then there exists a unique 𝑣∈ℋ^1,2(0,𝑇) such that 𝑑𝑣()()()𝑣(0)𝑑𝑡+𝐴𝑣+𝐵𝑣,𝑧+𝐵𝑧,𝑣+𝐵𝑣,𝑣=𝑔,𝑡≥0,=𝑣₀. (38) Moreover, sup_{𝑡∈[0,𝑇]}|𝑣(𝑡)|²≤𝐾²𝐿²,^𝑇₀|∇𝑣(𝑡)|²𝑑𝑡≤𝑀²,_𝑇0𝑣^′(𝑡)²_𝑉′𝑑𝑡≤𝑁²,^𝑇₀|𝑣(𝑡)|⁴_𝕃4(𝐷)𝑑𝑡≤2𝑇^1/2𝐾³𝐿³𝑀, (39) where 𝐾²=𝑒^{2∫𝑇0|𝑧(𝑠)|4𝕃4}𝑑𝑠,𝐿²=𝑣₀²+2^𝑇₀𝑔(𝑠)²_𝑉′𝑀𝑑𝑠,₂=𝑣₀²+9𝐾𝐿^𝑇₀|𝑧(𝑡)|²_{𝐿4(0,𝑇,𝕃4(𝐷))}+𝑇^1/4√2𝐾^3/2𝐿^1/2, (40) and the mapping 𝐿²(0,𝑇,𝑉^′)×𝐻∋(𝑔₀,𝑣₀)↦𝑣∈ℋ^1,2(0,𝑇) is analytic.

Proof. It can be shown by the same approach as the one in Proposition  8.7 in [11].

Lemma 14 (see [2, Proposition  10.1]). Let 𝑢∶[0,𝑇]→𝐵 be a continuous function whose left derivative 𝑑⁻𝑢𝑡𝑑𝑡⁰=lim_{𝜖→0,𝜖<0}𝑢𝑡₀+𝜖−𝑢𝑡₀𝜖 (41) exists at 𝑡₀∈[0,𝑇]. Then the function 𝛾(𝑡)=|𝑢(𝑡)|_𝐵, 𝑡∈[0,𝑇], is left differentiable at 𝑡₀ and 𝑑⁻𝛾𝑡𝑑𝑡⁰=min𝑥^∗,𝑑⁻𝑢𝑡𝑑𝑡⁰∶𝑥^∗∈𝜕𝑢𝑡_0𝐵. (42)

In order to apply the Yosida approximation for the solution of (11), we need to introduce some definitions of dissipative mapping (operator) (see [17] for details).

Definition 15. Let (𝐵,|⋅|_𝐵) be a separable Banach space, 𝐵^∗ be the dual space of 𝐵. The subdifferential 𝜕|𝑥|_𝐵 of norm |⋅|_𝐵 at 𝑥∈𝐵 is defined by the formula 𝜕|𝑥|_𝐵∶=𝑥^∗∈𝐵^∗∶𝑥+𝑦_𝐵−|𝑥|_𝐵≥𝑥^∗,𝑦,∀𝑦∈𝐵. (43)
A mapping 𝐹∶𝐷(𝐹)⊂𝐵→𝐵 is said to be dissipative, if for any 𝑥,𝑦∈𝐷(𝐹), there exists 𝑧^∗∈𝜕|𝑥−𝑦|_𝐵 such that 𝑧^∗,𝐹(𝑥)−𝐹𝑦≤0. (44)
A dissipative mapping 𝐹∶𝐷(𝐹)⊂𝐵→𝐵 is called an 𝑚-dissipative mapping or maximal dissipative if the image of 𝐼−𝜆𝐹 is equal to the whole space 𝐵 for some 𝜆>0 (and then for any 𝜆>0), that is, range(𝐼−𝜆𝐴)=𝐵,forsome𝜆>0. (45)
Assume that 𝐹 is an 𝑚-dissipative mapping. Then its resolvent 𝐽_𝛼 and respectively the Yosida approximations 𝐹_𝛼, 𝛼>0, are defined by 𝐽_𝛼𝑥=(𝐼−𝛼𝐹)⁻¹𝐹𝑥∈dom𝐹,_𝛼1𝑥=𝛼𝐽_𝛼𝑥−𝑥,∀𝑥∈dom𝐽_𝛼=range(𝐼−𝛼𝐹). (46)

Lemma 16 (see [2, Proposition  10.2]). Let 𝐹∶𝐷(𝐹)→𝐵 be an 𝑚-dissipative mapping on 𝐵. Then (1) for all 𝛼>0 and 𝑥,𝑦∈𝐵, |𝐽_𝛼(𝑥)−𝐽_𝛼(𝑦)|_𝐵≤|𝑥−𝑦|_𝐵;(2) the mapping 𝐹_𝛼, 𝛼>0, are dissipative and Lipschitz continuous: 𝐹_𝛼(𝑥)−𝐹_𝛼𝑦_𝐵≤2𝛼𝑥−𝑦_𝐵,∀𝑥,𝑦∈𝐵. (47) Moreover, |𝐹_𝛼(𝑥)|_𝐵≤|𝐹(𝑥)|_𝐵, for all 𝑥∈𝐷(𝐹); and(3)lim_𝛼→0𝐹_𝛼(𝑥)=𝑥, for all 𝑥∈𝐷(𝐹).

The following theorem is one of the main results of this paper, which will be proved by applying the well-known Yosida approach.

Theorem 17. For every 𝑢₀∈𝐻, under Assumption 1, the stochastic Boussinesq system (11) admits a unique cádlág mild solution 𝑢(𝑡).

Proof. Denote 𝑍_𝐴(𝜔) to be the stationary solution of Langevin equation (24). Let 𝑉=𝑈−𝑍_𝐴. Then (11) is converted into the following evolution equation with random coefficients: 𝑑𝑉=𝐴𝑉+𝐵𝑉+𝑍_𝐴,𝑉+𝑍_𝐴+𝑅𝑉+𝑍_𝐴𝑉(0)𝑑𝑡,𝑡≥0,=𝑈₀, (48) where (𝐴,𝐷(𝐴)) generates an analytic 𝐶₀-semigroup 𝑆 (see Section  2.2 in [2]). It follows from the proof of Theorem  10.1 in [2] that, for 𝛼>0, 𝛽>0, and sufficiently small 𝜂, the mappings 𝐴+𝜂 and 𝐵(⋅,⋅)+𝑅(⋅)+𝜂 are 𝑚-dissipative. Hence, the Yosida approximations of the 𝑚-dissipative mappings 𝐴+𝜂 and 𝐵(⋅,⋅)+𝑅(⋅)+𝜂 can be respectively denoted by 𝐴+𝜂_𝛽=1𝛽𝐼−𝛽𝐴+𝜂⁻¹,()−𝐼𝐵+𝑅+𝜂_𝛼=1𝛼𝐼−𝛼(𝐵+𝑅)+𝜂⁻¹−𝐼. (49)
Now consider the following random approximate equation: 𝑑⁻𝑌𝑑𝑡^𝛼,𝛽(𝑡)=𝐴+𝜂_𝛽𝑌_𝛼,𝛽+𝐵+𝑅+𝜂_𝛼𝑌_𝛼,𝛽+𝑍_𝐴(𝑡−)−2𝜂𝑌_𝛼,𝛽−𝜂𝑍_𝐴(),𝑌𝑡−_𝛼,𝛽(0)=𝑈₀. (50) It is easy to verify that ((𝐴+𝜂)_𝛽,𝐷((𝐴+𝜂)_𝛽)) generates an analytic 𝐶₀-semigroup 𝑆_𝛽. Notice that the Yosida approximate operators are Lipschitz. Therefore the random approximation equation (50) has a unique continuous solution 𝑌_𝛼,𝛽.
Next we will show that lim_𝛼→0lim_𝛽→0𝑌_𝛼,𝛽(𝑡)=𝑌(𝑡),𝑡≥0, (51) in 𝐻, and this limit is actually the mild solution of stochastic Boussinesq equation (48).
For the sake of simplification, we just present the estimations when 𝜂=0, and the remaining estimates can be obtained by the similar arguments for 𝜂≠0.
Let 𝑌_𝛼 be the solution of the integral equation: 𝑌_𝛼(𝑡)=𝑆(𝑡)𝑈₀+^𝑡₀𝑆()×𝐵𝑌𝑡−𝑠_𝛼(𝑠)+𝑍_𝐴(𝑠−),𝑌_𝛼(𝑠)+𝑍_𝐴()𝑌𝑠−+𝑅_𝛼(𝑠)+𝑍_𝐴(𝑠−)_𝛼𝑑𝑠. (52) Notice that the operator (𝐵(⋅,⋅)+𝑅(⋅))_𝛼 is Lipschitz continuous and 𝑍_𝐴 is cádlág. Hence, there exists a solution of random approximate equation (50), which is continuous in 𝐻.
For 𝛼>0 and 𝛽>0, direct computation implies 𝑌_𝛼−𝑌_𝛼,𝛽=𝑆(𝑡)𝑈₀−𝑆_𝛽(𝑡)+^𝑡₀𝑆(𝑡−𝑠)−𝑆_𝛽(×𝐵𝑌𝑡−𝑠)_𝛼(𝑠)+𝑍_𝐴(𝑠−),𝑌_𝛼(𝑠)+𝑍_𝐴()𝑌𝑠−+𝑅_𝛼(𝑠)+𝑍_𝐴(𝑠−)_𝛼+𝑑𝑠_𝑡0𝑆_𝛽(×𝐵𝑌𝑡−𝑠)_𝛼(𝑠)+𝑍_𝐴(𝑠−),𝑌_𝛼(𝑠)+𝑍_𝐴()𝑌𝑠−+𝑅_𝛼(𝑠)+𝑍_𝐴(𝑠−)_𝛼−𝐵𝑌_𝛼,𝛽(𝑠)+𝑍_𝐴(),𝑌𝑠−_𝛼,𝛽(𝑠)+𝑍_𝐴()𝑌𝑠−+𝑅_𝛼,𝛽(𝑠)+𝑍_𝐴(𝑠−)_𝛼𝑑𝑠. (53) Since both 𝐴 and 𝐵+𝑅 are 𝑚-dissipative. Therefore, there exists constant 𝑀, 𝜔, and 𝐶_𝛼 such that for all 𝑡≥0, 𝑉,𝑊∈𝐻, 𝑆_𝛽(𝑡)_{𝐿(𝐻,𝐻)}≤𝑀𝑒^𝜔𝑡,[𝐵(𝑉)+𝑅(𝑉)]_𝛼−[𝐵(𝑈)+𝑅(𝑈)]_𝛼≤𝐶_𝛼|𝑉−𝑈|_𝐻. (54) Then 𝑌_𝛼(𝑡)−𝑌_𝛼,𝛽(𝑡≤𝑆(𝑡)𝑈₀)−𝑆_𝛽(𝑡)𝑈₀+𝑀𝐶_𝛼^𝑡₀𝑒^{𝜔(𝑡−𝑠)}𝑌_𝛼(𝑠)−𝑌_𝛼,𝛽(𝑠+)𝑑𝑠_𝑡0𝑆_𝛽()(×𝐵𝑌𝑡−𝑠−𝑆𝑡−𝑠)_𝛼,𝛽(𝑠)+𝑍_𝐴(𝑠−),𝑌_𝛼,𝛽(𝑠)+𝑍_𝐴()𝑌𝑠−+𝑅_𝛼,𝛽(𝑠)+𝑍_𝐴(𝑠−)_𝛼𝑑𝑠. (55) By the Hille-Yosida theorem, it follows that 𝑆_𝛽(𝑡)𝑈₀⟶𝑆(𝑡)𝑈₀,as𝛽⟶0 (56) uniformly in 𝑡 on compact subsets 𝑈₀ of 𝐻.
Hence, it follows that 𝑌_𝛼(𝑡)−𝑌_𝛼,𝛽(𝑡)≤𝑀𝐶_𝛼^𝑡₀𝑌_𝛼(𝑠)−𝑌_𝛼,𝛽(𝑠)𝑑𝑠 (57) uniformly on bounded intervals as 𝛽→0.
By Gronwall inequality, we have lim_𝛽→0sup_𝑡≤𝑇𝑌_𝛼(𝑡)−𝑌_𝛼,𝛽(𝑡)=0,∀𝑇<∞. (58)
By Lemma 14, 𝑑⁻𝑌𝑑𝑡^𝛼,𝛽(𝑡𝑥)=min^∗,𝑑⁻𝑌𝑑𝑡^𝛼,𝛽(𝑡)∶𝑥^∗∈𝜕𝑌_𝛼,𝛽(𝑡𝑥)=min^∗,𝐴_𝛽𝑌_𝛼,𝛽(𝑡)+𝐵𝑌_𝛼,𝛽(𝑠)+𝑍_𝐴(𝑠−),𝑌_𝛼,𝛽(𝑠)+𝑍_𝐴()𝑌𝑠−+𝑅_𝛼,𝛽(𝑠)+𝑍_𝐴(𝑠−)_𝛼∶𝑥^∗∈𝜕𝑌_𝛼,𝛽(𝑡). (59) Recalling that both 𝐴_𝛽 and [𝐵(⋅,⋅)+𝑅(⋅)] are 𝑚-dissipative and 𝐴_𝛽 is linear, we obtain 𝑑⁻𝑌𝑑𝑡^𝛼,𝛽(𝑡)≤𝐵𝑌_𝛼,𝛽(𝑡)+𝑍_𝐴(𝑡−),𝑌_𝛼,𝛽(𝑠)+𝑍_𝐴()𝑌𝑡−+𝑅_𝛼,𝛽(𝑡)+𝑍_𝐴(𝑡−)_𝛼≤𝐵𝑌_𝛼,𝛽(𝑡)+𝑍_𝐴(𝑡−),𝑌_𝛼,𝛽(𝑠)+𝑍_𝐴()𝑌𝑡−+𝑅_𝛼,𝛽(𝑡)+𝑍_𝐴(𝑡−), (60) that is, 𝑌_𝛼,𝛽(𝑡)≤𝑈₀+^𝑡₀𝐵𝑌_𝛼,𝛽(𝑠)+𝑍_𝐴(𝑠−),𝑌_𝛼,𝛽(𝑠)+𝑍_𝐴()𝑌𝑠−+𝑅_𝛼,𝛽(𝑠)+𝑍_𝐴(𝑠−)𝑑𝑠,𝑡≥0. (61) It follows from the estimate (58) that, for any 𝛼>0, and 𝑡∈[0,𝑇], 𝑌_𝛼(𝑡)≤𝑈₀+^𝑡₀𝐵𝑌_𝛼,𝛽(𝑠)+𝑍_𝐴(𝑠−),𝑌_𝛼,𝛽(𝑠)+𝑍_𝐴()𝑌𝑠−+𝑅_𝛼,𝛽(𝑠)+𝑍_𝐴(𝑠−)𝑑𝑠. (62)
Similarly, by Lemma 16, for 𝑡∈[0,𝑇], 12𝑑⁻𝑌𝑑𝑡^𝛼,𝛽(𝑡)−𝑌_𝛾,𝛽(𝑡)²=𝑑⁻𝑌𝑑𝑡^𝛼,𝛽(𝑡)−𝑌_𝛾,𝛽(𝑡),𝑌_𝛼,𝛽(𝑡)−𝑌_𝛾,𝛽(𝑡=𝐴_𝛽)𝑌_𝛼,𝛽(𝑡)−𝐴_𝛽𝑌_𝛾,𝛽(𝑡)+(𝐵+𝑅)_𝛼×𝑌_𝛼,𝛽(𝑡)+𝑍_𝐴(𝜔)(𝑡−)−(𝐵+𝑅)_𝛾×𝑌_𝛾,𝛽(𝑡)+𝑍_𝐴(𝜔)(𝑡−),𝑌_𝛼,𝛽(𝑡)−𝑌_𝛾,𝛽(𝑡≤))(𝐵+𝑅_𝛼𝑌_𝛼,𝛽(𝑡)+𝑍_𝐴(𝜔)(𝑡−)−(𝐵+𝑅)_𝛾×𝑌_𝛾,𝛽(𝑡)+𝑍_𝐴(𝜔)(𝑡−),𝑌_𝛼,𝛽(𝑡)−𝑌_𝛾,𝛽(𝑡≤))𝛾+𝛼(𝐵+𝑅_𝛼𝑌_𝛼,𝛽(𝑡)+𝑍_𝐴(𝜔+))(𝑡−)(𝐵+𝑅_𝛾𝑌_𝛾,𝛽(𝑡)+𝑍_𝐴(𝜔)(𝑡−)²≤𝛾+𝛼(𝐵+𝑅)𝑌_𝛼,𝛽(𝑡)+𝑍_𝐴(𝜔)+)𝑌)(𝑡−(𝐵+𝑅_𝛾,𝛽(𝑡)+𝑍_𝐴(𝜔)(𝑡−)². (63) By the dissipation of the operators 𝐴, 𝐵, and 𝑅 and estimates (63), there exists a constant 𝐶>0 such that 12𝑑⁻𝑌𝑑𝑡^𝛼,𝛽(𝑡)−𝑌_𝛾,𝛽(𝑡)²≤𝐶𝛼+𝛾,𝑡∈[0,𝑇]. (64) Therefore 𝑌_𝛼,𝛽(𝑡)−𝑌_𝛾,𝛽(𝑡)²≤2𝐶𝛼+𝛾𝑇,𝑡∈[0,𝑇]. (65) By the estimate (58), 𝑌_𝛼(𝑡)−𝑌_𝛾(𝑡)²≤2𝐶𝛼+𝛾𝑇,𝑡∈[0,𝑇]. (66) Thus, 𝑌_𝛼(𝑡)→𝑌(𝑡) in 𝐻 uniformly on [0,𝑇] as 𝛼→0.
Next, we are going to show that the solution 𝑌_𝛼 of the Yosida approximations equation is a mild solution: 𝑌_𝛼(𝑡)=𝑆(𝑡)𝑈₀+^𝑡₀𝑆(𝑡−𝑠)(𝐵+𝑅)_𝛼𝑌_𝛼(𝑠)+𝑍_𝐴(𝑠)[].𝑑𝑠,𝑡∈0,𝑇 (67) By the reflexivity of 𝐻¹ and the estimate ‖𝑌_𝛼(𝑡)‖_𝐻1≤𝐶₂‖𝑈₀‖_𝐻1, 𝑡∈[0,𝑇], 𝛼>0, there exists a subsequence {𝑌_𝛼,𝑛}, which converges weakly in 𝐻¹ and weakly converges to the function 𝑌(𝑡) in 𝐻¹. Since {𝑌_𝛼,𝑛(𝑡)} is strong convergent in 𝐿², and ‖𝑌(𝑡)‖_𝐻¹≤𝐶₂𝑈_0𝐻1,𝑡∈[0,𝑇]. (68) Let ℎ∈𝐿², then 𝑌_𝛼(𝑡),ℎ_𝐿2=𝑆(𝑡)𝑈₀,ℎ_𝐿2+^𝑡₀(𝐵+𝑅)𝐽_𝛼𝑌_𝛼(𝑠)+𝑍_𝐴(𝑠),𝑆^∗(𝑡−𝑠)ℎ_𝐿2𝑑𝑠. (69) Moreover 𝐽_𝛼𝑌_𝛼(𝑠)+𝑍_𝐴(𝑠)⟶𝑌(𝑠)+𝑍_𝐴(𝑠),as𝛼⟶0. (70) Notice that (𝐵+𝑅)(𝐽_𝛼(𝑌_𝛼(𝑠)+𝑍_𝐴(𝑠))→(𝐵+𝑅)(𝑌_𝛼(𝑠)+𝑍_𝐴(𝑠)) weakly converges in 𝐿². So, letting 𝛼→0, we obtain ⟨𝑌(𝑡),ℎ⟩_𝐿²=𝑆(𝑡)𝑈₀,ℎ_𝐿2+^𝑡₀(𝑆(𝑡−𝑠)(𝐵+𝑅)(𝑌(𝑠)+𝑍_𝐴(𝑠)),ℎ_𝐿2𝑑𝑠. (71) It follows from the arbitrariness of ℎ that 𝑌(𝑡)=𝑆(𝑡)𝑈₀+^𝑡₀𝑆(𝑡−𝑠)(𝐵+𝑅)𝑌_𝛼(𝑠)+𝑍_𝐴(𝑠)[].𝑑𝑠,𝑡∈0,𝑇 (72) Thus, 𝑌(𝑡) is a mild solution of random Boussinesq equation (50).

Theorem 18. For any 𝑈₀∈𝐻, the map 𝜑∶𝕋×Ω×𝐻→𝐻 defined by the solution of stochastic Boussinesq equation (11) as 𝑈(𝑡)=Φ(𝑡,𝜗_𝑡(𝜔))𝑈₀ has the cocycle property; that is, the solution of stochastic Boussinesq equation (11) generates a random dynamical system (Ω,ℱ,ℙ,(𝜗_𝑡)_𝑡≥0,Φ).

Proof. From Theorem 17, stochastic Boussinesq equation (11) admits a unique solution 𝑉(𝑡,𝑍(𝜔)(𝑡),𝑥). Define the map Φ∶ℝ⁺Φ()((𝜔𝑡(𝜔0(𝜔).×Ω×𝐻⟶𝐻,𝑡,𝜔𝑥=𝑉𝑡⋅𝑍)())(𝑥−𝑍)())+𝑍)(𝑡+𝑠 (73)(i) By the similar argument of Theorem 17, every solution 𝑌_𝛼(𝑡) of the Yosida approximation equation (50) is measurable. Notice that 𝑌_𝛼(𝑡)→𝑌(𝑡) uniformly as 𝛼→0. Hence, the limit function 𝑌(𝑡) is also measurable. Thus, the mapping Φ is measurable.(ii) Obviously, Φ(0,𝜔)=𝐼.(iii) It suffices to verify that the cocycle property holds for the mapping Φ, that is, Φ(𝑡+𝑠,𝜔)𝑥=𝑉𝑡+𝑠,𝑍_𝐴(𝜔)(𝑡+𝑠)𝑥−𝑍_𝐴(𝜔)(0)+𝑍_𝐴(𝜔)(𝑡+𝑠). (74)
In fact, recalling that 𝑍_𝐴(𝜔)(𝑠)=𝑍_𝐴(𝜃_𝑠𝜔)(0), it follows that Φ𝑡,𝜃_𝑠𝜔[Φ(𝑠,𝜔)𝑥]=𝑉𝑡,𝑍_𝐴𝜃_𝑠𝜔(𝑡)Φ(𝑠,𝜔)𝑥−𝑍_𝐴𝜃_𝑠𝜔(0)+𝑍_𝐴𝜃_𝑠𝜔(𝑡)=𝑉𝑡,𝑍_𝐴𝜃_𝑠𝜔(𝑡)×𝑉𝑠,𝑍_𝐴(𝜔)(𝑠)𝑥−𝑍_𝐴(𝜔0)(𝜔𝑠)𝜃)(+𝑍)(−𝑍_𝑠𝜔(0)+𝑍𝜃_𝑠𝜔(𝑡)=𝑉𝑡,𝑍_𝐴𝜃_𝑠𝜔(𝑡)𝑉𝑠,𝑍_𝐴(𝜔𝑠)×)(𝑥−𝑍_𝐴(𝜔)(0)+𝑍_𝐴𝜃_𝑠𝜔(𝑡)=𝑉₁(𝑡). (75)
Moreover, 𝑉𝑡+𝑠,𝑍_𝐴(𝜔)(𝑡+𝑠)𝑥−𝑍_𝐴(𝜔)(0)=𝑉𝑡,𝑍_𝐴𝜃_𝑠𝜔(𝑡)𝑉𝑠,𝑍_𝐴(𝜔)(𝑠)𝑥−𝑍_𝐴(𝜔)(0)=𝑉₂(𝑡). (76)
Since 𝑉0,𝑍_𝐴𝜃_𝑠𝜔(0)𝑥−𝑍_𝐴𝜃_𝑠𝜔(0)=𝑥−𝑍_𝐴𝜃_𝑠𝜔(0). (77)
Thus, 𝑉₁(0)=𝑉𝑠,𝑍_𝐴(𝜔)(𝑠)𝑥−𝑍_𝐴(𝜔)(0)=𝑉0,𝑍_𝐴𝜃_𝑠𝜔(0)𝑉𝑠,𝑍_𝐴(𝜔𝑠)×)(𝑥−𝑍_𝐴(𝜔)(0)=𝑉₂(0),𝑑𝑉₁(𝑡)=()𝑑𝑡𝑑𝑉𝑡+𝑠,𝑍_𝐴(𝜔)().𝑑𝑡𝑡+𝑠 (78)
Therefore, we obtain 𝑑𝑉₁(𝑡)𝑑𝑡+𝐴𝑉₁(𝑡)+𝐵𝑉₁(𝑡)+𝑍_𝐴(𝜔)(𝑡+𝑠),𝑉₁(𝑡)+𝑍_𝐴(𝜔)𝑉)(𝑡+𝑠=−𝑅₁(𝑡)+𝑍_𝐴𝜃_𝑡+𝑠𝜔,𝑑𝑉₂(𝑡)𝑑𝑡+𝐴𝑉₂(𝑡)+𝐵𝑉₂(𝑡)+𝑍_𝐴𝜃_𝑠𝜔(𝑡),𝑉₂(𝑡)+𝑍_𝐴𝜃_𝑠𝜔(𝑡)=−𝑅𝑉₂(𝑡)+𝑍_𝐴𝜃_𝑠𝜔(𝑡). (79)
The uniqueness of the solution implies that almost surely 𝑉₁(𝑡)=𝑉₂(𝑡) holds, that is, Φ𝑡,𝜃_𝑠𝜔[Φ(𝑠,𝜔)𝑥]=Φ𝑡+𝑠,𝜃_𝑡+𝑠(𝜔)𝑥. (80)
Thus, the cocycle property for the mapping Φ holds.
By the definition of random dynamical systems [18], the solution mapping of the stochastic Boussinesq equation (11) generates a random dynamical system Φ. Thus, the proof of Theorem 18 is complete.

4. Discussion

In Section 3, we have studied the long-time behavior of stochastic Boussinesq equations (1) driven by subordinator Lévy noise and have shown the cocycle property of random dynamical systems generated by the mild solution of stochastic Boussinesq equation (1). To prove the existence of random attractor, it suffices to show the existence of random absorbing set and the compactness of random dynamical system Φ, we refer the similar argument to [13].

Here, we are also interested in the stochastic Boussinesq equations driven by Poisson noise and Wiener noise, and we are trying to show the existence of random dynamical systems. To the end, we consider the following stochastic Boussinesq equations driven by Lévy noises followed as 𝜕𝑢+()𝜕𝑡𝑢⋅∇𝑢−𝜈Δ𝑢+∇𝑝=𝜃𝑒₂+𝑏₁𝑑𝑡+𝑑𝑊¹(𝑡)+_𝑋𝑓(𝑥)𝑁¹(𝑑𝑡,𝑑𝑥),𝜕𝜃+()𝜕𝑡𝑢⋅∇𝜃−𝑘Δ𝜃=𝑢₂+𝑏₂𝑑𝑡+𝑑𝑊²(𝑡)+_𝑋𝑔(𝑥)𝑁₂(𝑑𝑡,𝑑𝑥),∇⋅𝑢=0,𝑢|_𝜕𝐷=0,𝑢(0)=𝑢₀,𝜃(0)=𝜃₀, (81) where 𝑊¹(⋅) and 𝑊²(⋅) are 𝐻-valued Brownian motion, 𝑏₁ and 𝑏₂ are constants vector in 𝐻, 𝑓 and 𝑔 are measurable mappings from some measurable space 𝑋 to 𝐻, and 𝑁₁ and 𝑁₂ are compensated Poisson measure on [0,∞)×𝑋 with intensity measure 𝑛𝜈₁ and 𝑛𝜈₂, respectively, where 𝜈₁ and 𝜈₂ are 𝜎-finite measure on ℬ(𝑋), 𝑓(𝑥), and 𝑔(𝑥)   satisfying _𝑈𝑓(𝑥)²𝑒^{𝛼|𝑓(𝑥)|}𝜈(𝑑𝑥)<∞,_𝑈𝑔(𝑥)²𝑒^{𝛽|𝑔(𝑥)|}𝜈(𝑑𝑥)<∞,∀𝛼>0,∀𝛽>0. (82)

Let 𝐷([0,𝑇],𝐻) be the space of all cádlág paths from [0,𝑇] to 𝐻 endowed with the uniform convergence topology. Since there are finite jumps when the character measure 𝜆(𝑍)<∞, we can rearrange the jump time of 𝑁(𝑑𝑡,𝑑𝑥) as 𝜎₁(𝜔)<𝜎₂(𝜔)<⋯. Since there is no jump on the interval [0,𝜎₁(𝜔)), just as the approach in [19], we can apply Banach fixed point theorem to prove that there exists a unique solution 𝜙(𝑡) in 𝐿²([0,𝜎₁(𝜔));𝑉)∩𝐷([0,𝜎₁(𝜔));𝐻). Define 𝜙⁽¹⁾(𝑡)=𝜙(𝑡),0≤𝑡<𝜎₁(𝜔),𝜙𝜎₁−+𝑓𝜙𝜎₁−,𝑃_𝜎1,𝑡=𝜎₁(𝜔). (83) On [𝜎₁(𝜔),𝜎₂(𝜔)), define 𝜙₀=𝜙⁽¹⁾𝜎₁1_(𝜎1<∞),𝜎₂=𝜎₂−𝜎₁1_(𝜎1<∞)+∞1_(𝜎1=∞),ℱ_𝑡=ℱ_𝜎1+𝑡,𝑃_𝑡=𝜃_𝜎1𝑃_𝑡1_(𝜎1<∞). (84) Similar to the argument in [11], since 𝑃_𝑡 is stationary Poisson point process on 𝑅⁺×𝑍 with intensity measure 𝜆(𝑑𝑥)𝑑𝑡, then 𝑃_𝑡 is also a stable Poisson point process on 𝑅⁺×𝑍 with intensity measure 𝜆(𝑑𝑥)𝑑𝑡. Define 𝜙⁽²⁾(𝑡)=𝜙⁽¹⁾(𝑡),0≤𝑡<𝜎₁(𝜔),𝜙⁽²⁾𝑡−𝜎₁,𝜎₁(𝜔)<𝑡<𝜎₂(𝜔),𝜙⁽²⁾𝜎₂−𝜎₁−+𝑓𝜙⁽²⁾𝜎₂−𝜎₁−,𝑃_𝜎2,𝑡=𝜎₂(𝜔),𝜙^(𝑛)(𝑡)=𝜙^(𝑛−1)(𝑡),𝑡<𝜎_𝑛−1(𝜔),𝜙^(𝑛)𝑡−𝜎_𝑛−1,𝜎_𝑛−1(𝜔)<𝑡<𝜎_𝑛(𝜔),𝜙^(𝑛)𝜎_𝑛−𝜎_𝑛−1−+𝑓𝜙^(𝑛)𝜎_𝑛−𝜎_𝑛−1−,𝑃_𝜎𝑛,𝑡=𝜎_𝑛. (85) Hence, 𝜙^(𝑛)(𝑡) is cádlág on [0,𝑇] such that 𝐵(𝜙^(𝑛),𝜙^(𝑛))∈𝐻 and 𝐴_𝑝(𝜙^(𝑛)∈𝐻, 𝑃 a.s. for all 𝑡≥0, and 𝑃^𝑡₀𝜙(𝑠)+𝐵𝜙(𝑠)+𝑧_𝐴(𝑠),𝜙(𝑠)+𝑧_𝐴(𝑠)+2𝜇₀𝑅𝜙(𝑠)+𝑧_𝐴(𝑠)𝑑𝑠<∞=1,∀𝑡>0. (86) Therefore, 𝜙^(𝑛)(𝑡) is a unique global weak solution of (81). We can verify the existence of random dynamical systems generated by the global weak solution of (81).

Acknowledgments

The authors would like to thank an anonymous referee for valuable comments. Supported by the NSF of China (no. 10971225, no. 11028102, no. 11271013, and no. 91130003), the SRF for ROCS, SEM, and the NSF Grant 1025422.