Random attractors for stochastic partly dissipative systems

We prove the existence of a global random attractor for a certain class of stochastic partly dissipative systems. These systems consist of a partial and an ordinary differential equation, where both equations are coupled and perturbed by additive white noise. The deterministic counterpart of such systems and their long-time behaviour have already been considered but there is no theory that deals with the stochastic version of partly dissipative systems in their full generality. We also provide several examples for the application of the theory.


Introduction
In this work, we study classes of stochastic partial differential equations (SPDEs), which are part of the general partly dissipative system where W 1,2 are cylindrical Wiener processes, the σ, f, g, h are given functions, B 1,2 are operator-valued, Δ is the Laplace operator, d > 0 is a parameter, the equation is posed on a bounded open domain D ⊂ R n , u 1,2 = u 1,2 (x, t) are the unknowns for (x, t) ∈ D × [0, T max ), and T max is the maximal existence time.
The term partly dissipative highlights the fact that only the first component contains the regularizing Laplace operator. In this work we analyse the case of additive noise and a certain coupling, more precisely, B 1 (x, u 1 , u 2 ) = B 1 , B 2 (x, u 1 , u 2 ) = B 2 , g(x, u 1 , u 2 ) = g(x, u 1 ), (1.2) We thank the anonymous referee for useful comments. CK and AN have been supported by a DFG grant in the D-A-CH framework (KU 3333/2-1). CK and AP acknowledge support by a Lichtenberg Professorship.
The Sobolev space W k,p (D) is a Banach space. H k 0 (D) denotes the space of functions in H k (D) = W k,2 (D) that vanish at the boundary (in the sense of traces).

Basics
Let D ⊂ R n be a bounded open set with regular boundary, set H := L 2 (D) and let U 1 , U 2 be two separable Hilbert spaces. We consider the following coupled, partly dissipative system with additive noise where u 1,2 = u 1,2 (x, t), (x, t) ∈ D × [0, T ], T > 0, W 1,2 are cylindrical Wiener processes on U 1 respectively U 2 , and Δ is the Laplace operator. Furthermore,  . Note that A is a self-adjoint operator that possesses a complete orthonormal system of eigenfunctions {e k } ∞ k=1 of L 2 (D). Within this work we always assume that there exists κ > 0 such that |e k (x)| 2 < κ for k ∈ N and x ∈ D. This holds for example when D = [0, π] n . For the deterministic reaction terms appearing in (2.1)-(2.2) we assume that:

Assumption 2.1. (Reaction terms)
(1) h ∈ C 2 (R n × R) and there exist δ 1 , δ 2 , δ 3 > 0, p > 2 such that (2.5) (2) f ∈ C 2 (R n × R × R) and there exist δ 4 > 0 and 0 < p 1 < p − 1 such that (3) σ ∈ C 2 (R n ) and there exist δ,δ > 0 such that (4) g ∈ C 2 (R n × R) and there exists δ 5 > 0 such that Random attractors for stochastic partly dissipative systems Page 5 of 37 35 In particular, Assumptions 2.1 (1) and (4) imply that there exist δ 7 , δ 8 > 0 such that (2.10) The Assumptions 2.1(1)-(4) are identical to those given in [20], except that in the deterministic case only a lower bound on σ was assumed. We always consider an underlying filtered probability space denoted as (Ω, F, (F t ) t≥0 , P) that will be specified later on. In order to guarantee certain regularity properties of the noise terms, we make the following additional assumptions: (1) We assume that B 2 : U 2 → H is a Hilbert-Schmidt operator. In particular, this implies that Q 2 := B 2 B * 2 is a trace class operator and B 2 W 2 is a Q 2 -Wiener process.
Let us now formulate problem (2.1)-(2.2) as an abstract Cauchy problem. We define the following space this becomes a separable Hilbert space. ·, · H denotes the corresponding scalar product. Furthermore, we let We define the following linear operator . Since all the reaction terms are twice continuously differentiable they obey in particular the Carathéodory conditions [34]. Thus, the corresponding Nemytskii operator is defined by where F : D(F) ⊂ H → H and D(F) := H. By setting we can rewrite the system (2.1)-(2.2) as an abstract Cauchy problem on the space H du = (Au + F(u)) dt + B dW, (2.11) with initial condition (2.12)

Mild solutions and stochastic convolution
We are interested in the concept of mild solutions to SPDEs. First of all, let us note the following. We have It is well known that A 1 generates an analytic semigroup on H and A 2 is a bounded multiplication operator on H. Hence, A is the generator of an analytic semigroup {exp (tA)} t≥0 on H as well, see [23, Chapter 3, Theorem 2.1]. Also note that A generates an analytic semigroup {exp (tA)} t≥0 on L p (D) for every p ≥ 1. In particular, we have for u ∈ L p (D) that for every α ≥ 0 there exists a constant C α > 0 such that where a > 0, see for instance [27,Theorem 37.5]. The domain D((−A) α ) can be identified with the Sobolev space W 2α,p (D) and thus we have in our setting

Remark 2.3.
Omitting the additive noise term in equation (2.11), we are in the deterministic setting of [20]. From there the existence of a global-in-time solution (u 1 , u 2 ) ∈ C([0, ∞), H) for every initial condition u 0 ∈ H already follows.
Let us now return to the stochastic Cauchy problem (2.11)-(2.12). We define is called stochastic convolution.
More precisely, we have (see [22,Proposition 3.1]) This is a well-defined H-valued Gaussian process. Furthermore, Assumptions 2.2 (1) and (2) ensure that W A (t) is mean-square continuous and F t -measurable, see [11].

Remark 2.5.
As W A is a Gaussian process, we can bound all its higher-order moments, i.e. for p ≥ 1 we have This follows from the Kahane-Khintchine inequality, see [29,Theorem 3.12]. for some T > 0, see [11,Theorem 7.7]. Hence, local in time existence for our problem is guaranteed by the classical SPDE theory.

Preliminaries
We now recall some basic definitions related to random attractors. For more information the reader is referred to the sources given in the introduction. Definition 3.1. (Metric dynamical system) Let (Ω, F, P) be a probability space and let θ = {θ t : Ω → Ω} t∈R be a family of P-preserving transformations (i.e. θ t P = P for t ∈ R), which satisfy for t, s ∈ R that (1) (t, ω) → θ t ω is measurable, Then (Ω, F, P, θ) is called a metric dynamical system. The metric dynamical system describes the dynamics of the noise.
for all s, t ∈ R + and for all ω ∈ Ω. We say that ϕ is a continuous or differen- We summarize some further definitions relevant for the theory of random attractors.

Definition 3.4. (Omega-limit set)
For a random set K we define the omegalimit set to be .

Definition 3.5. (Attracting and absorbing set)
Let A, B be random sets and let ϕ be a RDS.
• Let D be a collection of random sets (of non-empty subsets of V), which is closed with respect to set inclusion. A set B ∈ D is called D-absorbing/Dattracting for the RDS ϕ, if B absorbs/attracts all random sets in D.
Remark 3.6. Throughout this work we use a convenient criterion to derive the existence of an absorbing set. Let A be a random set. If for every where ρ(ω) > 0 for every ω ∈ Ω, then the ball centred in 0 with radius ρ(ω) + for a > 0, i.e. B(ω) := B(0, ρ(ω) + ), absorbs A.
where d(A) = sup a∈A a . We denote by T the set of all tempered subsets of V.
there is a set of full P-measure such that for all ω in this set we have Hence a random variable X is tempered when the stationary random process X(θ t ω) grows sub-exponentially.

Remark 3.9. A sufficient condition that a positive random variable
If θ is an ergodic shift, then the only alternative to (3.2) is i.e., the random process X(θ t ω) either grows sub-exponentially or blows up at least exponentially.

Definition 3.10. (Random attractor)
Suppose ϕ is a RDS such that there exists a random compact set A ∈ T which satisfies for any ω ∈ Ω Then A is said to be a T -random attractor for the RDS. Then there exists a unique T -random attractor A, which is given by We will use the above theorem to show the existence of a random attractor for the partly dissipative system at hand.

Associated RDS
We will now define the RDS corresponding to (2.11)-(2.12). We consider V = H := L 2 (D)×L 2 (D) and T is the set of all tempered subsets of H. In the sequel, we consider the fixed canonical probability space (Ω, F, P) corresponding to a two-sided Wiener process, more precisely endowed with the compact-open topology. The σ-algebra F is the Borel σalgebra on Ω and P is the distribution of the trace class Wiener processW (t) : , where we recall that B 1 and B 2 fulfil Assumptions 2.2. We identify the elements of Ω with the paths of these Wiener processes, more preciselỹ Furthermore, we introduce the Wiener shift, namely (3.5) is a metric dynamical system. Next, we consider the following equations The stationary solutions of (3.6)-(3.7) are given by Here, we observe that for t = 0 Random attractors for stochastic partly dissipative systems Page 11 of 37 35 More explicitly/or component-wise this reads as In the equations above no stochastic differentials appear, hence they can be considered path-wise, i.e., for every ω instead just for P-almost every ω. For every ω (3.8) is a deterministic equation, where z(θ t ω) can be regarded as a time-continuous perturbation. In particular, [6] guarantees that for ) is a solution to (2.1)-(2.4). In particular, we can conclude at this point that (2.1)-(2.4) has a global-in-time solution which belongs to C([0, ∞); H); see Remark 2.3. We define the corresponding solution operator ϕ : for all (t, ω, (u 0 1 , u 0 2 )) ∈ R + × Ω × H. Now, ϕ is a continuous RDS associated to our stochastic partly dissipative system. In particular, the cocycle property obviously follows from the mild formulation. In the following, we will prove the existence of a global random attractor for this RDS. Due to conjugacy, see [9,25] this gives us automatically a global random attractor for the stochastic partly dissipative system (2.1)-(2.4).

Bounded absorbing set
In the following we will prove the existence of a bounded absorbing set for the RDS (3.11). In the calculations we will make use of some versions of certain classical deterministic results several times. Therefore, we recall these results here for completeness and as an aid to follow the calculations later on.
where r, a 1 , a 2 , a 3 are positive constants. Then The temperedness of z 2 (ω) 2 2 then follows directly using Remark 3.9. Now, we consider the random variable z 1 (ω) p p . Note that using the so-called factorization method we have for (x, t) ∈ D × [0, T ] and α ∈ (0, 1/2) (see [11,Ch. 5 where we have used the formal representation W 1 (x, s) = where we have used Parseval's identity and the Itô isometry. Our assumption on the boundedness of the eigenfunctions {e k } ∞ k=1 yields together with Assumption 2.2 (3) that Hence, E |Y (x, τ )| 2m ≤ C m for C m > 0 and every m ≥ 1 (note that all odd moments of a Gaussian random variable are zero). Thus we have i.e., in particular for all p ≥ 1 we have Y ∈ L p (D × [0, T ]) P-a.s.. We now observe where we have used (2.13) and thus E sup i.e., temperedness of z 1 (ω) p p follows again with Remark 3.9. (2) Regarding again Assumption 2.2 (3) one can show in a similar way that z 1 ∈ W 1,p (D) and in particular also ∇z 1 (ω) p p is a tempered random variable for all p ≥ 1.

Remark 3.19.
Alternatively, one can introduce the Ornstein-Uhlenbeck processes z 1 and z 2 using integration by parts. We applied the factorization Lemma for the definition of z 1 in order to obtain regularity results for z 1 based on the interplay between the eigenvalues of the linear part and of the covariance operator of the noise. Using integration by parts, one infers that This expression can also be used in order to investigate the regularity of z 1 in a Banach space H as follows: Here one uses the Hölder-continuity of ω 1 in an appropriate function space in order to compensate the singularity in the previous formula. In our case, we need z 1 ∈ D(A α/2 ) = W α,p (D). Letting ω 1 ∈ D(A ε ) for ε ≥ 0 and using that ω 1 is β-Hölder continuous with β ≤ 1/2 one has which is well-defined if β + ε > α/2. Such a condition provides again an interplay between the time and space regularity of the stochastic convolution.
Based on the results regarding the stochastic convolutions we can now investigate the long-time behaviour of our system. The first step is contained in the next lemma, which establishes the existence of an absorbing set.
Proof. To show the existence of a bounded absorbing set, we want to make use of Remark 3.6, i.e. we need an a-priori estimate in H. We have for v = (v 1 , v 2 ) solution of (3.8) where we have used (2.7). We now estimate I 1 -I 3 separately. Deterministic constants denoted as C, C 1 , C 2 , . . . may change from line to line. Using (2.5) and (2.10) we calculate With (2.9) we compute Now, combining the estimates for I 2 and I 3 yields where we have used that for q = max{p 1 + 1, 2} < p there exists a constant C 2 such that Thus, Hence, in total we obtain (3.19) and thus d dt We replace ω by θ −t ω (note the P-preserving property of the MDS) and carry out a change of variables Now let D ∈ T be arbitrary and (u 0 Due to the temperedness of z 1 (ω) p p for p ≥ 1 and z 2 (ω) 2 2 , the improper integral above exists and ρ(ω) > 0 is a ω-dependent constant. As described in Remark 3.6, we can define for some > 0 Then B = {B(ω)} ω ∈ T is a T -absorbing set for the RDS ϕ with finite absorption time t T (ω) = sup D∈T t D (ω).
The random radius ρ(ω) depends on the restrictions imposed on the nonlinearity and the noise. These were heavily used in Lemma 3.20 in order to derive the expression 3.22 for ρ(ω). Regarding the structure of ρ(ω) we infer by Lemma 3.17 that ρ(ω) is tempered. Although we have now shown the existence of a bounded T -absorbing set for the RDS at hand, we need further steps. To show existence of a random attractor, we would like to make use of NoDEA Random attractors for stochastic partly dissipative systems Page 21 of 37 35 Theorem 3.11, i.e., we have to show existence of a compact T -absorbing set. This will be the goal of the next subsection.