IRREDUCIBILITY OF KURAMOTO-SIVASHINSKY EQUATION DRIVEN BY DEGENERATE NOISE

. In this paper, we study irreducibility of Kuramoto-Sivashinsky equation which is driven by an additive noise acting only on a ﬁnite number of Fourier modes. In order to obtain the irreducibility, we ﬁrst investigate the approximate controllability of Kuramoto-Sivashinsky equation driven by a ﬁnite-dimensional force, the proof is based on Agrachev-Sarychev type geometric control approach. Next, we study the continuity of solving operator for deterministic Kuramoto-Sivashinsky equation. Finally, combining the approximate controllability with continuity of solving operator, we establish the irreducibility of Kuramoto-Sivashinsky equation.

principle of stochastic KSE, [31] established the averaging principle for stochastic KSE. We refer readers to the references therein for more interesting results on stochastic KSE.
In this paper, we consider the irreducibility of stochastic KSE driven by highly degenerate noise where T = R/2πZ, β k , k = 1, 2, 3 are real valued mutually independent standard Wiener processes defined on a probability space (Ω, F, P) and b k , k The main contribution of this paper is the irreducibility of stochastic KSE driven by the highly degenerate noise. The irreducibility of stochastic partial differential equations (SPDEs) has attracted many authors' attention, see [6-8, 18, 22, 23, 43, 44] and references within. However, to the best knowledge of the author, the irreducibility of SPDEs is generally established for non-degenerate noise, the result for the case of degenerate noise is far less studied. The irreducibility is a fundamental concept in stochastic dynamic system, and it plays a crucial role in the research of ergodic theory, moderate deviation principle. More precisely, the main reason for the study of the irreducibility property is its relevance in ergodic theory, and in the analysis of the uniqueness and ergodicity of invariant measures. The property of irreducibility is at the core of the problem, it is sufficient to obtain the ergodic result by adding the strong Feller property, see the classical work [17,24]. Another application of the irreducibility is to establish moderate deviation principle of SPDEs, in general, the moderate deviation principle is obtained by verifying the irreducibility and some Lyapunov condition for SPDEs, see [43].
The proof of Theorem 1.2 relies on the approximate controllability of KSE (see Thm. 3.3) and the continuity of the mapping noise → solution along the controllers (see Prop. 4.2). It is well known that one usually solves a control problem to prove the irreducibility for SPDEs. However, due to the highly degenerate of the noise of (1.1), the control problem of the associated deterministic system is much harder than those in the usual case, the traditional methods and techniques in the existing literature are no longer feasible. The novelty of this paper is that we introduce the Agrachev-Sarychev type geometric control approach to overcome this difficulty. More precisely, with the help of asymptotic property of operator R and saturating property of subspace of H, we can prove the approximate controllability of the associated deterministic system with finite-dimensional control.
The control theory is known to be a useful tool in the study of stochastic systems with non-degenerate noise, see [17] and references therein for more interesting results on this topic. In recent years, stochastic system with highly degenerate noise is attracting more and more people's attention, this paper shows that the Agrachev-Sarychev approach is a powerful tool for the study of stochastic systems with highly degenerate noise. Let us mention that the method and framework in this paper are quite general and can be adapted to degenerate problems for more SPDEs.
The rest of this paper is organized as follows: In Section 2, we give some preliminary results which will be used in this paper. Section 3 is devoted to the approximate controllability of KSE driven by a finite-dimensional force. The proof of Theorem 1.2 is given in Section 4.

Preliminaries
If u ∈ L 2 (T), it can be written as u(x) = k∈Z u k e ikx with u k ∈ C and u −k = u k , the norm in H s (T) is defined by · s : For s = 0, · 0 := · . For any s ≥ 0 and u = k∈Z u k e ikx ∈ H s (T), we define In this paper, we will use the following inequalities.

Approximate controllability of KSE
We first consider the approximate controllability of the following KSE driven by a finite-dimensional force: where the function η plays the role of the control.
Here, we give examples of saturating spaces. I is called a generator if I ⊂ Z is a finite symmetric set containing 0 and any integer is a linear combination of elements of I with integer coefficients. We define the space H(I) = span{sin(mx), cos(mx) : m ∈ I}.

Well-posedness of generalised KSE
We consider the following generalised KSE: where C = C(T, s) is a positive constant that increases in the first variable.
The proof of Lemma 3.5 is done later in Section 3.5.
Proposition 3.6. For any T > 0 and s > 1 2 , let u 0 ∈ H s (T), ζ ∈ L 4 (0, T ; H s+2 (T)) and ϕ ∈ L 2 (0, T ; H s−2 (T)). Then system (3.2) admits a unique solution u ∈ X T,s . Moreover, let R be the mapping taking a triple (u 0 , ζ, ϕ) to the solution of (3.2), forû 0 ∈ H s (T) andφ ∈ L 2 (0, T ; H s−2 (T)), we have Proof of Proposition 3.6. Before we prove Proposition 3.6, we introduce the following important mathematical setting: Throughout the paper, the letter C denotes unessential positive constant whose value may change in different occasions, which may vary from line to line. We will write the dependence of constant on parameters explicitly if it is essential. First, we consider the existence and uniqueness of the solution for (3.2).
Since H s (T)(s > 1 2 ) is a Banach algebra, we get According to this estimate, we have For any v 1 , v 2 ∈ X θ,s , the same argument shows that we can choose θ sufficiently small such that then according to (3.5) and (3.6), we have This implies that Φ has a unique fixed point u ∈ B X θ,s (R) which will be the solution of (3.2) for small θ.
For s = 0, taking the scalar product in L 2 (T) of equation (3.2) with u, noting the facts that By Gronwall's inequality, (3.7) holds for s = 0. For s > 0, we can prove (3.7) by induction. More precisely, let k be any nonnegative integer, assume that if we have (3.7) for s = k, we claim that (3.7) holds for k < s ≤ k + 1.
To this purpose, taking the scalar product in L 2 (T) of equation (3.2) with ∂ 2s x u, we can obtain that (3.8) Our task is reduced to estimate (uu x + (uζ) x , ∂ 2s x u). Applying Young's inequality in Lemma 2.3 and interpolation inequality in Lemma 2.4, we have (3.9) We will prove that for any k < s ≤ k + 1, k is any nonnegative integer, it holds that Indeed, if k = 0, 0 < s ≤ 1, then 2s + 1 ≤ s + 2, then for any ε > 0, it holds that where we have applied the interpolation inequality in Lemma 2.4 to obtain u 1 u s+2 ≤ ε ∂ s+2 x u 2 + C(ε) u 2 . According to (3.7) for s = 0, we can choose ε small enough such that (3.10) holds.
If k ≥ 1, k < s ≤ k + 1, we have 2s − k + 1 ≤ s + 2, then it follows that Due to the assumption that (3.7) holds for s = k, the above estimate implies that (3.10) holds. Combining estimates (3.8)-(3.10), we conclude that Applying Gronwall's inequality again, we can obtain (3.7) for k < s ≤ k + 1. This implies that (3.7) holds for any s ≥ 0. Namely, we get the existence of the solution on [0, T ]. For the uniqueness, let u 1 , u 2 ∈ X T,s be two solutions of (3.2), then Repeating the above arguments, we get that which implies that v ≡ 0 in view of the Gronwall's inequality. Finally, we prove (3.4). To this purpose, letû be the solution of (3.2) with (û 0 , 0,φ) and w = u −û, then w is a solution of problem where w 0 = u 0 −û 0 and η = ϕ −φ. Applying similar methods as above, we can obtain that Thus, (3.4) follows from Gronwall's inequality. This ends the proof of Proposition 3.6.

Asymptotic property
The following asymptotic property plays an important role in the proof of approximate controllability.
Proposition 3.7. Let T > 0 and s > 1 2 . For any u 0 , η ∈ H s+2 (T), ζ ∈ H s+4 (T), h ∈ L 2 (0, T ; H s−2 (T)), there is a number δ 0 ∈ (0, 1) such that for any δ ∈ (0, δ 0 ), the following limit holds Proof. Let us take any δ > 0 and consider the equation By Proposition 3.6, (3.12) has a unique solution in X T,s . Make a time substitution and consider the functions Then v is a solution of the following system Taking the scalar product in L 2 (T) of equation (3.13) with v, we can obtain that Then taking the scalar product in L 2 (T) of equation (3.13) with ∂ 2s x v, it follows that x v).

P. GAO
The first term in N (w, ζ, v) can be estimated as follows: Since H s (T)(s > 1 2 ) is an algebra, it follows from the interpolation inequality in Lemma 2.4 that Therefore, for any s > 1 2 , By the similar method as above, we can deduce that Next, we consider (δ 1 2 (ζv) x , ∂ 2s x v). Since s > 1 2 , with the help of the interpolation inequality in Lemma 2.4, we have Then, consideration is given to the remaining terms in N (w, ζ, v), Combining the above estimates, we have Proceeding as in the proof of Proposition 2.4 in [37], we can choose δ 0 ∈ (0, 1) so small such that for any δ < δ 0 , The proof of Proposition 3.7 is complete.

Proof of Theorem 3.3
The proof of Theorem 3.3 is analogous to Theorem 3.3 in [37], the proof is divided into four steps.
Step 1. Controllability in small time to u 0 + H 0 . Let us assume for the moment that u 0 ∈ H s+2 (T). For any η ∈ H 0 , applying Proposition 3.7 for the couple (η, 0), we see that It follows from the above limit that for any ε > 0, there is a time 0 < θ 1 < T such thatη = θ −1 1 η ∈ L 2 (0, T ; H) and Step 2. Controllability in small time to u 0 + H N . We argue by induction. Assume that the approximate controllability of (3.1) to the set u 0 + H N −1 is already proved. Let η 1 ∈ H N be of the form for some integer n ≥ 1 and vectors η, ζ 1 , · · · , ζ n ∈ H N −1 . Applying Proposition 3.7 for the couple (0, ζ 1 ), we see that Using the equality and the limit (3.17), we obtain Combining this with the fact that η, ζ 1 ∈ H N −1 , the induction hypothesis, and Proposition 3.6, we can find a small time θ 2 > 0 and a controlη 1 ∈ L 2 (0, T ; H) such that Iterating this argument successively for the vectors ζ 2 , · · · , ζ n , we construct a small time θ > 0 and a control η 1 ∈ L 2 (0, T ; H) satisfying where we used (3.16). This proves the approximate controllability in small time to any point in u 0 + H N .
Step 3. Global controllability in small time. Now let u 1 ∈ H s (T) be arbitrary. As H ∞ is dense in H s (T), there is an integer N ≥ 1 and pointû 1 ∈ u 0 + H N such that By the results of Steps 1 and 2, for any ε > 0, there is a time θ > 0 and a controlη ∈ L 2 (0, T ; H) satisfying Combining this with (3.18), we get approximate controllability in small time to u 1 . Due to Proposition 3.6 and the fact that the space H s+2 (T) is dense in H s (T), we can also obtain small time approximate controllability starting from u 0 ∈ H s (T).
Step 4. Global controllability in fixed time T . Applying the result of Step 3, for any ε > 0, there is a time T 1 > 0 and a control η 1 ∈ L 2 (0, Take v 1 = R T1 (u 0 , h + η 1 ). According to Proposition 3.6, we can find τ > 0 such that for t ∈ [0, τ ], Define a control function then, it follows that If T 1 + τ ≥ T , then the proof is complete. Otherwise, take v 2 = R T1+τ (u 0 , h + η 2 ), by the result of Step 3, there is a time T 2 > 0 and a control η 3 ∈ L 2 (0, T 2 ; H) satisfying Take v 3 = R T2 (v 2 , h + η 3 ), applying again Proposition 3.6, for the same τ , if t ∈ [0, τ ], we have Define the control function we can see that Again, if T 1 + T 2 + 2τ ≥ T , then the proof is complete. Otherwise, after a finite number of iterations, we complete the proof of Theorem 3.3.

Proof of Proposition 3.4
The proof is divided into four steps.
Indeed, the case m = 0 is obvious. If m = 0, we have Step 2. We prove cos(l + m)x ∈ H 1 for l, m ∈ I.  By the definition of H 1 and the results in Step 1, we can obtain that According to (3.19), we can get the expressions of cos(lx) cos(mx) and sin(lx) sin(mx), this means that cos(lx) cos(mx), sin(lx) sin(mx) ∈ H 1 .
Step 4. Note that for any m ∈ I, sin(−mx), cos(−mx) ∈ H, thus by the results in Step 2 and Step 3, we have cos(l ± m)x, sin(l ± m)x ∈ H 1 for l, m ∈ I.
Since I is a generator, repeating the above steps, we derive span{sin(mx), cos(mx) : m ∈ Z} ⊂ H ∞ , this implies that H(I) is saturating. The proof of Proposition 3.4 is complete.

Proof of Lemma 3.5
Let y be the solution to the system Taking the scalar product in L 2 (T) of equation (3.21) with y + ∂ 2s x y, it follows that this yields x y, f )dr). Since we have Plugging this estimate into (3.22), it is shown that Combining the above estimates, we conclude that y X T ,s ≤ C( y 0 s + f L 2 (0,T ;H s−2 (T) ).
If we take f = 0 in (3.21), the solution y to (3.

Proof of Theorem 1.2
In this section, we will establish the irreducibility of stochastic KSE.

Continuity of solving operator for deterministic KSE
We consider the following integral equation: (4.1) For s ∈ R, we define spaces By the same argument as in Exercise 2.1.27 of [33], we know that (4.1) has a unique solution u ∈ C([0, T ], L 2 (T)) ∩ L 2 (0, T ; H 2 (T)) if u 0 ∈ L 2 (T) and g ∈ V 2 .
Proposition 4.1. For any T > 0, the solution of equation (4.1) satisfies the following estimates: Proof. (i). Making the substitution u(t) = y(t) + g(t), then y satisfies the equation The right hand side can be estimated as follows: .
Applying (4.5) and Gronwall's inequality, we can obtain that This implies (4.3). The proof of Proposition 4.1 is complete.
The proof of Proposition 4.2 is complete.
The proof of Theorem 1.2 is complete.