THE PROBABILISTIC CAUCHY PROBLEM FOR THE FOURTH ORDER SCHR¨ODINGER EQUATION WITH SPECIAL DERIVATIVE NONLINEARITIES

. Chen and Zhang [7] consider the probabilistic Cauchy problem of the fourth order Schr¨odinger equation where P m is a homogeneous polynomial of degree m . The almost sure local well-posedness and small data global existence were obtained in H s ( R d ) with the regularity threshold s c − 1 / 2 when d ≥ 3, where s c := d/ 2 − 2 / ( m − 1) is the scaling critical regularity. For the lower regularity threshold ( d − 1) s c /d with m = 2 and s c − min { 1 ,d/ 4 } with m ≥ 3, we get the corresponding well-posedness of the following fourth order nonlinear Schr¨odinger equation on R d ( d ≥ 2) with random initial data.

We establish the almost sure local well-posedness and small data global existence of (1.1) for random initial data in H s with s ∈ (s d,m ∨ 0, s c ], where The 4NLS, including its special forms, arises in deep water wave dynamics, plasma physics, optical communications, cf. [11,18,19,20]. The well-posedness of 4NLS with different nonlinearities was widely studied in scaling critical or supercritical regime by several authors, cf. [13,17,23,26,27]. 4NLS (1.1) is ill-posed below the scaling critical regularity in some cases, cf. [9,24].
It seems that the decay estimates (2.6) are only available for u. In this paper, the nonlinearities of (1.1) have special structure which can cancel out the worse interaction such that the decay estimates (2.6) are available for (1.1). Thus, we expect to lower the regularity threshold of the random data Cauchy problem of (1.1).
Applying the decay estimates (2.6), Strichartz estimates and the truncated bilinear estimates exploited by us, we establish the almost sure supercritical wellposedness of (1.1) in H s (R d ) for s ∈ (s d,m , s c ], where s d,m is in (1.3). Obviously, s d,m is smaller than the regularity threshold in [7]. Because the singularity near the frequency 0 occurs in Strichartz estimates of (1.1) when ε = 0, 1, we split the main proof into the low and high frequency sections. We also use symmetries method that we exploited in [7] to simplify our main proof.
1.1. Randomization procedure. We give the randomization of initial data based on the uniform decomposition of the frequency space. In [29], Wang, Zhao and Guo first applied the frequency uniform decomposition operators ψ(D − n) to study nonlinear evolution equations with initial data in modulation spaces which contain a class of super-critical data in Sobolev spaces, where ψ ∈ S(R d ) satisfies supp ψ ⊂ [−1, 1] d and Let φ ∈ H s be a complex-valued function for some s ∈ R. Wiener randomization of φ is defined as follows Here, {g n (ω)} n∈Z d is a sequence of independent mean zero complex-valued random variables on a probability space (Ω, F, P ). The real and imaginary parts of g n (ω) are independent and endowed with probability distributions µ 1 n and µ 2 n , respectively. Throughout this paper, we assume that there exists c > 0 such that for any γ ∈ R, n ∈ Z d , j = 1, 2. This condition is satisfied by the standard complexvalued Gaussian random variables and the standard Bernoulli random variables. It follows from Bernstein's inequality that We can see that there is no regularity increasement for the Bernstein estimate of ψ(D − n)(cf. [28]). Thus, the frequency uniform decomposition can improve space integrability and then improve the Strichartz-type estimates of random data (1.4) in a way. Moreover, by [5,Lemma 3.1], there exists C > 0 such that which means that the summation of g n (ω)c n in L p can be controlled by the l 2 -norm of {c n } n∈Z d (Since 1 ⊂ 2 , (1.6) has gained some regularity). These two points are crucial for us to consider the supercritical random data.
1.2. Main results. The almost sure local well-posedness of (1.1) for random initial data reads as follows.
Then for each 0 < T 1, there exists a set Ω T ⊂ Ω with the following properties: The next theorem states almost sure global well-posedness and scattering of (1.1) for random small data. If ε = −1, there exists C, c > 0 and a set Ω φ ⊂ Ω with the following properties:

Functional framework.
We introduce the precise functional framework used in the proofs of Theorems 1.1 and 1.2. For 2 ≤ q, r ≤ ∞, a pair (q, r) is called biharmonic admissible and Strichartz admissible if respectively. We only present the improved Strichartz-type estimates for the free solution of (1.1) with random initial data. The Strichartz-type estimates are given in [7]. One can see [2, Proposition 2 and Lemma 4] and [15, Lemma 2.4] for the proof.
x (R d ) and φ ω is its randomization defined by (1.4). Then the following results hold: (ii) Let (q, r) be biharmonic admissible with q, r < ∞ and r ≤r < ∞, then there exist C, c > 0 such that be Strichartz admissible with q, r < ∞ and r ≤r < ∞, then there exist C, c > 0 such that Specially, if ε = −1, we also have We apply U p , V p spaces to hold the solutions to (1.1). U p , V p spaces serve as a development of the Bourgain spaces, and have been very effective in establishing well-posedness of various dispersive PDEs. More details can be seen in [12,14].
The definition of U p , V p , U p S , V p S and V p −,rc,S are the same as the ones presented in [7], we omit the details. We give the definition of Z s , Y s spaces.
respectively. Here, P N is acting on the space variable.
There are the following embeddings: for p > 2, Given an interval I ⊂ R, we define the local-in-time versions of these spaces as restriction norms. For example, we define the Z s (I)-norm by The dual property of the Z sc norm is as follows. One can refer to [14] for the proof.
We present some important estimates in the U p , V p settings.
Exploring the frequency interaction, we get the following truncated bilinear estimates.
where δ > 0 can be arbitrarily small.
Proof of Lemma 2.5. We use I to represent By the proof of [7, Lemma 2.9 ], we may assume N 3 = N min and only need to show The idea is essential due to the proof of [8, Lemma 1]. We divide suppϕ N2 into disjoint cubes {Q 2,j } j∈Z + of side N 3 with center ξ 2,j , and choose a series of cubes Q 1,j of side 5N 3 with center ξ 1,j := −ξ 2,j . Then Similar calculations as that in [7] give dξdτ = Jdξ 1 1 dξ 2 , where J N 3 2 . It follows from Cauchy-Schwarz inequality, canonical transform and (2.15) that Here, Cauchy-Schwarz inequality and (2.15) lead to the last inequality.
Then (1.1) with random initial data is equivalent to the following perturbed 4NLS Thus it suffices to investigate (3.1). We establish the nonlinear estimates of (3.1) in this section. Denote z W s the maximum of all the norms of z, ∂z used in the proof of Lemmas 3.1, 3.2 and 3.3 which have the following forms: where (q 1 , r 1 ), (q 3 , r 3 ) are the improved Strichartz admissible pairs; (q 2 , r 2 ), (q 4 , r 4 ) are the improved biharmonic admissible pairs. For R > 0, put Recall ΓN m (u) in (1.8). We state the following global-in-time and local-in-time nonlinear estimates.  (1.3)). Let φ ∈ H s (R d ), φ ω be its randomization defined in (1.4). .
The following Lemma gives that the dual formula (2.5) is available for the truncated nonlinearities of (1.1).

Now we show Lemma 3.1 via dual formula (2.5).
Proof of Lemma 3.1. We only prove (3.4) since (3.5) is similar. Set v = v · χ [0,T ) and z = z · χ [0,T ) with T < ∞. We split the proof into the low frequency section and the high frequency section. Referring to the proof of Lemma 3.1 in [7], we only need to treat the high frequency section and it suffices to show N0,··· ,Nm where 0 ≤ l ≤ m, N max = max{N 0 , N 1 , · · · , N m } and N max ≥ 10m.
For the left side of (3.7), denote (τ j , ξ j ) ∈ R 1+d the frequency of P Nj v and P Nj z.
Denote R j := Q j P Nj with Q j ∈ {Q ≤M , Q >M } (see (1.10) for the definition). By (2.12), Q >M z = Q >M S(t)φ ω = 0. In order to show (3.7), it suffices to show that for some 0 ≤ k 0 ≤ l which satisfies R k0 = Q N 4 max P N k 0 , we have where 0 ≤ l ≤ m. For a set R ⊂ (2 N0 ) m+1 (for example, R is defined by N 0 ∼ N l ≥ · · · ≥ N 1 , N m ≥ · · · ≥ N l+1 ), we use the notation I R as By the symmetries method that we used in [7], we only need to show (3.9) in the following cases: (3.11) For N max ≥ 10m, we show (3.9) according to m = 2 and m ≥ 3.
Case 2: N 0 ∼ N m N l . If Q k = Q N 4 0 , k = 1, · · · , l, we may assume for ω ∈ E s R and s > max{0, s c − 1}. If l = m − 1, for P Nm z, we substitute L 2+δ t with L 2 t . Second, assume 2 ≤ d < 4. Referring to the above case: d ≥ 4, we have for ω ∈ E s R and s > s c − d 4 . Case 3: N m ∼ N m−1 ≥ N 0 , N l . We assume l ≤ m − 2 otherwise this is reduced to the case N m ∼ N l ≥ N 0 .
If Q 0 = Q N 4 m , we can argue via the same inequalities as that in the case Q 0 = Q N 4 m of Case 2: N m ∼ N 0 N l . For Q k = Q N 4 m (k = 1, · · · , l), we assume Q 1 = Q N 4 l and argue in the following two cases.