GLOBAL DYNAMICS FOR THE STOCHASTIC KDV EQUATION WITH WHITE NOISE AS INITIAL DATA

. We study the stochastic Korteweg-de Vries equation (SKdV) with an additive space-time white noise forcing, posed on the one-dimensional torus. In particular, we construct global-in-time solutions to SKdV with spatial white noise initial data. Due to the lack of an invariant measure, Bourgain’s invariant measure argument is not applicable to this problem. In order to overcome this diﬃculty, we implement a variant of Bourgain’s argument in the context of an evolution system of measures and construct global-in-time dynamics. Moreover, we show that the white noise measure with variance 1 + t is an evolution system of measures for SKdV with the white noise initial data.

1. Introduction 1.1.Main result.The main objective of the present paper is to explain how techniques developed to study invariance of certain measures (in our case, a spatial white noise) under the flow of Hamiltonian partial differential equations (PDEs) can be combined with the analysis of stochastic perturbations of these equations to construct global-in-time solutions in a probabilistic setting.
In particular, we consider the following Cauchy problem for the stochastic Korteweg-de Vries equation (SKdV) on the one-dimensional torus T = R/(2πZ): x u + u∂ x u = ξ u| t=0 = u 0 . (1.1) Here, ξ denotes an additive (Gaussian) space-time white noise forcing whose space-time covariance is (formally) given by for x 1 , x 2 ∈ T and t 1 , t 2 ∈ R + with δ denoting the Dirac delta function.In particular, we study (1.1) with a spatial white noise 1 on T, independent of the forcing ξ, as initial data.
More concretely, we take u 0 = u ω 0 of the form: where {g n } n∈Z is a family of independent standard complex-valued Gaussian random variables conditioned that g −n = g n , n ∈ Z.The main difficulty of this problem comes from the roughness of the noise and the white noise initial data, such that the solution u(t) to (1.1) belong to H s (T) \ H − 1 2 (T), s < − 1 2 , almost surely.Here, H s (T) denotes the L 2 -based Sobolev space defined by the norm: , The well-posedness issue of SKdV with an additive forcing: where φ is a bounded operator on L 2 , has been studied both on the real line and on the torus [20,21,51,22,37].In the periodic setting, de Bouard, Debussche, and Tsutsumi [22] proved local well-posedness of (1.4) on T when φ is a Hilbert-Schmidt operator from L 2 (T) to H s (T) for s > − 1 2 , barely missing the case of an additive space-time white noise.This local well-posedness result in [22] was obtained via a contraction argument, based on the Fourier restriction norm method (namely, utilizing the X s,b -spaces) adapted to the Besov space, utilizing the endpoint Besov regularity of the Brownian motion [13,54,1].With an additional assumption that φ is Hilbert-Schmidt from L 2 (T) to L 2 (T), they also proved global well-posedness of (1.4) in L 2 (T).In [37], the first author improved this result and proved local well-posedness of (1.4) even when φ = Id (thus reducing to (1.1)), thus handling the case of an additive space-time white noise. 3We point out that the argument in [37] is based on an approximation argument, in particular, not based on a contraction argument.Below, we will describe the approach in [37] more in detail; see Section 3. Our main goal is to construct global-in-time dynamics for (1.1) with the spatial white noise u ω 0 in (1.3) as initial data.
Before proceeding further, let us go over the known well-posedness results for the (deterministic) KdV on T: x u + u∂ x u = 0. (1.5) In [4], Bourgain introduced the so-called Fourier restriction norm method, utilizing the X s,b -spaces defined by the norm: and proved local well-posedness of (1.5) in L 2 (T) via a fixed point argument, immediately yielding global well-posedness in L 2 (T) thanks to the conservation of the L 2 -norm.Subsequently, Kenig, Ponce, and Vega [31] (also see [14]) improved Bourgain's result and proved local well-posedness of (1.5) in H − 1 2 (T) by establishing the following bilinear estimate: for s ≥ − 1 2 and b = 1 2 under the (spatial) mean-zero assumption on u and v.In [14], Colliander, Keel, Staffilani, Takaoka, and Tao then proved the corresponding global wellposedness result in H − 1 2 (T) via the I-method.The KdV equation (1.5) is also known to be one of the simplest completely integrable PDEs, and there are well-posedness results for (1.5), exploiting the completely integrable structure of the equation.In [6], Bourgain proved global well-posedness of (1.5) in the class M(T) of finite Borel measures λ on T, assuming that its total variation λ is sufficiently small.His proof was based on partially iterating the Duhamel formulation of (1.5) and establishing bilinear and trilinear estimates, assuming an a priori uniform bound of the form: on the Fourier coefficients of the solution u.Then, he established the global-in-time a priori bound (1.8), using the complete integrability.In [30], Kappeler and Topalov proved global well-posedness of (1.5) in H −1 (T) via the inverse spectral method.See also [33].
For SKdV (1.1) with a random perturbation, such an integrable structure is destroyed and thus the approaches based on the complete integrability of KdV are no longer applicable.Nonetheless, in [37], the first author adapted Bourgain's approach [6], based on a partial iteration of the Duhamel formulation (= the mild formulation) of (1.1), and proved local well-posedness of (1.1).In particular, he bypassed the assumption (1.8) by employing the Fourier restriction norm method adapted to the "Fourier-Besov" space b s p,∞ (T) introduced in [36], defined by the norm: which captures the spatial regularity of the space-time white noise when sp < −1; see Proposition 3.4 in [36]. 4Here, Z ≥0 = N ∪ {0}, and {|n| ∼ 2 j } means {2 j−1 < |n| ≤ 2 j } when j ≥ 1 and {|n| ≤ 1} when j = 0. Note that, by taking p > 2 (but close to 2), we can take s > − 1 2 , still satisfying sp < −1, which is crucial in establishing relevant nonlinear estimates.In Section 3, we go over some aspects of the local well-posedness argument from [37].
We now state our main result, which extends the solution constructed in [37] globally in time in the case of the white noise initial data.We say that u is a solution to (1.1) if it satisfies the following Duhamel formulation (= the mild formulation): where S(t) = e −t∂ 3 x denotes the linear KdV propagator (= the Airy propagator) and W denotes a cylindrical Wiener process on L 2 (T): where {β n } n∈Z is defined by β n (t) = ξ, 1 [0,t] • e n x,t .Here, •, • x,t denotes the duality pairing on T × R + .As a result, we see that {β n } n∈Z is a family of mutually independent complex-valued Brownian motions conditioned that β −n = β n , n ∈ Z.In particular, β 0 is a standard real-valued Brownian motion, and we have x,t = t (1.12) for any n ∈ Z.Note that the space-time white noise ξ in (1.1) is a distributional time derivative of the cylindrical Wiener process W in (1.11).The third term on the right-hand side of (1.10) is the so-called stochastic convolution, representing the effect of the stochastic forcing.
In the following, we set for some small δ 1 , δ 2 > 0 such that sp < −1.Given α ≥ 0, 5 we say that a distributionvalued random variable X on T (and its law, denoted by µ α ) is a (spatial) white noise on T with variance α if where u ω 0 is the white noise (with variance 1) in (1.3).Theorem 1.1.The stochastic KdV equation (1.1) with an additive space-time white noise forcing is globally well-posed with white noise initial data.More precisely, there exist small δ 1 , δ 2 > 0 such that, with probability 1, there exists a unique global-in-time solution u to (1.1), belonging to the class C(R + ; b s p,∞ (T)) with s and p as in (1.13), with the white noise initial data u ω 0 in (1.3).Moreover, for any t ≥ 0, we have Law(u(t)) = µ 1+t . (1.15) Namely, u(t) is a white noise with variance 1 + t.
The proof of Theorem 1.1 is based on a variant of Bourgain's invariant measure argument [5] in the context of an evolution system of measures [19,18], which is a natural generalization of the concept of invariant measures for an autonomous dynamical system.Let us give a somewhat formal definition of an evolution system of measures.Let Φ t 1 ,t 2 = Φ ω t 1 ,t 2 , t 2 ≥ t 1 ≥ 0, be a solution map for a given autonomous (random) dynamical system, sending the data ϕ at time t 1 to the solution Φ t 1 ,t 2 ϕ at time t 2 .Then, we define the transition semigroup P t 1 ,t 2 by for a bounded measurable function F on the phase space M.Then, we say that6 a family {ρ t } t∈R + of probability measures on M is an evolution system of measures indexed by for any bounded continuous function F on M and t 2 ≥ t 1 ≥ 0. Note that (1.17) is equivalent to ρ t 2 = P * t 1 ,t 2 ρ t 1 for any t 2 ≥ t 1 ≥ 0. If there exists an invariant measure ρ, then by setting ρ t = ρ, t ∈ R + , the family {ρ t } t∈R + is obviously an evolution system of measures.It is in this sense that the notion of an evolution system of measures is a generalization of the notion of an invariant measure.
Given t ∈ R + , let µ 1+t be the white noise of variance 1 + t defined in (1.14).Then, the following corollary follows from (1.15) and the flow property for t 3 ≥ t 2 ≥ t 1 ≥ 0 of the solution map to SKdV (1.1) constructed in Theorem 1.1.
Corollary 1.2.Let µ 1+t be the white noise measure with variance 1 + t as in (1.14).Then, the family {µ 1+t } t∈R + is an evolution system of measures for SKdV (1.1) with the white noise initial data u ω 0 in (1.3).
denotes the family of probability measures on M.However, we simply refer to the family {ρt} t∈R + of measures as an evolution system of measures.
Furthermore, we have the following corollary to Theorem 1.1.
Corollary 1.3.(i) Given α ≥ 0, let u ω 0,α be a white noise on T with variance α given by where {g n } n∈Z is as in (1.3).Then, with probability 1, there exists a unique global-in-time solution u to (1.1), with u| t=0 = u ω 0,α .Moreover, for any t ≥ 0, we have where µ α+t is as in (1.14).Namely, u(t) is a white noise with variance α + t.
(ii) Let w 0 be a deterministic function in L 2 (T) and α > 0.Then, with probability 1, there exists a unique global-in-time solution u to (1.1) with u| t=0 = w 0 + √ α u ω 0,α , where u ω 0 is the white noise on T with variance α as in (1.3).
Part (i) of Corollary 1.3 directly follows from Theorem 1.1 together with the flow property (1.18) and the time translation invariance (in law) of SKdV (1.1).See also Remark 1.5.Part (ii) of Corollary 1.3 follows from Corollary 1.3 (i) and the Cameron-Martin theorem [9] by noting that L 2 (T) is the Cameron-Martin space of µ α = Law( √ αu ω 0 ).See [42] for a further discussion.
Thanks to the time reversibility of the KdV equation, Theorem 1.1 and Corollary 1.3 also hold for negative times (where the variances 1 + t in (1.15) and α + t in (1.19) are replaced by 1 + |t| and α + |t|, respectively.For simplicity of the presentation, however, we only consider positive times in the remaining part of the paper.Moreover, in the following discussion, in considering a stochastic flow on a time interval [t 1 , t 2 ], it is understood that random initial data at time t 1 and a stochastic forcing on [t 1 , t 2 ] are independent (which is justified by (1.2)).
1.2.Outline of the proof.Let us now describe some aspects of the proof of Theorem 1.1.Except in the small data regime (including a small perturbation of a known global solution), one usually needs to exploit conservation laws in order to construct global-in-time solutions to nonlinear dispersive PDEs.A remarkable intuition by Bourgain in [5] was to use (formal) invariance of a Gibbs measure as a replacement of a conservation law to construct global-intime solutions with the Gibbsian initial data.More precisely, he used the rigorous invariance of the truncated Gibbs measures for the associated truncated dynamics and combined it with a PDE approximation argument to construct the desired global-in-time invariant Gibbs dynamics.This argument, known as Bourgain's invariant measure argument, has been applied to many dispersive PDEs with random initial data (and stochastic forcing), in particular over the last fifteen years.See the survey papers [38,3,58] for a further discussion on this topic and the references therein.See also [26,39,40] for more recent results in the context of stochastic dispersive PDEs.We point out that Bourgain's invariant measure argument has also been applied to globalize solutions to stochastic parabolic PDEs; see, for example, [28,46,45].
In the current problem at hand, due to the lack of a damping term, there is no invariant measure for SKdV (1.1), and thus Bourgain's invariant measure argument is not applicable.It is, however, easy to see, at a formal level, (as explained below) that SKdV (1.1) with the white noise initial data (1.3) possesses a (formal) evolution system of measures {µ 1+t } t∈R + , where µ 1+t is a white noise measure with variance 1 + t defined in (1.14).See also Proposition 1.4.Our main strategy is then to use this (formal) evolution system of measures {µ 1+t } t∈R + as a replacement of a (formal) invariant measure in Bourgain's invariant measure argument (and hence as a replacement of a conservation law in the deterministic setting).
Before proceeding further, let us provide a heuristic argument for the claim that {µ 1+t } t∈R + is an evolution system of measures for SKdV (1.1) with the white noise initial data.First, view the SKdV dynamics (1.1) as a superposition of the deterministic KdV (1.5) and (at the level of infinitesimal generators).On the one hand, the white noise (with any variance) is known to be invariant under the flow of the deterministic KdV (1.5); see [52,36,38,43,32].On the other hand, the stochastic flow (1.20) with a white noise initial data (with any variance) increases the variance by the length of the time interval under consideration.Then, the claim follows, at least at a purely formal level, from these observations together with the Lie-Trotter product formula [53,Section VIII.8]: (which holds, for example, for finite-dimensional matrices A, B).We point out that the Lie-Trotter product formula (1.21) is not directly applicable to our problem, and the core of the proof of Theorem 1.1 consists of justifying this heuristic argument by an approximation argument, which we explain next.
• Truncated SKdV dynamics.Given N ∈ N, let P N denotes the Dirichlet projection on (spatial) frequencies {|n| ≤ N }.Then, consider the following truncated SKdV equation: where u ω 0 is the white noise given in (1.3).Note that the truncation appears only on the nonlinearity, but not on the noise or the initial data.With P ⊥ N = Id −P N , set u N = P N u N and u ⊥ N = P ⊥ N u N .Then, the truncated SKdV dynamics (1.22) decouples into the finite-dimensional nonlinear dynamics for the low frequency part u N = P N u N : and the linear dynamics for the high frequency part (1.24) It is easy to see that both (1.23)  denote the transition semigroups for (1.23) and (1.24), respectively, defined as in (1.16), where the expectation is taken over the noise restricted to the time interval [t 1 , t 2 ].We also use Φ N t 1 ,t 2 and P N t 1 ,t 2 to denote the solution map and the transition semigroup for the truncated SKdV (1.22).
Given α ≥ 0, let µ α be the white noise measure (with variance α) as in (1.14).Then, we can write µ α as where µ N,low α = (P N ) * µ α and µ N,high α = (P ⊥ N ) * µ α the pushforward image measures of µ α under P N and P ⊥ N , respectively.Note that µ N,low α and µ N,high α are nothing but the white noise measures (with variance α) on E N = span{e inx : |n| ≤ N } and E ⊥ N = span{e inx : |n| > N }, respectively, where the latter span is taken over the space D ′ (T) of distributions on T.
The high frequency dynamics (1.23) is linear and it is easy to verify that By writing it on the Fourier side, we see that the low frequency dynamics (1.23) is nothing but a finite-dimensional system of SDEs, which can be viewed as the superposition of the finite-dimensional KdV dynamics: and the linear stochastic dynamics: (1.28) While the former (1.27) preserves the white noise µ N,low α (with any variance), the latter (1.28) increases the variance of the white noise initial data by the length of the time interval under consideration.Then, in view of the Lie-Trotter product formula (1.21), we see that Putting (1.26) and (1.29) together, we then obtain the following proposition.
We present the proof of Proposition 1.4 in Section 2. As for the low frequency part of the claim, instead of decomposing the low frequency dynamics (1.23) into (1.27) and (1.28) and applying the Lie-Trotter product formula (1.21), we verify (1.29) by directly showing that µ N,low 1+t is the unique solution to the Kolmogorov forward equation (= the Fokker-Planck equation).
Remark 1.5.Let α ≥ 0. A straightforward modification of the proof of Proposition 1.4 yields , which is the key ingredient for proving Corollary 1.3 (i), replacing Proposition 1.4.
Once we obtain Proposition 1.4, we use ideas from Bourgain's invariant measure argument [5] together with the nonlinear analysis in [37], and establish a probabilistic uniform (in N ) growth bound on the solutions to the truncated SKdV (1.22).See Proposition 4.1.Finally, Theorem 1.1 follows from a PDE approximation argument and this probabilistic uniform growth bound.See Section 5.
• Mean-zero assumption: Recall that the bilinear estimate (1.7) holds only for (spatial) mean-zero functions, namely, the spatial means of u(t) and v(t) are zero for any t ∈ R. In the case of the deterministic KdV (1.5), if initial data u 0 has non-zero mean α 0 , then the following Galilean transformation: as in [15] together with the conservation of the (spatial) mean under KdV transforms KdV with a non-zero mean into the mean-zero KdV (so that the bilinear estimate (1.7) is applicable).In the case of SKdV with an additive noise, the spatial mean of a solution is no longer conserved.Nonetheless, in [22,37], a similar transformation was employed to reduce SKdV with an additive noise to the mean-zero case.The transformation in this case depends not only on the mean of the initial condition but also on the Brownian motion β 0 at the zeroth frequency in (1.11).See [22,37] for details.
For conciseness of the presentation, we impose the following mean-zero assumption in the remaining part of the paper.
• We assume that the white noise initial data u ω 0 in (1.3) and the space-time white noise ξ in (1.1) and (1.22) have spatial mean-zero.This means that the random initial data is now given by where Z * = Z \ {0}, and the stochastic forcing ξ is given by the distributional time derivative of Namely, we have ξ = P =0 ξ, where P =0 is the projection onto the non-zero (spatial) frequencies.This assumption together with the presence of the derivative on the nonlinearity u∂ x u = 1 2 ∂ x u 2 implies that a solution u to SKdV (1.1) has spatial mean zero as long as it exists.
It is understood that all the functions/distributions have spatial mean zero in the following.The required modifications to handle the general case (i.e. with the white noise u ω 0 in (1.3) and the space-time white noise ξ without the projection P =0 ) are straightforward and hence we omit details.See [37] for details.
We conclude this introduction by stating several remarks.
Remark 1.6.The usual application of Bourgain's invariant measure argument provides a growth bound 7 on a solution by √ log t for t ≫ 1, where the implicit constant is random.In the current SKdV problem, we instead obtain a growth bound on a solution by (something slightly faster than) √ t log t for t ≫ 1, where the extra factor √ t comes from the fact that the variance of the white noise at time t grows like ∼ t.See Remark 5.2.
Remark 1.7.(i) As mentioned above, by applying the I-method, Colliander, Keel, Staffilani, Takaoka, and Tao [14] proved global well-posedness of the deterministic KdV (1.5) in H − 1 2 (T).It would be of interest to apply the I-method to study global well-posedness of SKdV (1.1) with general deterministic initial data.In [11], the first author with Cheung and Li adapted the I-method to the stochastic setting and proved global well-posedness, below the energy space, of the stochastic nonlinear Schrödinger equation (SNLS) on R 3 with additive stochastic forcing, white in time and correlated in space.On the one hand, the I-method is suitable for controlling an L 2 -based Sobolev norm.On the other hand, the only known local well-posedness result of SKdV (1.1) is in the Fourier-Besov space b s p,∞ (at this point), and thus there is a non-trivial difficulty in adapting the I-method to this problem.
(ii) In [34], Killip, Vişan, and Zhang exploited the complete integrable structure of the deterministic KdV (1.5) and established a global-in-time a priori bound for solutions to KdV in H s (T), s ≥ −1.This a priori bound was given by a sum of suitable rescaled perturbation determinants, (each of which is given as an infinite series).It would also be of interest to investigate if their approach can be adapted to the current stochastic setting (and moreover to the Fourier-Besov setting, using the ideas in [49]).
Remark 1.8.Consider the following SNLS on T: where ξ is a complex-valued space-time white noise on T × R + , with the complex-valued white noise initial data: where {g n } n∈Z is a family of independent standard complex-valued Gaussian random variables.(Here, we do not impose the condition g −n = g n .)Due to the low regularity of the initial data and the forcing, we need to renormalize the nonlinearity in (1.32) to that considered in [12,27,25,50].In the following discussion, we suppress this renormalization issue.
Let us first consider the (deterministic) nonlinear Schrödinger equation (NLS) on T: (1.34) 7 At least in the setting of [5].In the singular setting, we have a growth bound by a suitable power of log t.See, for example, Section 5 in [44].
Given α > 0, let µ α = Law( √ α u ω 0 ) with u ω 0 as in (1.33) be the (complex) white noise measure with variance α.Formally, we have See [38,43].Then, in view of the conservation of the L 2 -norm under (1.34) and the fact that NLS (1.34) is Hamiltonian, we expect that the white noise measure µ α is invariant under the NLS dynamics.In [43], the first two authors with Valkó proved formal invariance of the white noise measure under NLS (1.34) in the sense that the white noise measure is a weak limit of invariant measures for NLS (1.34).In the same paper, they also conjectured invariance of the white noise under NLS (1.34).This conjecture remains as a challenging open problem to date, in particular due to the critical nature of the well-posedness issue for (1.34) (and also for (1.32)) with white noise initial data; see [25,23].See also [48] for invariance of the white noise measure under the fourth order NLS on T, where Let us now turn our attention to SNLS (1.32).As in the SKdV case, by viewing (1.32) as a superposition of the deterministic NLS (1.34) and the stochastic flow i∂ t u = ξ together with the conjectural invariance of the white noise under NLS (1.34), we arrive at the following conjecture.
Conjecture 1.The family {µ 1+t } t∈R + of the white noise measures with variance 1 + t is an evolution system of measures for SNLS (1.32) with the white noise initial data u ω 0 in (1.33).This conjecture is of importance not only from the viewpoint of mathematical analysis but also from the viewpoint of applications due to the importance of SNLS (1.32) (and NLS (1.34)) in nonlinear fiber optics.A straightforward modification of the proof of Proposition 1.4 shows that, for any N ∈ N, the family {µ 1+t } t∈R + is an evolution system of measure for the following truncated SNLS: with the white noise initial data u ω 0 in (1.33).The main obstacle for proving Conjecture 1 is the local well-posedness issue as in the case of NLS (1.34) with the white noise initial data.

Finite-dimensional approximations and their distributions
In the remaining part of the paper, we work on a probability space (Ω, F, P) supporting • A family {g n } n∈N of independent standard complex-valued Gaussian random variables: Here, {Re g n , Im g n } n∈N is a family of independent real-valued Gaussian random variables with mean 0 and variance 1 2 .We then set g −n = g n , n ∈ N. The random variables g n are used to define the spatial white noise u ω 0 on T in (1.30) which we use as initial data for (1.1) and (1.22).
• A family {β n } n∈N of independent complex-valued Brownian motions, satisfying (1.12): The Brownian motions β n serve to define the driving space-time white noise appearing in (1.1) as well as its truncated version (1.22).
We emphasize that we only work with (spatial) mean-zero functions/distributions in the following.Given α ≥ 0, let u ω 0,α be a white noise on T with variance α given by where {g n } n∈Z is as in (2.1), and set In this section, we study the truncated SKdV (1.22) and present the proof of Proposition 1.4.In view of the discussion in Section 1, it suffices to prove (1.26) and (1.29) for the high and low frequency dynamics, respectively.
We first consider the high frequency dynamics (1.24): By working on the Fourier side, we see that (2.4) is a system of decoupled linear SDEs for each frequency.In particular, (2.4) is globally well-posed and the solution to (2.4) is given by for general initial data u ⊥ N (0) = P ⊥ N u ⊥ N (0).In particular, when the initial data is given by P ⊥ N u ω 0 with u ω 0 in (1.30), we have Then, from the independence of I n and II n , we conclude that (P N,high 0,t (2.5) Therefore, from (2.5) and the flow property of the solution map Φ N,high t 1 ,t 2 for (2.4) (analogous to (1.18)), we conclude (1.26).
Let us now turn our attention to the low frequency dynamics (1.23): Proof.By writing (2.6) in the Duhamel formulation, we have where W is as in (1.31).Note that u N (t) = P N u N (t) as long as the solution u N exists.By Bernstein's inequality ([57, Appendix A]), we have which allows us to control the second term on the right-hand side of (2.7).By the unitarity of S(t) on L 2 (T) and the basic property of a Wiener integral, we have T N.
In particular, we have for some almost surely finite random constant C(ω) > 0. Hence, we conclude from a standard contraction argument in C([0, T ]; L 2 (T)) with (2.8) and (2.9) that (2.6) is locally well-posed.Furthermore, the solution exists globally in time as long as its L 2 (T)-norm remains bounded.As observed in [22, Theorem 1.5] and [21, Section 3.2], a simple argument, using Ito's formula, Doob's martingale inequality, and the L 2 -conservation of the truncated KdV equation (1.27), provides the following bound: for any finite T > 0, where • HS(L 2 ;L 2 ) denotes the Hilbert-Schmidt norm from L 2 (T) to L 2 (T).From this a priori bound, we conclude global well-posedness of (2.6).
In the following, we study the evolution of the distribution of the solution u N (t) to the low frequency dynamics (2.6).Let p n (t) = Re u N (n, t) and q n (t) = Im u N (n, t) for 1 ≤ |n| ≤ N .Since u N is real-valued, we have Then, by writing (2.6) on the Fourier side, we obtain the following finite-dimensional system of SDEs for (p, q) = (p 1 , . . ., p N , q 1 , . . ., q N ): for n = 1, . . ., N , where P n and Q n are defined by (2.11) Define A(p, q) = A(p 1 , . . ., p N , q 1 , . . ., q N ) by (2.12) Then, we have (2.13) Let x = (x 1 , . . ., x 2N ) = (p 1 , . . ., p N , q 1 , . . ., q N ).In the following, we briefly go over the derivation of the Kolmogorov forward equation for the evolution of the density of the distribution for (2.14) See, for example, [55,17].Recalling from (1.12) that E[(Re , we see that the Kolmogorov operator L for (2.10) is given by where A(x) is given by Lemma 2.2.Let f 0 (x) be a density of the distribution for U (0).Then, the density f (x, t) of the distribution for U (t) satisfies the following Kolmogorov forward equation on R 2N : where the vector field A(x) is as in (2.16).
Proof.This is classical, so we only provide a sketch.Consider where L is as in (2.15) where U (t) is as in (2.14) and the expectation on the right-hand side is taken with respect to the vector ȳ = U (t) conditioned that x = U (0).Hence, it follows from (2.20) and (2.19) that Therefore, the density f (ȳ, t) of U (t) is given by Now, it follows from (2.19), (2.20), and Ito's formula (see, for example, the proof of Proposition 9.9 in [17]), we see where L t is the formal adjoint of L given by Note that, in the computation of L t , we used (2.13): div ȳA(ȳ) = 0. Hence, we conclude that (ȳ, t) → p(x, ȳ, t) satisfies (∂ t −L t ȳ)p(x, ȳ, t) = 0, and therefore, we conclude from (2.21) that f (ȳ, t) satisfies (∂ t − L t ȳ )f (ȳ, t) = 0.
We are now ready to prove (1.26).Let γ α be the density for the normal distribution on R with mean 0 and variance α 2 > 0: Then, in the current setting, the density of the distribution for P N u ω 0 with u ω 0 as in (1.30) is given by The following lemma shows that the solution u N (t) to (2.6) with initial data u ω 0,α = √ αu ω 0 in (2.2) is distributed by the (mean-zero) white noise measure µ α+t in (2.3), which in particular proves (1.26).
Lemma 2.3.For any α > 0, the function f N,α given by is the unique solution to (2.17).
Proof.Uniqueness is classical (see [17,Theorem 9.16]).Hence, we only need to check that f N,α is a solution to (2.17).
A direct computation shows for n = 1, . . ., N .Hence, it suffices to prove it suffices to check A(x) • x = 0. Recalling x = (x 1 , . . ., x 2N ) = (p 1 , . . ., p N , q 1 , . . ., q N ), it follows from (2.11) and (2.16) that where F x denotes the Fourier transform.In the second step, we used the definition: p n (t) = Re u N (n, t) and q n (t) = Im u N (n, t) together with (2.6) and (2.10).By Parseval's identity with the fact that u N is real-valued and u N = P N u N , we then have We point out that, in view of (2.11) and (2.12), A(x) • x = 0 in (2.22) is equivalent to the conservation of ℓ 2 -norm (= the Euclidean distance in R 2N ) for the deterministic system: for n = 1, . . ., N (which in turn is equivalent to the conservation of the L 2 -norm for the finite-dimensional KdV (1.27)).This concludes the proof of Lemma 2.3 Remark 2.4.A slight modification of the computations in the proof of Lemma 2.3 shows the truncated white noise is invariant under the finite-dimensional KdV dynamics (1.27).
As a corollary to Proposition 1.4, we obtain the following tail estimate on the size of solutions u N (t) to (1.22).Lemma 2.5.Let s < 0 and finite p > 1 such that sp < −1.Given α > 0, let u N be the solution to (1.22) with initial data u ω 0,α = √ α u ω 0 in (2.2).Then, we have (2.23) for any t ∈ R + and λ > 0, where the constants C, c > 0 are independent of α > 0.

Review of the local well-posedness argument for SKdV
In this section, we go over the local well-posedness argument in [37] and collect useful estimates.
3.1.Function spaces.We first recall the definition of the X s,b -spaces adapted to the space b s p,∞ (T) defined in (1.9).Given s ∈ R and 1 ≤ p, q ≤ ∞, define the space X s,b p,q (T × R) by the norm: In terms of the interaction representation v(t) = S(−t)u(t), we have where FL b,q (R) denotes the Fourier-Lebesgue space defined by the norm: From (1.9), we have b 0 p,∞ (Z) ⊃ ℓ p (Z) ⊃ ℓ 2 (Z) for p ≥ 2, and thus we have for p ≥ 2, where X s,b is the standard X s,b -space defined in (1.6).We also have u provided that δ > p−2 4p (with p ≥ 2).See [37, eq. ( 17)].See also the embedding (5.2) below.Given an interval I ⊂ R + , we define the restriction space X s,b p,q (I) of X s,b p,q to the interval I by When I = [0, T ], we also set X s,b,T p,q = X s,b p,q ([0, T ]).When b > 1 2 , it follows from the Riemann-Lebesgue lemma that for any s ∈ R and 1 ≤ p ≤ ∞.When q = 2, in order to capture the temporal regularity of the stochastic convolution: where W is as in (1.11), we need to take b < 1 2 (see Lemma 3.4 below), for which the embedding (3.6) fails.When q = 1, we have the following embedding: and thus we use X s,0 p,1 (I) as an auxiliary function space.We now recall the basic linear estimate for KdV; given s ∈ R and 0 ≤ b < 1 2 , we have ) is the restriction space defined in (3.5).Next, we recall the L 4 -Strichartz estimate due to Bourgain [4,Proposition 7.15] (see also [56,Proposition 6.4]): Lastly, we define the Fourier-Lebesgue space FL s.p (T) in the spatial variable by the norm: Then, for s ∈ R and 1 ≤ p, q ≤ ∞, we define the X s,b -spaces adapted to the Fourier-Lebesgue spaces by the norm: Trivially, we have Given an interval I ⊂ R + , we define the restriction space Y s,b p,q (I) as in (3.5).

Partially iterated Duhamel formulation.
In this subsection, we discuss the partially iterated Duhamel formulation used in [37].First, we consider the deterministic KdV (1.5) considered in [6].By writing it in the Duhamel formulation, we have where N (u 1 , u 2 ) is given by Note that the Fourier transform u 1 u 2 (n, τ ) can be written in the convolution form: Henceforth, we denote by (n, τ ), (n 1 , τ 1 ), and (n 2 , τ 2 ) the space-time frequency variables for the Fourier transforms of N (u 1 , u 2 ), u 1 , and u 2 in (3.15), respectively.In particular, we have By assuming that the initial data u 0 has spatial mean 0, it follows that u(t) also has mean 0. Furthermore, in view of the derivative on the nonlinearity, we may assume that n, n 1 , n 2 = 0. We also denote the modulations by (3.17) Recall the following algebraic relation [4]: Then, under (3.16), we have In our setting, we need to take b < 1 2 to capture the temporal regularity of the stochastic convolution Ψ in (3.7).On the other hand, the crucial bilinear estimate (1.7) holds only for b = 1 2 .In order to overcome this difficulty, we decompose the nonlinearity into three pieces, depending on the sizes of the modulations σ 0 , σ 1 , and σ 2 .Define the sets M j , j = 0, 1, 2, by (3.19) For j = 0, 1, 2, let N j (u 1 , u 2 ) be the contribution of N (u 1 , u 2 ) on M j , and thus we have The standard bilinear estimate (1.7) allows us to estimate N 0 (u 1 , u 2 ) even when b < 1 2 ; see [37, eq. ( 46)].As for N j (u 1 , u 2 ), j = 1, 2, however, the bilinear estimate fails for temporal regularity b < 1 2 (in X s,b p,q for any s ∈ R and 1 ≤ p, q ≤ ∞) since, in this case, we do not have a sufficient power for the largest modulation σ j to control the derivative loss in the nonlinearity.See [31].
This issue was circumvented in [6,37,38] by partially iterating the Duhamel formulation (3.14) and writing it as Namely, for j = 1, 2, we replaced the jth entry in N j (u, u) (where the maximum modulation is given by σ j ) by its Duhamel formulation (3.14).It follows from the definition of N j (see (3.19)) that there is no contribution from the linear solution S(t)u 0 in iterating the Duhamel formulation, since its space-time Fourier transform is supported on {τ = n 3 }, namely, S(t)u 0 has zero modulation, and thus from the definition of N j , we have N 1 (S(t)u 0 , u) = N 2 (u, S(t)u 0 ) = 0.
In the context of SKdV (1.1) and its Duhamel formulation (1.10): the discussion above leads to where Ψ is the stochastic convolution in (3.7).In [37], the first author studied this new formulation (3.22) and establish an a priori bound on solutions (with smooth initial data and (spatially) smooth noise), which allowed him to construct a solution (1.10) by an approximation argument.Lastly, we state an analogous formulation for the truncated SKdV (1.22).By writing (1.22) in the Duhamel formulation, we have where N N (u N , u N ) is given by Then, by partially iterating the Duhamel formulation as above, we rewrite (3.23) as where

3.3.
Local well-posedness and an a priori bound.In this subsection, we collect the useful nonlinear estimates on the iterated formulation (3.22) from [37], and establish an a priori bound for solutions to the truncated SKdV (1.22).We first recall the following local well-posedness result of SKdV (1.1) from [37].
In the following, we recall some of the nonlinear estimates from [37].In the remaining part of the paper, we fix small δ, δ 0 > 0, satisfying the hypothesis in Theorem 3.1, and set as in [6,37].The following discussion applies to both the original SKdV (1.1) and its truncated version (1.22).In order to treat them in a uniform manner, we take N ∈ Z ≥0 and set Note that, in the following, all the estimates hold, uniformly in N ∈ Z ≥0 .
Given T > 0, we define the random quantity L ω (T ) by which is a pathwise8 upper bound for the (3.24).See Appendix B. We point out that, while the analysis in Appendix B (see (B.2) and (B.4)) yields the spatial regularity − 1 2 − δ, we use a slightly worse spatial regularity for the definition of L ω (T ) in (3.26) (so that the estimate (5.8) below holds, allowing us to gain a decay in N ).Note that, in contrast to [37], we defined L ω (T ) on the "long" interval [0, T ].This will be useful in Sections 4 and 5 when we iterate the localin-time argument on many small subintervals of [0, T ] but with a fixed driving space-time white noise.From (3.26) and Remark 3.5 below, we have for any T > 0 and 1 ≤ r < ∞, provided that δ > 0 is sufficiently small such that 11 16 With this notation, the main nonlinear estimate [37, eq. ( 73)] (see also Appendix B) reads (with some small θ > 0) for any T > 0 and 0 < T 1 ≤ min(1, T ), provided that Here, in importing (3.28) from [37], we use the fact that P N is bounded on relevant function spaces, uniformly in N ∈ N. The nonlinear estimate (3.28) together with an analogous difference estimate ([37, eq. ( 74)]) allows us to construct a solution u ∈ X −α,α,T 1 p,2 to (1.1) as a limit of smooth solutions.
We now state an a priori bound on a solution u N to the truncated SKdV (1.22).
Lemma 3.2.Let N ∈ N.Then, there exist absolute constants γ > 0 and C * , c * > 0 such that, given any T > 0, we have for any time interval I = [t 0 , t 0 + T 1 ] ⊂ [0, T ] of length T 1 ≤ 1 and any solution u N to the truncated SKdV (1.22), provided that Here, the constants γ > 0 and C * , c * > 0 are independent of N ∈ N.
Remark 3.3.In order use a continuity argument in the proof of Lemma 3.2 presented above, we need the continuity of the X −α,α p,2 ([t 0 , t 1 ])-norm with respect to the right endpoint t 1 .While it may be possible to check this directly (see, for example, [2, Appendix A] and [27,Lemma 8.1]), let us use the following equivalence: where the norm on the left-hand side is defined in (3.5), and study the latter norm.Recall that the equivalence (3.36) holds since the temporal regularity b = α = 1 2 − δ is below 1 2 (see [16, eq.(3.5)]).Such equivalence also holds for the general X s,b p,q ([t 0 , t 1 ]) for 0 ≤ b < q−1 q ; see [10].
Given small h > 0, from the triangle inequality, we have and thus it suffices to show that the right-hand side of (3.37) tends to 0 as h → 0. In view of the definition (3.1), such a claim follows once we prove lim h→0 for a function f ∈ H α (R).Obviously, we have lim h→0 1 [t 1 ,t 1 +h] f L 2 = 0. Using the physical side characterization of the homogeneous Sobolev norm, we have where I , II, and III are defined by By the dominated convergence theorem with the fact that f ∈ H α (R), we see that lim h→∞ I (h) = 0.As for II(h), integration in τ yields where the second step follows from Hardy's inequality ([57, Lemma A.2]) since 0 ≤ α < 1  2 .Noting that III(h) = II(h), we see that the term III(h) also satisfies the bound above.Also, the case h < 0 follows from an analogous consideration.Putting everything together, we conclude (3.38).See also Lemma 4.4 in [7].
We conclude this section by stating a lemma on growth of the stochastic convolution Ψ in (3.7) over long time intervals.We point out that analogous regularity results were obtained in [37, Propositions 4.1 and 4.5] but they are only for short times.Lemma 3.4.Let s < 0 and 1 ≤ p, q < ∞ such that sp < −1.
(i) Let (b − 1)q < −1.Given any 1 ≤ r < ∞ and T ≥ 1, we have for any interval I ⊂ [0, T ] with length |I| ≤ 1, where the implicit constant is independent of r and T .
(ii) Given any 1 ≤ r < ∞ and T ≥ 1, we have where the implicit constant is independent of r and T .
We present the proof of Lemma 3.4 in Appendix A.
Remark 3.5.We point out that the bound (3.39) holds only for intervals I of short lengths.Indeed, a slight modification of the proof yields the following estimate for I = [0, T ]: where the right-hand side is much worse than those in (3.39) and (3.40).See Remark A.1.

Probabilistic uniform growth bound
Given N ∈ N, let u N be the global solution to the truncated SKdV (1.22) with the meanzero white noise initial data u ω 0 in (1.30).Our main goal in this section is to establish the following probabilistic growth bound on the solution u N to (1.1) whose proof is based on a variant of Bourgain's invariant measure argument in the current setting of an evolution system of measures (Proposition 1.4).Proposition 4.1.Let α and p be as in (3.25) and Theorem 3.1, respectively.Given any T ≫ 1 and 0 < ε ≪ 1, there exists a set Ω T,ε (N ) such that P(Ω T,ε (N ) c ) < ε and on Ω T,ε (N ), where the constant C > 0 is independent of N ∈ N, T ≫ 1, and ε ≪ 1.

Approximation argument
In this section, we present the proof of Theorem 1.1.We first establish the following 'almost' almost sure global well-posedness of SKdV (1.1) via an approximation argument.
Given N ∈ N, let u N be the global solution to the truncated SKdV (1.22) with the mean-zero white noise initial data u ω 0 in (1.30), and let u be the solution to SKdV (1.1) with the mean-zero white noise initial data u ω 0 in (1.30), whose local existence is guaranteed by Theorem 3.1.
As compared to Theorem 3.1, we need an extra restriction δ > p−2 3p in order to obtain a decay in N .See (5.2) and (5.12).
As for the difference of the linear solutions, it follows from (3.5) (with T 1 ≤ 1) and (3.9) that Hence, from (5.18) and (5.22), we have (5.23) under the assumptions (3.29) and (5.19).We point out that the estimates (5.21) and (5.23) hold true on a general time interval of length T 1 .
where Ω k , k = 1, . . ., 4, are as in (4.2), (4.5), (4.7), and (4.14), respectively.In particular, if necessary, we have made T 1 smaller such that (4.11) is satisfied.In the following, it is understood that we work on Ω T,ε (N ) and that all the estimates are restricted to Ω T,ε (N ), where the value of N may increase in each step.
Proceeding iteratively, we conclude that, on the jth interval where the latter is as in Proposition 4.1, we conclude from (5.30) that, on Ω T,ε , we have . This concludes the proof of Proposition 5.1.
We now present the proof of Theorem 1.1.We first note that the claimed almost sure global well-posedness of SKdV (1.1) with the white noise initial data immediately follows from the 'almost' almost sure global well-posedness result established in Proposition 5.1; see [16,2].Indeed, define Σ ⊂ Ω by where Ω T,ε is as in Proposition 5.1.Then, we have for any N ≥ N * * .This in particular implies that, for each ω ∈ Ω T,ε , the solution u = u(ω) to SKdV (1.1) is the limit of u N = u N (ω) in C([0, T ]; b −α p,∞ (T)).Hence, given t ∈ R + , it follows from the discussion above that, for each ω ∈ Σ, u N (t; ω) − u(t; ω) b −α p,∞ −→ 0 as N → ∞.This in particular implies convergence in law of u N (t) to u(t).Recalling that Law(u N (t)) = µ 1+t for any N ∈ N, we then conclude that Law(u(t)) = µ 1+t .
This concludes the proof of Theorem 1.1.In this way, we can obtain a growth bound which is only slightly faster than √ t log t, t ≫ 1 (but the random constant C(ω) gets worse).
(ii) It follows from [37,Proposition 4.5] that the stochastic convolution is continuous in time with values in b s p,∞ (T) when sp < −1, at least locally in time.In the following, we estimate its growth in a direct manner by following the argument in [41,Lemma 3.4].