Stochastic nonlinear Schrödinger equations on tori

Abstract

We consider the stochastic nonlinear Schrödinger equations (SNLS) posed on d-dimensional tori with either additive or multiplicative stochastic forcing. In particular, for the one-dimensional cubic SNLS, we prove global well-posedness in L 2 (T). As for other power-type nonlinearities, namely (i) (super)quintic when d = 1 and (ii) (super)cubic when d ≥ 2, we prove local well-posedness in all scaling-subcritical Sobolev spaces and global well-posedness in the energy space for the defocusing, energy-subcritical problems.Date: March 8, 2018. 2010 Mathematics Subject Classification. 60H15. Key words and phrases. stochastic nonlinear Schrödinger equations; well-posedness. 1. The multiplicative noise given by the Stratonovich product u • φξ with real-valued ξ is relevant in physical applications, as it conserves the mass of u (i.e. t → u(t) 2 L 2x (T d ) is constant) almost surely. Our analysis can handle either the Itô or the Stratonovich product, and we choose to work with the former for the sake of simpler exposition.Here, P ≤N is the Littlewood-Paley projection onto frequencies {n ∈ Z d : |n| ≤ N }, p ≥ 2(d+2) d , and ε > 0 is an arbitrarily small quantity 3 . However, such Strichartz estimates are not strong enough for a fixed point argument in mixed Lebesgue spaces for the deterministic NLS on T d . To overcome this problem, we shall employ the Fourier restriction norm method by means of X s,b -spaces defined via the norms(1.7)The indices s, b ∈ R measure the spatial and temporal regularities of functions u ∈ X s,b , and F t,x denotes Fourier transform of functions defined on R × T d . This harmonic analytic 2. Here, W s,r (T d ) denotes the L r -based Sobolev space defined by the Bessel potential norm:, where n := 1 + |n| 2 . When r = 2, we have H s (T d ) = W s,2 (T d ).3. More recently, Killip and Vişan [25] removed the arbitrarily small loss of ε derivatives in (1.6) when p > 2(d+2) d . However, we do not need this scale-invariant improvement in our results.

Martingale solutions for the stochastic nonlinear Schrödinger equation in the energy space

“…Hence, their argument cannot be transfered to higher dimensions. After this work was finished, we learned about a recent paper [12] by Cheung and Mosincat. Using the additional structure in the special case of the d-dimensional torus M = T d and algebraic nonlinearities, i.e.…”

Stochastic Nonlinear Schrödinger Equation With Almost Space–time White Noise

“…Then it is easy to see that φ ∈ HS(L 2 ; H s ) if and only if s < α − 1 2 . In particular, when α > 1 2 , the L 2 well-posedness theory in [11] is readily applicable and we conclude that (1.1) is globally well-posed in L 2 (T) in this case. When α ≤ 1 2 , however, the stochastic convolution lies almost surely outside L 2 (T) (for fixed t 0), which causes a serious issue in studying (1.1) with rough noises.…”

“…In particular, it was shown in [11] that (1.1) is locally well-posed in L 2 (T), provided that φ ∈ HS(L 2 ; L 2 ). The argument in [11] is based on (a slight modification of) the L 2 -local theory by Bourgain [6] and controlling the stochastic convolution in the relevant X s,b -norm (see Lemma 3.1 below). A standard application of Ito's lemma combined with the conservation of the L 2 -norm for the deterministic NLS (1.5) yields an a priori bound on the L 2 -norm of a solution and thus global well-posedness of (1.1) in L 2 (T).…”

“…A standard application of Ito's lemma combined with the conservation of the L 2 -norm for the deterministic NLS (1.5) yields an a priori bound on the L 2 -norm of a solution and thus global well-posedness of (1.1) in L 2 (T). See [11] for details. See also [19] for a related argument in the context of the stochastic Kortewegde Vries (KdV) equation.…”

Weak martingale solutions for the stochastic nonlinear Schrödinger equation driven by pure jump noise

“…, N and an R N − valued Lévy process L(t) := (L 1 (t), · · · , L N (t)) with pure jump defined as Before we describe our approach and state our result in detail, we would like to give a general overview of the literature on the stochastic NLS. In the two previous decades, existence and uniqueness results for the stochastic NLS with Gaussian noise have been treated in many articles, most notably [dBD99], [dBD03], [BRZ14], [BRZ16], [Hor18b] in the R d -setting, [BM14] for general 2D compact manifolds and [CM18] for the d-dimensional torus T d . In these articles, the authors applied Strichartz estimates in a fixed point argument based on the mild formulation.…”