Scattering for Stochastic Nonlinear Schrödinger Equations

Abstract

We study the scattering behavior of global solutions to stochastic nonlinear Schrödinger equations with linear multiplicative noise. In the case where the quadratic variation of the noise is globally finite and the nonlinearity is defocusing, we prove that the solutions scatter at infinity in the pseudo-conformal space and in the energy space respectively, including the energy-critical case. Moreover, in the case where the noise is large, non-conservative and has infinite quadratic variation, we show that the solutions scatter at infinity with high probability for all energy-subcritical exponents.2010 Mathematics Subject Classification. 60H15, 35Q55, 35P25. in the L 2 case is proved in [31] by using the stochastic Strichartz estimates from [10]. In addition, we refer the reader to [22] for results on Schrödinger equations with potential perturbed by a temporal white noise, to [10,11] for results on compact manifolds, and to [16,21,24] for Schrödinger equations with modulated dispersion.In this paper, we are mainly concerned with the asymptotic behavior of solutions to stochastic nonlinear Schrödinger equations. More precisely, we focus on the scattering property of solutions, which is of physical importance and, roughly speaking, means that solutions behave asymptotically like those to linear Schrödinger equations.There is an extensive literature on scattering in the deterministic case. Let α(d) denote the Strauss exponent, see (1.10) below. In the defocusing case, scattering was proved in the pseudo-conformal space (i.e., {u ∈ H 1 :For small initial data, scattering was also obtained for α ∈ (1 + 4/d, 1 + 4/(d − 2)) in [45,15]. Moreover, in the energy space H 1 , the scattering property of solutions was first proved for the inter-critical case where α ∈ (1 + 4/d, 1 + 4/(d − 2)) in [28]. In the much more difficult energy-critical case α = 1 + 4/(d − 2), global well-posedness and scattering are established in [17] for d = 3, in [41] for d = 4, and in [46] for d ≥ 5. Moreover, global well-posedness and scattering in L 2 are proved in [25] for the masscritical exponent α = 1 + 4/d, d ≥ 3. In the focusing case, there exists a threshold for global well-posedness, scattering and blow-up; we refer to [26,33,34,36,37] and references therein.In the framework of stochastic mechanics, developed by E. Nelson [38], there are also several works devoted to potential scattering, in terms of diffusions instead of wave functions. See, e.g., [12,13,43].However, to the best of our knowledge, there are few results on the scattering problem for stochastic nonlinear Schrödinger equations (1.1). One interesting question is that, whether the scattering property is preserved under thermal fluctuations? Furthermore, in the regime where the deterministic system fails to scatter, will the input of some large noise have the effect to improve scattering with high probability?One major challenge here lies in establishing global-in-time Strichartz estimates for (1.1), which actually measure the dispersion and are closely related to those of ti...

Multi solitary waves to stochastic nonlinear Schrödinger equations

“…The scattering phenomena are also exhibited in the stochastic case. We refer to [39] for the H 1 -subcritical and critical cases, and [33,34,35,36,61] for the L 2 -critical case.…”

“…Our strategy of proof is mainly based on the rescaling approach and the modulation method. The rescaling approach in [39] relies on two types of Doss-Sussman transforms, which enable us to study the large time behavior of solutions by transforming the original equation to random Schrödinger equations, for which the sharper pathwise analysis can be performed. This method is actually quite robust for many other stochastic partial differential equations, see, e.g., [1,4,24] and references therein.…”

“…Let us mention that, the pathwise analysis in [39] is based on the stability of scattering which, however, is quite difficult for the multi-solitons in the subcritical case ( [45]) and even fails in the critical case. Instead, we construct multi-solitions in a direct way by using the modulation method and analysing the Lyapunov type functional constructed by Martel, Merle and Tsai [45].…”

“…(ii). In comparison with the scattering results in [39,61], where the solutions behave asymptotically like a free Schrödinger flow in the defocusing (sub)critical cases, the asymptotics (1.20) in Theorem 1.3 gives a different asymptotic behavior in the focusing case, namely, the solutions do not scatter at infinity and may even propagate as any finitely many decoupled solitary waves.…”

Optimal bilinear control of stochastic nonlinear Schrödinger equations: mass-(sub)critical case

“…For the global well-posedness of (1.1) in the mass-subcritical case (i.e., 1 < α < 1+ 4 d ), see [5,6,18,19,34,36]. See also [15] for the compact manifold setting, and [12,13,14,20,21] for the study of martingale solutions.…”