Strong convergence rate of finite difference approximations for stochastic cubic Schrödinger equations

Abstract

In this paper, we derive a strong convergence rate of spatial finite difference approximations for both focusing and defocusing stochastic cubic Schrödinger equations driven by a multiplicative Q-Wiener process. Beyond the uniform boundedness of moments for high order derivatives of the exact solution, the key requirement of our approach is the exponential integrability of both the exact and numerical solutions. By constructing and analyzing a Lyapunov functional and its discrete correspondence, we derive the uniform boundedness of moments for high order derivatives of the exact solution and the first order derivative of the numerical solution, which immediately yields the well-posedness of both the continuous and discrete problems. The latter exponential integrability is obtained through a variant of a criterion given by [Cox, Hutzenthaler and Jentzen, arXiv:1309.5595]. As a by-product of this exponential integrability, we prove that the exact and numerical solutions depend continuously on the initial data and obtain a large deviation-type result on the dependence of the noise with first order strong convergence rate.

Analysis of a Splitting Scheme for Damped Stochastic Nonlinear Schrödinger Equation with Multiplicative Noise

“…As a consequence, the solution of (1) is shown to be exponentially stable. On the other hand, we show the exponential integrability properties of exact and numerical solutions by an exponential integrability lemma established in [9,Corollary 2.4]; see also [11,Lemma 3.1]. This type of exponential integrability is also useful to get the strongly continuous dependence on initial data of both exact and numerical solutions and to deduce Gaussian tail estimations of these solutions (see e,.g.…”

“…In [15] and [4,16] it was proved that the stochastic NLS equation admits a unique solution in H and H 1 , respectively. Recently, [8,11] gave the global well-posedness of the one-dimensional stochastic NLS equation in H 2 . In this paper, we focus on strong and weak approximations of the following one-dimensional damped stochastic nonlinear equation with multiplicative noise:…”

On global existence and blow-up for damped stochastic nonlinear Schrödinger equation

“…Moreover, we can get the following exponential integrability of u, which is useful for studying the continuous dependence on initial data and noise, exponential tail estimate of the solution, strong and weak convergence rates of numerical approximations (see e.g. [7,8,9]). Again by the Gagliardo-Nirenberg inequality (6) and the Sobolev embedding theorem, we obtain…”

“…with α depending on u 0 , a and Q, we obtain the strong continuous dependence on the initial data in one dimensional case, which is not a trivial property for stochastic partial differential equation with non-global coefficients, see [6,8] and references therein. We would like to mention that this exponential integrability is useful for studying the continuous dependence on noises, exponential tail estimate of the solution, strong and weak convergence rates of numerical approximations, see [6,7,8,9,15] and references therein. Next we consider the influence of damped term and noise on the blow-up.…”

“…With the help of the rescaling transformation, [1] obtains that the local well-posedness in H 1 for σ < 2 (d−2) + with some decay conditions on the noise, and that the global existence when λ = −1, σ < 2 (d−2) + or λ = 1, σ < 2 d . The global existence of the solution in H 2 is presented in [8] for one dimensional stochastic NLS equation driven by linear multiplicative noises. For the global existence of the solution of the NLS equation in critical case, there exist much more results in the deterministic case than those in stochastic case.…”