The paper deals with the numerical solution of the nonlinear Ito stochastic differential equations (SDEs) appearing in the unravelling of quantum master equations. We first develop an exponential scheme of weak order 1 for general globally Lipschitz SDEs governed by Brownian motions. Then, we proceed to study the numerical integration of a class of locally Lipschitz SDEs. More precisely, we adapt the exponential scheme obtained in the first part of the work to the characteristics of certain finite-dimensional nonlinear stochastic Schrodinger equations. This yields a numerical method for the simulation of the mean value of quantum observables. We address the rate of convergence arising in this computation. Finally, an experiment with a representative quantum master equation illustrates the good performance of the new scheme.
This paper is devoted to the study of noise effects on blow-up solutions to stochastic nonlinear Schrödinger equations. It is a continuation of our recent work \cite{BRZ14}, where the (local) well-posedness is established in $H^1$, also in the non-conservative critical case. Here we prove that in the non-conservative focusing mass-(super)critical case, by adding a large multiplicative Gaussian noise, with high probability one can prevent the blow-up on any given bounded time interval $[0,T]$, $0<T<\9$. Moreover, in the case of spatially independent noise, the explosion even can be prevented wi…
We discuss numerical aspects related to a new class of nonlinear Stochastic Differential Equations in the sense of McKean, which are supposed to represent non conservative nonlinear Partial Differential equations (PDEs). We propose an original interacting particle system for which we discuss the propagation of chaos. We consider a time-discretized approximation of this particle system to which we associate a random function which is proved to converge to a solution of a regularized version of a nonlinear PDE.
The paper "The Stochastic Nonlinear Schrödinger Equation in $H^{1}$" \cite{debouard2003} gives an existence proof for a stochastic nonlinear Schrödinger equation with multiplicative noise. We point out two mistakes that draw the validity of the proof into question.
A stochastic mode reduction strategy is applied to multiscale models with a deterministic energy-conserving fast sub-system. Specifically, we consider situations where the slow variables are driven stochastically and interact with the fast sub-system in an energy-conserving fashion. Since the stochastic terms only affect the slow variables, the fast-subsystem evolves deterministically on a sphere of constant energy. However, in the full model the radius of the sphere slowly changes due to the coupling between the slow and fast dynamics. Therefore, the energy of the fast sub-system becomes an a…
We consider a stationary sequence $(X_n)$ constructed by a multiple stochastic integral and an infinite-measure conservative dynamical system. The random measure defining the multiple integral is non-Gaussian, infinitely divisible and has a finite variance. Some additional assumptions on the dynamical system give rise to a parameter $β\in(0,1)$ quantifying the conservativity of the system. This parameter $β$ together with the order of the integral determines the decay rate of the covariance of $(X_n)$. The goal of the paper is to establish limit theorems for the partial sum process of $(X_n)$.…
We consider the stochastic NLS with nonlinear Stratonovic noise for initial values in $L^2(R^d)$ and prove local existence and uniqueness of a mild solution for subcritical and critical nonlinearities. The proof is based on deterministic and stochastic Strichartz estimates. In the subcritical case, we prove that the solution is global, if we impose an additional assumption on the nonlinear noise.
In this paper we prove global existence and uniqueness of solutions to the stochastic logarithmic Schrödinger equation with linear multiplicative noise. Our approach is mainly based on the rescaling approach and the method of maximal monotone operators. In addition, uniform estimates of solutions in the energy space $H^1(\mathbb{R}^d)$ and in an appropriate Orlicz space are also obtained here.
We study the nonlinear Schrödinger equation with linear damping, i.e. a zero order dissipation, and additive noise. Working in $R^d$ with d = 2 or d = 3, we prove the uniqueness of the invariant measure when the damping coefficient is sufficiently large.
Here is investigated the bilinear optimal control problem of quantum mechanical systems with final observation governed by a stochastic nonlinear Schrödinger equation perturbed by a linear multiplicative Wiener process. The existence of an open loop optimal control and first order Lagrange optimality conditions are derived, via Skorohod's representation theorem, Ekeland's variational principle and the existence for the linearized dual backward stochastic equation. Moreover, our approach in particular applies to the deterministic case.
We study the effective approximation for a nonlocal stochastic Schrodinger equation with a rapidly oscillating, periodically time-dependent potential. We use the natural diffusive scaling of heterogeneous system and study the limit behaviour as the scaling parameter tends to 0. This is motivated by data assimilations with non-Gaussian uncertainties. The nonlocal operator in this stochastic partial differential equation is the generator of a non-Gaussian Levy-type process (i.e., a class of anomalous diffusion processes), with non-integrable jump kernel. With help of a two-scale convergence tech…
We prove the existence and the uniqueness of a solution to the stochastic NSLE on a two-dimensional compact riemannian manifold. Thus we generalize a recent work by Burq, Gérard and Tzvetkov in the deterministic setting, and a series of papers by de Bouard and Debussche, who have examined similar questions in the case of the flat euclidean space with random perturbation. We prove the existence and the uniqueness of a local maximal solution to stochastic nonlinear Schrödinger equations with multiplicative noise on a compact d-dimensional riemannian manifold. Under more regularity on the noise, …
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 exponen…
In this paper, we consider the well-posedness of the weakly damped stochastic nonlinear Schrödinger(NLS) equation driven by multiplicative noise. First, we show the global existence of the unique solution for the damped stochastic NLS equation in critical case. Meanwhile, the exponential integrability of the solution is proved, which implies the continuous dependence on the initial data. Then, we analyze the effect of the damped term and noise on the blow-up phenomenon. By modifying the associated energy, momentum and variance identity, we deduce a sharp blow-up condition for damped stochastic…
This paper is devoted to the well-posedness of stochastic nonlinear Schrödinger equations in the energy space H1(Rd), which is a natural continuation of our recent work [1]. We consider both focusing and defocusing nonlinearities and prove global well-posedness in H1(Rd), including also the pathwise continuous dependence on initial conditions, with exponents exactly the same as in the deterministic case. In particular, this work improves earlier results in [4]. Moreover, the local existence, uniqueness and blowup alternative are also established for the energy-critical case. The approach prese…
Stochastic conservation laws are often challenging when it comes to proving existence of non-negative solutions. In a recent work by J. Fischer and G. Grün (2018, Existence of positive solutions to stochastic thin-film equations, SIAM J. Math. Anal.), existence of positive martingale solutions to a conservative stochastic thin-film equation is established in the case of quadratic mobility. In this work, we focus on a larger class of mobilities (including the linear one) for the thin-film model. In order to do so, we need to introduce nonlinear source potentials, thus obtaining a non-conservati…
The present article is devoted to well-posedness by noise for the continuity equation. Namely, we consider the continuity equation with non-linear and partially degenerate stochastic perturbations in divergence form. We prove the existence and uniqueness of entropy solutions under hypotheses on the velocity field which are weaker than those required in the deterministic setting. This extends related results of [Flandoli, Gubinelli, Priola; Invent. Math., 2010] applicable for linear multiplicative noise to a non-linear setting. The existence proof relies on a duality argument which makes use of…
We establish a new version of the stochastic Strichartz estimate for the stochastic convolution driven by jump noise which we apply to the stochastic nonlinear Schrödinger equation with nonlinear multiplicative jump noise in the Marcus canonical form. With the help of the deterministic Strichartz estimates, we prove the existence and uniqueness of a global solution to stochastic nonlinear Schrödinger equation in $L^2(\RR^n)$ with either focusing or defocusing nonlinearity in the full subcritical range of exponents as in the deterministic case.
In this article, we study Stochastic mass critical nonlinear Schrödinger equations with a multiplicative noise in 3D with a slight time decay ($\langle t \rangle^{-ε}$), and prove associated space-time bound and scattering behavior.
By starting from the stochastic Schrödinger equation and quantum trajectory theory, we introduce memory effects by considering stochastic adapted coefficients. As an example of a natural non-Markovian extension of the theory of white noise quantum trajectories we use an Ornstein-Uhlenbeck coloured noise as the output driving process. Under certain conditions a random Hamiltonian evolution is recovered. Moreover, we show that our non-Markovian stochastic Schrödinger equations unravel some master equations with memory kernels.
In this paper, we establish the existence and uniqueness of solutions of stochastic nonlinear Schrödinger equations with additive jump noise in $L^2(\mathbb{R}^d)$. Our results cover all either focusing or defocusing nonlinearity in the full subcritical range of exponents as in the deterministic case.
We study the stochastic cubic nonlinear Schrödinger equation (SNLS) with an additive noise on the one-dimensional torus. In particular, we prove local well-posedness of the (renormalized) SNLS when the noise is almost space-time white noise. We also discuss a notion of criticality in this stochastic context, comparing the situation with the stochastic cubic heat equation (also known as the stochastic quantization equation).
We prove pathwise uniqueness for solutions of the nonlinear Schrödinger equation with conservative multiplicative noise on compact 3D manifolds. In particular, we generalize the result by Burq, Gérard and Tzvetkov (N. Burq, P. Gérard, and N. Tzvetkov. Strichartz inequalities and the nonlinear Schrödinger equation on compact manifolds. American Journal of Mathematics, 126 (3):569--605, 2004) to the stochastic setting. The proof is based on deterministic and stochastic Strichartz estimates and the Littlewood-Paley decomposition.
We here show how the methods recently applied by [DW16] to solve the stochastic nonlinear Schrödinger equation on $\mathbb{T}^2$ can be enhanced to yield solutions on $\mathbb{R}^2$ if the non-linearity is weak enough. We prove that the solutions remains localized on compact time intervals which allows us to apply energy methods on the full space.
We continue our study for the stochastic defocusing mass crtical nonlinear Schrödinger equation with conservative multiplicative noise, and show that it is globally well-posed for arbitrary initial data in $L_ω^{\infty}L_{x}^{2}$. The main ingredients are several stability type results for deterministic (modified) NLS, which have their own interest. We also give some results on other stochastic NLS type models.
In this paper, we prove the global existence and uniqueness of the solution of the stochastic logarithmic Schrödinger (SlogS) equation driven by additive noise or multiplicative noise. The key ingredient lies on the regularized stochastic logarithmic Schrödinger (RSlogS) equation with regularized energy and the strong convergence analysis of the solutions of (RSlogS) equations. In addition, temporal Hölder regularity estimates and uniform estimates in energy space $\mathbb H^1(\mathcal O)$ and weighted Sobolev space $L^2_α(\mathcal O)$ of the solutions for both SlogS equation and RSlogS equati…
We study optimal bilinear control problems for stochastic nonlinear Schrödinger equations in both the mass subcritical and critical case. For general initial data of the minimal L2 regularity, we prove the existence and first order Lagrange condition of an open loop control. Furthermore, we obtain uniform estimates of (backward) stochastic solutions in new spaces of type U2 and V2, adapted to evolution operators related to linear Schrödinger equations with lower order perturbations. In particular, we obtain a new temporal regularity of rescaled (backward) stochastic solutions, which is the key…
We consider a stochastic nonlinear Schrödinger equation with multiplicative noise in an abstract framework that covers subcritical focusing and defocusing stochastic NLS in $H^1$ on compact manifolds and bounded domains. We construct a martingale solution using a modified Faedo-Galerkin-method based on the Littlewood-Paley-decomposition. For 2d manifolds with bounded geometry, we use Strichartz estimates to show pathwise uniqueness.
Bielefeld Universität（比勒费尔德大学）；Max Planck Institute for Mathematics in the Sciences（马克斯·普朗克数学科学研究所）；Institute of Mathematics of NAS of Ukraine（乌克兰国家科学院数学研究所）
The convergence of stochastic interacting particle systems in the mean-field limit to solutions of conservative stochastic partial differential equations is established, with optimal rate of convergence. As a second main result, a quantitative central limit theorem for such SPDEs is derived, again, with optimal rate of convergence. The results apply, in particular, to the convergence in the mean-field scaling of stochastic gradient descent dynamics in overparametrized, shallow neural networks to solutions of SPDEs. It is shown that the inclusion of fluctuations in the limiting SPDE improves th…
We give decay estimates of the solution to the linear Schrödinger equation in dimension $d \geq 3$ with a small noise which is white in time and colored in space. As a consequence, we also obtain certain asymptotic behaviour of the solution. The proof relies on the bootstrapping argument used by Journé-Soffer-Sogge for decay of deterministic Schrödinger operators.
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 $L^2(\mathbb{T})$. As for other power-type nonlinearities, namely (i) (super)quintic when $d = 1$ and (ii) (super)cubic when $d \geq 2$, 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.
In this short note, we present a construction for the log-log blow up solutions to focusing mass-critical stochastic nonlinear Schröidnger equations with multiplicative noises. The solution is understood in the sense of controlled rough path as in \cite{SZ20}.
Following closely the construction of the Schrodinger bridge, we build a new class of Stochastic Volatility Models exactly calibrated to market instruments such as for example Vanillas, options on realized variance or VIX options. These models differ strongly from the well-known local stochastic volatility models, in particular the instantaneous volatility-of-volatility of the associated naked SVMs is not modified, once calibrated to market instruments. They can be interpreted as a martingale version of the Schrodinger bridge. The numerical calibration is performed using a dynamic-like version…
We prove the global existence of solution to the small data mass critical stochastic nonlinear Schrödinger equation in $d=1$. We further show the stability of the solution under perturbation of initial data. Our construction starts with the existence of the solution to the truncated subcritical problem. We then obtain uniform bounds on these solutions that enable us to reach criticality and then remove the truncation.
In this article, we study the Anderson Hamiltonian in the full space and prove wellposedness of nonlinear stochastic wave equation and NLS with polynomial nonlinearities.
In this paper, we present a globalization argument for stochastic nonlinear dispersive PDEs with additive noises by adapting the $I$-method (= the method of almost conservation laws) to the stochastic setting. As a model example, we consider the defocusing stochastic cubic nonlinear Schrödinger equation (SNLS) on $\mathbb R^3$ with additive stochastic forcing, white in time and correlated in space, such that the noise lies below the energy space. By combining the $I$-method with Ito's lemma and a stopping time argument, we construct global-in-time dynamics for SNLS below the energy space.
We study the Cauchy problem for Schrödinger type stochastic partial differential equations with uniformly bounded coefficients on a curved space. We give conditions on the coefficients, on the drift and diffusion terms, on the Cauchy data, and on the spectral measure associated with the noise, such that the Cauchy problem admits a unique function-valued mild solution in the sense of Da Prato and Zabczyc.
We consider the stochastic 2-dimensional Cahn-Hilliard equation which is driven by the derivative in space of a space-time white noise. We use two different approaches to study this equation. First we prove that there exists a unique solution $Y$ to the shifted equation (see (1.4) below), then $X:=Y+{Z}$ is the unique solution to stochastic Cahn-Hilliard equaiton, where ${Z}$ is the corresponding O-U process. Moreover, we use Dirichlet form approach in \cite{Albeverio:1991hk} to construct the probabilistically weak solution the the original equation (1.1) below. By clarifying the precise relat…
We show robustness of various truncated and subcritical approximations to the stochastic defocusing mass-critical nonlinear Schrödinger equation (NLS) in dimension $d=1$, whose solution was constructed in [FX18] with one particular such approximation. The key ingredient in the proof is a uniform bound of the solutions to the family of deterministic mass-subcritical defocusing NLS.
