The present paper focuses on the stochastic nonlinear Schrodinger equation with polynomial nonlinearity, and a zero-order (no derivatives involved) linear damping. Here, the random forcing term appears as a mix of a nonlinear noise in the Ito sense and a linear multiplicative noise in the Stratonovich sense. We prove the Laplace principle for the family of solutions to the stochastic system in a suitable Polish space, using the weak convergence framework of Budhiraja and Dupuis. This analysis is nontrivial, since it requires uniform estimates for the solutions of the associated controlled stochastic equation in the underlying solution space in order to verify the weak convergence criterion. The Wentzell Freidlin type large deviation principle is proved using Varadhan's lemma and Bryc's converse to Varadhan's lemma. The local well-posedness of the skeleton equation (deterministic controlled system) is established by employing the Banach fixed point theorem, and the global well posedness is established via Yosida approximation. We show that the conservation law holds in the absence of the linear damping and Ito noise. The well posedness of the stochastic controlled equation is also nontrivial in this case. We use a truncation method, a stopping time argument, and the Yosida technique to get the global well-posedness of the stochastic controlled equation.
Large Deviations for the Nonlinear Schrödinger Equation with Randomized Quasi-Periodic Initial Data in Higher Dimensions: Subcritical Case
Institute of Mathematics, Jilin University（吉林大学数学研究所）；School of Mathematics and Statistics, Center for Mathematics and Interdisciplinary Sciences, Northeast Normal University（东北师范大学数学与统计学院、数学与交叉科学中心）
Doubly Damped Stochastic Parallel Translations and Hessian Formulas
本文研究了与时间无关的薛定谔方程解的Hessian，目的是得到尽可能多的一类完备黎曼流形，使估计$C(\frac 1 t +\frac {d^2}{t^2})$成立.为此，我们引入了双阻尼随机平行迁移方程，研究了它们，并对它们作了指数估计，导出了一个二阶Feynman-Kac公式，得到了所需的估计.我们在这里的目的是解释的直觉，基本技术，和公式，可能是有用的其他研究。
