Large Deviations Principles for Symplectic Discretizations of Stochastic Linear Schrödinger Equation

In this paper, we consider the large deviations principles (LDPs) for stochastic linear Schrödinger equation and its symplectic discretizations. These numerical discretizations are spatial semi-discretization based on spectral Galerkin method, further full with schemes in temporal direction. First, by means of abstract Gärtner–Ellis theorem, prove that observable $B_{T}=\frac {u(T)}{T}$ , T > 0 exact solution u is exponentially tight satisfies an LDP $L^{2}(0, \pi ; \mathbb C)$ . Then, present LDPs both $\{{B^{M}_{T}}\}_{T>0}$ discretization $\{u^{M}\}_{M\in N}$ $\{{B^{M}_{N}}\}_{N\in $\{{u^{M}_{N}}\}_{M,N\in where ${B^{M}_{T}}=\frac {u^{M}(T)}{T}$ ${B^{M}_{N}}=\frac {{u^{M}_{N}}}{N\tau }$ discrete approximations BT. Further, show can weakly asymptotically preserve {BT}T> 0. results ability to equation, first provide effective approach approximating rate function infinite dimensional space