Large deviations principles for symplectic discretizations of stochastic linear Schr\"odinger Equation

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