Large deviations principles for symplectic discretizations of stochastic linear Schrödinger Equation

Abstract

In this paper, we consider the large deviations principles (LDPs) for the stochastic linear Schrödinger 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 Gärtner-Ellis theorem, we prove that the observable BT = u(T ) T , T > 0 of the exact solution u is exponentially tight and satisfies an LDP on L 2 (0, π; C). Then, we present the LDPs for both {B M T }T >0 of the spatial discretization {u M } M ∈N and {B M N } N∈N of the full discretization {u M N } M,N∈N , whereNτ are the discrete approximations of BT . Further, we show that both the semi-discretization {u M } M ∈N and the full discretization {u M N } M,N∈N based on temporal symplectic schemes can weakly asymptotically preserve the LDP of {BT }T >0 . These results show the ability of symplectic discretizations to preserve the LDP of the stochastic linear Schrödinger equation, and first provide an effective approach to approximating the LDP rate function in infinite dimensional space based on the numerical discretizations.

Numerically asymptotical preservation of the large deviations principles for invariant measures of Langevin equations

“…For an ergodic numerical method for (1.1) that possesses a unique invariant measure µ h ν,ǫ , two natural questions are: whether µ h ν,ǫ also satisfies the LDP with the rate function I h as ǫ → 0 or ν → ∞; if it does, whether I h can approximate well the rate function of {µ ν,ǫ } ǫ>0 or {µ ν,ǫ } ν>0 . Following the ideas in [6,5], we introduce the concept of asymptotical preservation of numerical methods for the LDP of {µ ν,ǫ } ǫ>0 or {µ ν,ǫ } ν>0 in the sense that the rate function I h converges pointwise to the rate function of {µ ν,ǫ } ǫ>0 or {µ ν,ǫ } ν>0 as h → 0. Roughly speaking, that a numerical method {P n , Q n } n≥0 asymptotically preserves the LDP of {µ ν,ǫ } means that the exponential decay of µ ν,ǫ (A) can be well approximated by µ h ν,ǫ (A), provided that h is small enough, for a given measurable set A ⊆ R 2 .…”