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18 pages, 290 KB

Open AccessArticle

The Adapted Solutions for Backward Stochastic Schrödinger Equations with Jumps

by Li Yang and Lin Liu

Mathematics 2025, 13(5), 820; https://doi.org/10.3390/math13050820 - 28 Feb 2025

Viewed by 451

This study considers a class of backward stochastic semi-linear Schrödinger equations with Poisson jumps in

R^{d}

or in its bounded domain of a

C^{2}

boundary, which is associated with a stochastic control problem of nonlinear Schrödinger equations driven by Lévy noise. [...] Read more.

This study considers a class of backward stochastic semi-linear Schrödinger equations with Poisson jumps in

R^{d}

or in its bounded domain of a

C^{2}

boundary, which is associated with a stochastic control problem of nonlinear Schrödinger equations driven by Lévy noise. The approach to establish the existence and uniqueness of solutions is mainly based on the complex Itô formula, the Galerkin’s approximation method, and the martingale representation theorem. Full article

22 pages, 1381 KB

Open AccessArticle

Kernel-Based Approximation of the Koopman Generator and Schrödinger Operator

by Stefan Klus, Feliks Nüske and Boumediene Hamzi

Entropy 2020, 22(7), 722; https://doi.org/10.3390/e22070722 - 30 Jun 2020

Cited by 49 | Viewed by 7910

Abstract

Many dimensionality and model reduction techniques rely on estimating dominant eigenfunctions of associated dynamical operators from data. Important examples include the Koopman operator and its generator, but also the Schrödinger operator. We propose a kernel-based method for the approximation of differential operators in reproducing kernel Hilbert spaces and show how eigenfunctions can be estimated by solving auxiliary matrix eigenvalue problems. The resulting algorithms are applied to molecular dynamics and quantum chemistry examples. Furthermore, we exploit that, under certain conditions, the Schrödinger operator can be transformed into a Kolmogorov backward operator corresponding to a drift-diffusion process and vice versa. This allows us to apply methods developed for the analysis of high-dimensional stochastic differential equations to quantum mechanical systems. Full article

(This article belongs to the Special Issue Machine Learning for Prediction, Data Assimilation, and Uncertainty Quantification of Dynamical Systems)

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