Filtering of Stochastic Nonlinear Wave Equations

TL;DR

This work addresses filtering for a broad class of stochastic nonlinear wave equations arising in quantum dynamics and laser propagation, unifying Itô-calculus and white-noise approaches. It develops measure-valued nonlinear filters, proves existence and uniqueness, and analyzes first-order approximations via linearization that yield infinite-dimensional operator Riccati equations and an infinite-dimensional Kalman filter. The framework is demonstrated on laser- and quantum-inspired models (stochastic paraxial/Klein-Gordon, Maxwell-Dirac, sine-Gordon) with comprehensive Itô and white-noise formulations, including Zakai and FKK equations and backward Kolmogorov arguments for uniqueness. The results provide a rigorous foundation for estimation in nonlinear wave systems under Gaussian and Lévy perturbations, with potential impact on quantum dynamics and high-power laser technologies.

Abstract

In this paper we will develop linear and nonlinear filtering methods for a large class of nonlinear wave equations that arise in applications such as quantum dynamics and laser generation and propagation in a unified framework. We consider both stochastic calculus and white noise filtering methods and derive measure-valued evolution equations for the nonlinear filter and prove existence and uniqueness theorems for the solutions. We will also study first order approximations to these measure-valued evolutions by linearizing the wave equations and characterize the filter dynamics in terms of infinite dimensional operator Riccati equations and establish solvability theorems.

Theorems & Definitions (38)

• Definition 1.1
• Definition 1.2
• Theorem 1.1
• Remark 1.1
• Theorem 1.2
• Theorem 1.3
• Theorem 1.4
• Theorem 1.5
• Lemma 2.1
• Lemma 2.2