A weighted identity for stochastic partial differential operators and its applications

TL;DR

This paper establishes a weighted identity for stochastic PDEs that unifies approaches to controllability, observability, and inverse problems, enabling derivation of known estimates and new inverse problem solutions.

Contribution

It introduces a novel weighted identity for stochastic PDE operators, providing a unified framework for analyzing controllability, observability, and inverse problems, including new results for stochastic complex Ginzburg-Landau equations.

Findings

Unified approach to controllability and observability for stochastic PDEs

Derivation of Carleman estimates for various stochastic equations

New inverse problem results for stochastic complex Ginzburg-Landau equations

Abstract

In this paper, a pointwise weighted identity for some stochastic partial differential operators (with complex principal parts) is established. This identity presents a unified approach in studying the controllability, observability and inverse problems for some deterministic/stochastic partial differential equations. Based on this identity, one can deduce all the known Carleman estimates and observability results, for some deterministic partial differential equations, stochastic heat equations, stochastic Schr\"odinger equations and stochastic transport equations. Meanwhile, as its new application, we study an inverse problem for linear stochastic complex Ginzburg-Landau equations.