An Internal Observability Estimate for Stochastic Hyperbolic Equations

TL;DR

This paper develops a new internal observability estimate for linear stochastic hyperbolic equations using a novel Carleman estimate, addressing the challenges of stochastic process adaptedness.

Contribution

It introduces a global Carleman estimate for stochastic hyperbolic equations, advancing the understanding of observability in stochastic PDEs.

Findings

Established a new Carleman estimate for stochastic hyperbolic equations

Derived an internal observability estimate in the $L^2$-space

Addressed the adaptedness issue in stochastic analysis

Abstract

This paper is addressed to establishing an internal observability estimate for some linear stochastic hyperbolic equations. The key is to establish a new global Carleman estimate for forward stochastic hyperbolic equations in the $L^2$-space. Different from the deterministic case, a delicate analysis of the adaptedness for some stochastic processes is required in the stochastic setting.